From 9742bfb0f7ccf7cc2b55c58f6ad51359977dfa73 Mon Sep 17 00:00:00 2001 From: samuel niederer Date: Mon, 7 Mar 2022 15:44:31 +0100 Subject: name und thema geandert --- buch/papers/kra/main.tex | 52 ++++++++++++++++++++++++------------------------ 1 file changed, 26 insertions(+), 26 deletions(-) (limited to 'buch') diff --git a/buch/papers/kra/main.tex b/buch/papers/kra/main.tex index 559d85c..fcee25b 100644 --- a/buch/papers/kra/main.tex +++ b/buch/papers/kra/main.tex @@ -3,34 +3,34 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:kra}} -\lhead{Thema} +\chapter{Kalman, Riccati und Abel\label{chapter:kra}} +\lhead{Kalman, Riccati und Abel} \begin{refsection} -\chapterauthor{Hans Muster} + \chapterauthor{Samuel Niederer} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} + Ein paar Hinweise für die korrekte Formatierung des Textes + \begin{itemize} + \item + Absätze werden gebildet, indem man eine Leerzeile einfügt. + Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. + \item + Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende + Optionen werden gelöscht. + Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. + \item + Beginnen Sie jeden Satz auf einer neuen Zeile. + Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen + in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt + anzuwenden. + \item + Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren + Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. + \end{itemize} -\input{papers/kra/teil0.tex} -\input{papers/kra/teil1.tex} -\input{papers/kra/teil2.tex} -\input{papers/kra/teil3.tex} + \input{papers/kra/teil0.tex} + \input{papers/kra/teil1.tex} + \input{papers/kra/teil2.tex} + \input{papers/kra/teil3.tex} -\printbibliography[heading=subbibliography] + \printbibliography[heading=subbibliography] \end{refsection} -- cgit v1.2.1 From 403b8888ab0702f4d4cf4c7df24adc8c3fa45ab0 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Patrik=20M=C3=BCller?= Date: Mon, 7 Mar 2022 17:19:47 +0100 Subject: Start paper about Laguerre polynomials --- buch/papers/laguerre/definition.tex | 48 +++++++++++++++++++++++++++++++++ buch/papers/laguerre/eigenschaften.tex | 8 ++++++ buch/papers/laguerre/main.tex | 36 +++++++------------------ buch/papers/laguerre/packages.tex | 2 +- buch/papers/laguerre/quadratur.tex | 29 ++++++++++++++++++++ buch/papers/laguerre/transformation.tex | 31 +++++++++++++++++++++ buch/papers/laguerre/wasserstoff.tex | 29 ++++++++++++++++++++ 7 files changed, 156 insertions(+), 27 deletions(-) create mode 100644 buch/papers/laguerre/definition.tex create mode 100644 buch/papers/laguerre/eigenschaften.tex create mode 100644 buch/papers/laguerre/quadratur.tex create mode 100644 buch/papers/laguerre/transformation.tex create mode 100644 buch/papers/laguerre/wasserstoff.tex (limited to 'buch') diff --git a/buch/papers/laguerre/definition.tex b/buch/papers/laguerre/definition.tex new file mode 100644 index 0000000..5f6d8bd --- /dev/null +++ b/buch/papers/laguerre/definition.tex @@ -0,0 +1,48 @@ +% +% definition.tex +% +% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule +% +\section{Definition +\label{laguerre:section:definition}} +\rhead{Definition} + +\begin{align} + x y''(x) + (1 - x) y'(x) + n y(x) + = + 0 + \label{laguerre:dgl} +\end{align} + +\begin{align} + L_n(x) + = + \sum_{k=0}^{n} + \frac{(-1)^k}{k!} + \begin{pmatrix} + n \\ + k + \end{pmatrix} + x^k + \label{laguerre:polynom} +\end{align} + +\begin{align} + x y''(x) + (\alpha + 1 - x) y'(x) + n y(x) + = + 0 + \label{laguerre:generell_dgl} +\end{align} + +\begin{align} + L_n^\alpha (x) + = + \sum_{k=0}^{n} + \frac{(-1)^k}{k!} + \begin{pmatrix} + n + \alpha \\ + n - k + \end{pmatrix} + x^k + \label{laguerre:polynom} +\end{align} diff --git a/buch/papers/laguerre/eigenschaften.tex b/buch/papers/laguerre/eigenschaften.tex new file mode 100644 index 0000000..b7597e5 --- /dev/null +++ b/buch/papers/laguerre/eigenschaften.tex @@ -0,0 +1,8 @@ +% +% eigenschaften.tex +% +% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule +% +\section{Eigenschaften +\label{laguerre:section:eigenschaften}} +\rhead{Eigenschaften} \ No newline at end of file diff --git a/buch/papers/laguerre/main.tex b/buch/papers/laguerre/main.tex index 207e8d7..1fe0f8b 100644 --- a/buch/papers/laguerre/main.tex +++ b/buch/papers/laguerre/main.tex @@ -1,36 +1,20 @@ % -% main.tex -- Paper zum Thema +% main.tex -- Paper zum Thema Laguerre-Polynome % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:laguerre}} -\lhead{Thema} +\chapter{Laguerre-Polynome\label{chapter:laguerre}} +\lhead{Laguerre-Polynome} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Patrik Müller} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} +Hier kommt eine Einleitung. -\input{papers/laguerre/teil0.tex} -\input{papers/laguerre/teil1.tex} -\input{papers/laguerre/teil2.tex} -\input{papers/laguerre/teil3.tex} +\input{papers/laguerre/definition} +\input{papers/laguerre/eigenschaften} +\input{papers/laguerre/quadratur} +\input{papers/laguerre/transformation} +\input{papers/laguerre/wasserstoff} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/laguerre/packages.tex b/buch/papers/laguerre/packages.tex index 6fbe890..ab55228 100644 --- a/buch/papers/laguerre/packages.tex +++ b/buch/papers/laguerre/packages.tex @@ -6,5 +6,5 @@ % if your paper needs special packages, add package commands as in the % following example -%\usepackage{packagename} +\usepackage{derivative} diff --git a/buch/papers/laguerre/quadratur.tex b/buch/papers/laguerre/quadratur.tex new file mode 100644 index 0000000..8ab1af5 --- /dev/null +++ b/buch/papers/laguerre/quadratur.tex @@ -0,0 +1,29 @@ +% +% quadratur.tex +% +% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule +% +\section{Gauss-Laguerre Quadratur +\label{laguerre:section:quadratur}} + +\begin{align} + \int_a^b f(x) w(x) + \approx + \sum_{i=1}^N f(x_i) A_i + \label{laguerre:gaussquadratur} +\end{align} + +\begin{align} + \int_{0}^{\infty} f(x) e^{-x} dx + \approx + \sum_{i=1}^{N} f(x_i) A_i + \label{laguerre:laguerrequadratur} +\end{align} + +\begin{align} + A_i + = + \frac{x_i}{(n + 1)^2 \left[ L_{n + 1}(x_i)\right]^2} + \label{laguerre:quadratur_gewichte} +\end{align} + diff --git a/buch/papers/laguerre/transformation.tex b/buch/papers/laguerre/transformation.tex new file mode 100644 index 0000000..4de86b6 --- /dev/null +++ b/buch/papers/laguerre/transformation.tex @@ -0,0 +1,31 @@ +% +% transformation.tex +% +% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule +% +\section{Laguerre Transformation +\label{laguerre:section:transformation}} +\begin{align} + L \left\{ f(x) \right\} + = + \tilde{f}_\alpha(n) + = + \int_0^\infty e^{-x} x^\alpha L_n^\alpha(x) f(x) dx + \label{laguerre:transformation} +\end{align} + +\begin{align} + L^{-1} \left\{ \tilde{f}_\alpha(n) \right\} + = + f(x) + = + \sum_{n=0}^{\infty} + \begin{pmatrix} + n + \alpha \\ + n + \end{pmatrix}^{-1} + \frac{1}{\Gamma(\alpha + 1)} + \tilde{f}_\alpha(n) + L_n^\alpha(x) + \label{laguerre:inverse_transformation} +\end{align} \ No newline at end of file diff --git a/buch/papers/laguerre/wasserstoff.tex b/buch/papers/laguerre/wasserstoff.tex new file mode 100644 index 0000000..caaa6af --- /dev/null +++ b/buch/papers/laguerre/wasserstoff.tex @@ -0,0 +1,29 @@ +% +% wasserstoff.tex +% +% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule +% +\section{Radialer Schwingungsanteil eines Wasserstoffatoms +\label{laguerre:section:radial_h_atom}} + +\begin{align} + \nonumber + - \frac{\hbar^2}{2m} + & + \left( + \frac{1}{r^2} \pdv{}{r} + \left( r^2 \pdv{}{r} \right) + + + \frac{1}{r^2 \sin \vartheta} \pdv{}{\vartheta} + \left( \sin \vartheta \pdv{}{\vartheta} \right) + + + \frac{1}{r^2 \sin^2 \vartheta} \pdv[2]{}{\varphi} + \right) + u(r, \vartheta, \varphi) + \\ + & - + \frac{e^2}{4 \pi \epsilon_0 r} u(r, \vartheta, \varphi) + = + E u(r, \vartheta, \varphi) + \label{laguerre:pdg_h_atom} +\end{align} -- cgit v1.2.1 From 98036160c9165dbef0fcb237d5cdcf7f804748c3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Patrik=20M=C3=BCller?= Date: Mon, 7 Mar 2022 17:22:21 +0100 Subject: Change title, name, added some formulas and structured paper --- buch/papers/laguerre/teil0.tex | 22 ----------------- buch/papers/laguerre/teil1.tex | 55 ------------------------------------------ buch/papers/laguerre/teil2.tex | 40 ------------------------------ buch/papers/laguerre/teil3.tex | 40 ------------------------------ 4 files changed, 157 deletions(-) delete mode 100644 buch/papers/laguerre/teil0.tex delete mode 100644 buch/papers/laguerre/teil1.tex delete mode 100644 buch/papers/laguerre/teil2.tex delete mode 100644 buch/papers/laguerre/teil3.tex (limited to 'buch') diff --git a/buch/papers/laguerre/teil0.tex b/buch/papers/laguerre/teil0.tex deleted file mode 100644 index a0a215b..0000000 --- a/buch/papers/laguerre/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{laguerre:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{laguerre:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/laguerre/teil1.tex b/buch/papers/laguerre/teil1.tex deleted file mode 100644 index 744d505..0000000 --- a/buch/papers/laguerre/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{laguerre:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{laguerre:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{laguerre:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{laguerre:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{laguerre:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/laguerre/teil2.tex b/buch/papers/laguerre/teil2.tex deleted file mode 100644 index d514042..0000000 --- a/buch/papers/laguerre/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{laguerre:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{laguerre:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/laguerre/teil3.tex b/buch/papers/laguerre/teil3.tex deleted file mode 100644 index 120067d..0000000 --- a/buch/papers/laguerre/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{laguerre:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{laguerre:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - -- cgit v1.2.1 From 63a61e6c7f7383bfff291573526a5160c9e71690 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 7 Mar 2022 21:13:34 +0100 Subject: fix laguerre/Makefile.inc --- buch/papers/laguerre/Makefile.inc | 10 ++++++---- 1 file changed, 6 insertions(+), 4 deletions(-) (limited to 'buch') diff --git a/buch/papers/laguerre/Makefile.inc b/buch/papers/laguerre/Makefile.inc index e83a069..1eb5034 100644 --- a/buch/papers/laguerre/Makefile.inc +++ b/buch/papers/laguerre/Makefile.inc @@ -7,8 +7,10 @@ dependencies-laguerre = \ papers/laguerre/packages.tex \ papers/laguerre/main.tex \ papers/laguerre/references.bib \ - papers/laguerre/teil0.tex \ - papers/laguerre/teil1.tex \ - papers/laguerre/teil2.tex \ - papers/laguerre/teil3.tex + papers/laguerre/definition.tex \ + papers/laguerre/eigenschaften.tex \ + papers/laguerre/quadratur.tex \ + papers/laguerre/transformation.tex \ + papers/laguerre/wasserstoff.tex + -- cgit v1.2.1 From b2975538a4f250b286c5bf0fee910ace5deb4887 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 8 Mar 2022 07:02:38 +0100 Subject: nlwave -> parzyl --- buch/papers/common/Makefile.inc | 6 ++-- buch/papers/common/addbibresources.tex | 2 +- buch/papers/common/addpackages.tex | 2 +- buch/papers/common/addpapers.tex | 2 +- buch/papers/common/includes.inc | 4 +-- buch/papers/common/paperlist | 2 +- buch/papers/nlwave/Makefile | 9 ------ buch/papers/nlwave/Makefile.inc | 14 --------- buch/papers/nlwave/main.tex | 36 ---------------------- buch/papers/nlwave/packages.tex | 10 ------- buch/papers/nlwave/references.bib | 35 ---------------------- buch/papers/nlwave/teil0.tex | 22 -------------- buch/papers/nlwave/teil1.tex | 55 ---------------------------------- buch/papers/nlwave/teil2.tex | 40 ------------------------- buch/papers/nlwave/teil3.tex | 40 ------------------------- buch/papers/parzyl/Makefile | 9 ++++++ buch/papers/parzyl/Makefile.inc | 14 +++++++++ buch/papers/parzyl/main.tex | 36 ++++++++++++++++++++++ buch/papers/parzyl/packages.tex | 10 +++++++ buch/papers/parzyl/references.bib | 35 ++++++++++++++++++++++ buch/papers/parzyl/teil0.tex | 22 ++++++++++++++ buch/papers/parzyl/teil1.tex | 55 ++++++++++++++++++++++++++++++++++ buch/papers/parzyl/teil2.tex | 40 +++++++++++++++++++++++++ buch/papers/parzyl/teil3.tex | 40 +++++++++++++++++++++++++ 24 files changed, 270 insertions(+), 270 deletions(-) delete mode 100644 buch/papers/nlwave/Makefile delete mode 100644 buch/papers/nlwave/Makefile.inc delete mode 100644 buch/papers/nlwave/main.tex delete mode 100644 buch/papers/nlwave/packages.tex delete mode 100644 buch/papers/nlwave/references.bib delete mode 100644 buch/papers/nlwave/teil0.tex delete mode 100644 buch/papers/nlwave/teil1.tex delete mode 100644 buch/papers/nlwave/teil2.tex delete mode 100644 buch/papers/nlwave/teil3.tex create mode 100644 buch/papers/parzyl/Makefile create mode 100644 buch/papers/parzyl/Makefile.inc create mode 100644 buch/papers/parzyl/main.tex create mode 100644 buch/papers/parzyl/packages.tex create mode 100644 buch/papers/parzyl/references.bib create mode 100644 buch/papers/parzyl/teil0.tex create mode 100644 buch/papers/parzyl/teil1.tex create mode 100644 buch/papers/parzyl/teil2.tex create mode 100644 buch/papers/parzyl/teil3.tex (limited to 'buch') diff --git a/buch/papers/common/Makefile.inc b/buch/papers/common/Makefile.inc index d32c902..1e699cc 100644 --- a/buch/papers/common/Makefile.inc +++ b/buch/papers/common/Makefile.inc @@ -11,7 +11,7 @@ PAPERFILES = \ papers/000template/main.tex \ papers/lambertw/main.tex \ papers/fm/main.tex \ - papers/nlwave/main.tex \ + papers/parzyl/main.tex \ papers/fresnel/main.tex \ papers/kreismembran/main.tex \ papers/sturmliouville/main.tex \ @@ -99,7 +99,7 @@ PAPER_DIRECTORIES = \ 000template \ lambertw \ fm \ - nlwave \ + parzyl \ fresnel \ kreismembran \ sturmliouville \ @@ -117,7 +117,7 @@ PAPER_MAKEFILEINC = \ papers/000template/Makefile.inc \ papers/lambertw/Makefile.inc \ papers/fm/Makefile.inc \ - papers/nlwave/Makefile.inc \ + papers/parzyl/Makefile.inc \ papers/fresnel/Makefile.inc \ papers/kreismembran/Makefile.inc \ papers/sturmliouville/Makefile.inc \ diff --git a/buch/papers/common/addbibresources.tex b/buch/papers/common/addbibresources.tex index be4802c..0d60231 100644 --- a/buch/papers/common/addbibresources.tex +++ b/buch/papers/common/addbibresources.tex @@ -6,7 +6,7 @@ \addbibresource{papers/000template/references.bib} \addbibresource{papers/lambertw/references.bib} \addbibresource{papers/fm/references.bib} -\addbibresource{papers/nlwave/references.bib} +\addbibresource{papers/parzyl/references.bib} \addbibresource{papers/fresnel/references.bib} \addbibresource{papers/kreismembran/references.bib} \addbibresource{papers/sturmliouville/references.bib} diff --git a/buch/papers/common/addpackages.tex b/buch/papers/common/addpackages.tex index 00e564a..c97ce85 100644 --- a/buch/papers/common/addpackages.tex +++ b/buch/papers/common/addpackages.tex @@ -6,7 +6,7 @@ \input{papers/000template/packages.tex} \input{papers/lambertw/packages.tex} \input{papers/fm/packages.tex} -\input{papers/nlwave/packages.tex} +\input{papers/parzyl/packages.tex} \input{papers/fresnel/packages.tex} \input{papers/kreismembran/packages.tex} \input{papers/sturmliouville/packages.tex} diff --git a/buch/papers/common/addpapers.tex b/buch/papers/common/addpapers.tex index 6327dc3..9e53036 100644 --- a/buch/papers/common/addpapers.tex +++ b/buch/papers/common/addpapers.tex @@ -6,7 +6,7 @@ \input{papers/000template/main.tex} \input{papers/lambertw/main.tex} \input{papers/fm/main.tex} -\input{papers/nlwave/main.tex} +\input{papers/parzyl/main.tex} \input{papers/fresnel/main.tex} \input{papers/kreismembran/main.tex} \input{papers/sturmliouville/main.tex} diff --git a/buch/papers/common/includes.inc b/buch/papers/common/includes.inc index e5b4a63..ad8af23 100644 --- a/buch/papers/common/includes.inc +++ b/buch/papers/common/includes.inc @@ -1,7 +1,7 @@ include papers/000template/Makefile.inc include papers/lambertw/Makefile.inc include papers/fm/Makefile.inc -include papers/nlwave/Makefile.inc +include papers/parzyl/Makefile.inc include papers/fresnel/Makefile.inc include papers/kreismembran/Makefile.inc include papers/sturmliouville/Makefile.inc @@ -19,7 +19,7 @@ TEXFILES = \ $(dependencies-000template) \ $(dependencies-lambertw) \ $(dependencies-fm) \ - $(dependencies-nlwave) \ + $(dependencies-parzyl) \ $(dependencies-fresnel) \ $(dependencies-kreismembran) \ $(dependencies-sturmliouville) \ diff --git a/buch/papers/common/paperlist b/buch/papers/common/paperlist index e2ef4c3..6eab61d 100644 --- a/buch/papers/common/paperlist +++ b/buch/papers/common/paperlist @@ -1,7 +1,7 @@ 000template lambertw fm -nlwave +parzyl fresnel kreismembran sturmliouville diff --git a/buch/papers/nlwave/Makefile b/buch/papers/nlwave/Makefile deleted file mode 100644 index d2c7958..0000000 --- a/buch/papers/nlwave/Makefile +++ /dev/null @@ -1,9 +0,0 @@ -# -# Makefile -- make file for the paper nlwave -# -# (c) 2020 Prof Dr Andreas Mueller -# - -images: - @echo "no images to be created in nlwave" - diff --git a/buch/papers/nlwave/Makefile.inc b/buch/papers/nlwave/Makefile.inc deleted file mode 100644 index e9d59b2..0000000 --- a/buch/papers/nlwave/Makefile.inc +++ /dev/null @@ -1,14 +0,0 @@ -# -# Makefile.inc -- dependencies for this article -# -# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -# -dependencies-nlwave = \ - papers/nlwave/packages.tex \ - papers/nlwave/main.tex \ - papers/nlwave/references.bib \ - papers/nlwave/teil0.tex \ - papers/nlwave/teil1.tex \ - papers/nlwave/teil2.tex \ - papers/nlwave/teil3.tex - diff --git a/buch/papers/nlwave/main.tex b/buch/papers/nlwave/main.tex deleted file mode 100644 index 12fccc2..0000000 --- a/buch/papers/nlwave/main.tex +++ /dev/null @@ -1,36 +0,0 @@ -% -% main.tex -- Paper zum Thema -% -% (c) 2020 Hochschule Rapperswil -% -\chapter{Thema\label{chapter:nlwave}} -\lhead{Thema} -\begin{refsection} -\chapterauthor{Hans Muster} - -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} - -\input{papers/nlwave/teil0.tex} -\input{papers/nlwave/teil1.tex} -\input{papers/nlwave/teil2.tex} -\input{papers/nlwave/teil3.tex} - -\printbibliography[heading=subbibliography] -\end{refsection} diff --git a/buch/papers/nlwave/packages.tex b/buch/papers/nlwave/packages.tex deleted file mode 100644 index 7f5be16..0000000 --- a/buch/papers/nlwave/packages.tex +++ /dev/null @@ -1,10 +0,0 @@ -% -% packages.tex -- packages required by the paper nlwave -% -% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil -% - -% if your paper needs special packages, add package commands as in the -% following example -%\usepackage{packagename} - diff --git a/buch/papers/nlwave/references.bib b/buch/papers/nlwave/references.bib deleted file mode 100644 index 976d046..0000000 --- a/buch/papers/nlwave/references.bib +++ /dev/null @@ -1,35 +0,0 @@ -% -% references.bib -- Bibliography file for the paper nlwave -% -% (c) 2020 Autor, Hochschule Rapperswil -% - -@online{nlwave:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} -} - -@book{nlwave:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} -} - -@article{nlwave:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} -} - diff --git a/buch/papers/nlwave/teil0.tex b/buch/papers/nlwave/teil0.tex deleted file mode 100644 index cbbf458..0000000 --- a/buch/papers/nlwave/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{nlwave:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{nlwave:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/nlwave/teil1.tex b/buch/papers/nlwave/teil1.tex deleted file mode 100644 index f64aee9..0000000 --- a/buch/papers/nlwave/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{nlwave:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{nlwave:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{nlwave:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{nlwave:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{nlwave:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/nlwave/teil2.tex b/buch/papers/nlwave/teil2.tex deleted file mode 100644 index b93d8b4..0000000 --- a/buch/papers/nlwave/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{nlwave:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{nlwave:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/nlwave/teil3.tex b/buch/papers/nlwave/teil3.tex deleted file mode 100644 index c28424a..0000000 --- a/buch/papers/nlwave/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{nlwave:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{nlwave:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/parzyl/Makefile b/buch/papers/parzyl/Makefile new file mode 100644 index 0000000..d2c7958 --- /dev/null +++ b/buch/papers/parzyl/Makefile @@ -0,0 +1,9 @@ +# +# Makefile -- make file for the paper nlwave +# +# (c) 2020 Prof Dr Andreas Mueller +# + +images: + @echo "no images to be created in nlwave" + diff --git a/buch/papers/parzyl/Makefile.inc b/buch/papers/parzyl/Makefile.inc new file mode 100644 index 0000000..e9d59b2 --- /dev/null +++ b/buch/papers/parzyl/Makefile.inc @@ -0,0 +1,14 @@ +# +# Makefile.inc -- dependencies for this article +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +dependencies-nlwave = \ + papers/nlwave/packages.tex \ + papers/nlwave/main.tex \ + papers/nlwave/references.bib \ + papers/nlwave/teil0.tex \ + papers/nlwave/teil1.tex \ + papers/nlwave/teil2.tex \ + papers/nlwave/teil3.tex + diff --git a/buch/papers/parzyl/main.tex b/buch/papers/parzyl/main.tex new file mode 100644 index 0000000..12fccc2 --- /dev/null +++ b/buch/papers/parzyl/main.tex @@ -0,0 +1,36 @@ +% +% main.tex -- Paper zum Thema +% +% (c) 2020 Hochschule Rapperswil +% +\chapter{Thema\label{chapter:nlwave}} +\lhead{Thema} +\begin{refsection} +\chapterauthor{Hans Muster} + +Ein paar Hinweise für die korrekte Formatierung des Textes +\begin{itemize} +\item +Absätze werden gebildet, indem man eine Leerzeile einfügt. +Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. +\item +Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende +Optionen werden gelöscht. +Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. +\item +Beginnen Sie jeden Satz auf einer neuen Zeile. +Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen +in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt +anzuwenden. +\item +Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren +Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. +\end{itemize} + +\input{papers/nlwave/teil0.tex} +\input{papers/nlwave/teil1.tex} +\input{papers/nlwave/teil2.tex} +\input{papers/nlwave/teil3.tex} + +\printbibliography[heading=subbibliography] +\end{refsection} diff --git a/buch/papers/parzyl/packages.tex b/buch/papers/parzyl/packages.tex new file mode 100644 index 0000000..7f5be16 --- /dev/null +++ b/buch/papers/parzyl/packages.tex @@ -0,0 +1,10 @@ +% +% packages.tex -- packages required by the paper nlwave +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +% if your paper needs special packages, add package commands as in the +% following example +%\usepackage{packagename} + diff --git a/buch/papers/parzyl/references.bib b/buch/papers/parzyl/references.bib new file mode 100644 index 0000000..976d046 --- /dev/null +++ b/buch/papers/parzyl/references.bib @@ -0,0 +1,35 @@ +% +% references.bib -- Bibliography file for the paper nlwave +% +% (c) 2020 Autor, Hochschule Rapperswil +% + +@online{nlwave:bibtex, + title = {BibTeX}, + url = {https://de.wikipedia.org/wiki/BibTeX}, + date = {2020-02-06}, + year = {2020}, + month = {2}, + day = {6} +} + +@book{nlwave:numerical-analysis, + title = {Numerical Analysis}, + author = {David Kincaid and Ward Cheney}, + publisher = {American Mathematical Society}, + year = {2002}, + isbn = {978-8-8218-4788-6}, + inseries = {Pure and applied undegraduate texts}, + volume = {2} +} + +@article{nlwave:mendezmueller, + author = { Tabea Méndez and Andreas Müller }, + title = { Noncommutative harmonic analysis and image registration }, + journal = { Appl. Comput. Harmon. Anal.}, + year = 2019, + volume = 47, + pages = {607--627}, + url = {https://doi.org/10.1016/j.acha.2017.11.004} +} + diff --git a/buch/papers/parzyl/teil0.tex b/buch/papers/parzyl/teil0.tex new file mode 100644 index 0000000..cbbf458 --- /dev/null +++ b/buch/papers/parzyl/teil0.tex @@ -0,0 +1,22 @@ +% +% einleitung.tex -- Beispiel-File für die Einleitung +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Teil 0\label{nlwave:section:teil0}} +\rhead{Teil 0} +Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam +nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam +erat, sed diam voluptua \cite{nlwave:bibtex}. +At vero eos et accusam et justo duo dolores et ea rebum. +Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum +dolor sit amet. + +Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam +nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam +erat, sed diam voluptua. +At vero eos et accusam et justo duo dolores et ea rebum. Stet clita +kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit +amet. + + diff --git a/buch/papers/parzyl/teil1.tex b/buch/papers/parzyl/teil1.tex new file mode 100644 index 0000000..f64aee9 --- /dev/null +++ b/buch/papers/parzyl/teil1.tex @@ -0,0 +1,55 @@ +% +% teil1.tex -- Beispiel-File für das Paper +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Teil 1 +\label{nlwave:section:teil1}} +\rhead{Problemstellung} +Sed ut perspiciatis unde omnis iste natus error sit voluptatem +accusantium doloremque laudantium, totam rem aperiam, eaque ipsa +quae ab illo inventore veritatis et quasi architecto beatae vitae +dicta sunt explicabo. +Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit +aut fugit, sed quia consequuntur magni dolores eos qui ratione +voluptatem sequi nesciunt +\begin{equation} +\int_a^b x^2\, dx += +\left[ \frac13 x^3 \right]_a^b += +\frac{b^3-a^3}3. +\label{nlwave:equation1} +\end{equation} +Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, +consectetur, adipisci velit, sed quia non numquam eius modi tempora +incidunt ut labore et dolore magnam aliquam quaerat voluptatem. + +Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis +suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? +Quis autem vel eum iure reprehenderit qui in ea voluptate velit +esse quam nihil molestiae consequatur, vel illum qui dolorem eum +fugiat quo voluptas nulla pariatur? + +\subsection{De finibus bonorum et malorum +\label{nlwave:subsection:finibus}} +At vero eos et accusamus et iusto odio dignissimos ducimus qui +blanditiis praesentium voluptatum deleniti atque corrupti quos +dolores et quas molestias excepturi sint occaecati cupiditate non +provident, similique sunt in culpa qui officia deserunt mollitia +animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. + +Et harum quidem rerum facilis est et expedita distinctio +\ref{nlwave:section:loesung}. +Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil +impedit quo minus id quod maxime placeat facere possimus, omnis +voluptas assumenda est, omnis dolor repellendus +\ref{nlwave:section:folgerung}. +Temporibus autem quibusdam et aut officiis debitis aut rerum +necessitatibus saepe eveniet ut et voluptates repudiandae sint et +molestiae non recusandae. +Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis +voluptatibus maiores alias consequatur aut perferendis doloribus +asperiores repellat. + + diff --git a/buch/papers/parzyl/teil2.tex b/buch/papers/parzyl/teil2.tex new file mode 100644 index 0000000..b93d8b4 --- /dev/null +++ b/buch/papers/parzyl/teil2.tex @@ -0,0 +1,40 @@ +% +% teil2.tex -- Beispiel-File für teil2 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Teil 2 +\label{nlwave:section:teil2}} +\rhead{Teil 2} +Sed ut perspiciatis unde omnis iste natus error sit voluptatem +accusantium doloremque laudantium, totam rem aperiam, eaque ipsa +quae ab illo inventore veritatis et quasi architecto beatae vitae +dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit +aspernatur aut odit aut fugit, sed quia consequuntur magni dolores +eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam +est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci +velit, sed quia non numquam eius modi tempora incidunt ut labore +et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima +veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, +nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure +reprehenderit qui in ea voluptate velit esse quam nihil molestiae +consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla +pariatur? + +\subsection{De finibus bonorum et malorum +\label{nlwave:subsection:bonorum}} +At vero eos et accusamus et iusto odio dignissimos ducimus qui +blanditiis praesentium voluptatum deleniti atque corrupti quos +dolores et quas molestias excepturi sint occaecati cupiditate non +provident, similique sunt in culpa qui officia deserunt mollitia +animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis +est et expedita distinctio. Nam libero tempore, cum soluta nobis +est eligendi optio cumque nihil impedit quo minus id quod maxime +placeat facere possimus, omnis voluptas assumenda est, omnis dolor +repellendus. Temporibus autem quibusdam et aut officiis debitis aut +rerum necessitatibus saepe eveniet ut et voluptates repudiandae +sint et molestiae non recusandae. Itaque earum rerum hic tenetur a +sapiente delectus, ut aut reiciendis voluptatibus maiores alias +consequatur aut perferendis doloribus asperiores repellat. + + diff --git a/buch/papers/parzyl/teil3.tex b/buch/papers/parzyl/teil3.tex new file mode 100644 index 0000000..c28424a --- /dev/null +++ b/buch/papers/parzyl/teil3.tex @@ -0,0 +1,40 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Teil 3 +\label{nlwave:section:teil3}} +\rhead{Teil 3} +Sed ut perspiciatis unde omnis iste natus error sit voluptatem +accusantium doloremque laudantium, totam rem aperiam, eaque ipsa +quae ab illo inventore veritatis et quasi architecto beatae vitae +dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit +aspernatur aut odit aut fugit, sed quia consequuntur magni dolores +eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam +est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci +velit, sed quia non numquam eius modi tempora incidunt ut labore +et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima +veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, +nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure +reprehenderit qui in ea voluptate velit esse quam nihil molestiae +consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla +pariatur? + +\subsection{De finibus bonorum et malorum +\label{nlwave:subsection:malorum}} +At vero eos et accusamus et iusto odio dignissimos ducimus qui +blanditiis praesentium voluptatum deleniti atque corrupti quos +dolores et quas molestias excepturi sint occaecati cupiditate non +provident, similique sunt in culpa qui officia deserunt mollitia +animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis +est et expedita distinctio. Nam libero tempore, cum soluta nobis +est eligendi optio cumque nihil impedit quo minus id quod maxime +placeat facere possimus, omnis voluptas assumenda est, omnis dolor +repellendus. Temporibus autem quibusdam et aut officiis debitis aut +rerum necessitatibus saepe eveniet ut et voluptates repudiandae +sint et molestiae non recusandae. Itaque earum rerum hic tenetur a +sapiente delectus, ut aut reiciendis voluptatibus maiores alias +consequatur aut perferendis doloribus asperiores repellat. + + -- cgit v1.2.1 From 1caf2628298cf35f88025d348954efa6243a8a34 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 8 Mar 2022 07:42:33 +0100 Subject: fix cover --- buch/common/teilnehmer.tex | 16 ++++++++-------- buch/papers/parzyl/Makefile | 4 ++-- buch/papers/parzyl/Makefile.inc | 16 ++++++++-------- buch/papers/parzyl/main.tex | 12 ++++++------ buch/papers/parzyl/packages.tex | 2 +- buch/papers/parzyl/references.bib | 8 ++++---- buch/papers/parzyl/teil0.tex | 4 ++-- buch/papers/parzyl/teil1.tex | 10 +++++----- buch/papers/parzyl/teil2.tex | 4 ++-- buch/papers/parzyl/teil3.tex | 4 ++-- 10 files changed, 40 insertions(+), 40 deletions(-) (limited to 'buch') diff --git a/buch/common/teilnehmer.tex b/buch/common/teilnehmer.tex index c8a28fc..ec6e915 100644 --- a/buch/common/teilnehmer.tex +++ b/buch/common/teilnehmer.tex @@ -4,21 +4,21 @@ % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % Joshua Bär, % E -Selvin Blöchlinger, % E +%Selvin Blöchlinger, % E Marc Benz, % MSE -Manuel Cattaneo%, % MSE -\\ +Manuel Cattaneo, % MSE Fabian Dünki, % E -Robin Eberle, % E +\\ +%Robin Eberle, % E Enez Erdem, % B -Nilakshan Eswararajah%, % B +Nilakshan Eswararajah, % B +Réde Hadouche%, % E \\ -Réde Hadouche, % E David Hugentobler, % E Alain Keller, % E -Yanik Kuster%, % E +Yanik Kuster, % E +Marc Kühne%, % B \\ -Marc Kühne, % B Erik Löffler, % E Kevin Meili, % M-I Andrea Mozzini Vellen%, % E diff --git a/buch/papers/parzyl/Makefile b/buch/papers/parzyl/Makefile index d2c7958..3578f26 100644 --- a/buch/papers/parzyl/Makefile +++ b/buch/papers/parzyl/Makefile @@ -1,9 +1,9 @@ # -# Makefile -- make file for the paper nlwave +# Makefile -- make file for the paper parzyl # # (c) 2020 Prof Dr Andreas Mueller # images: - @echo "no images to be created in nlwave" + @echo "no images to be created in parzyl" diff --git a/buch/papers/parzyl/Makefile.inc b/buch/papers/parzyl/Makefile.inc index e9d59b2..fc182e1 100644 --- a/buch/papers/parzyl/Makefile.inc +++ b/buch/papers/parzyl/Makefile.inc @@ -3,12 +3,12 @@ # # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -dependencies-nlwave = \ - papers/nlwave/packages.tex \ - papers/nlwave/main.tex \ - papers/nlwave/references.bib \ - papers/nlwave/teil0.tex \ - papers/nlwave/teil1.tex \ - papers/nlwave/teil2.tex \ - papers/nlwave/teil3.tex +dependencies-parzyl = \ + papers/parzyl/packages.tex \ + papers/parzyl/main.tex \ + papers/parzyl/references.bib \ + papers/parzyl/teil0.tex \ + papers/parzyl/teil1.tex \ + papers/parzyl/teil2.tex \ + papers/parzyl/teil3.tex diff --git a/buch/papers/parzyl/main.tex b/buch/papers/parzyl/main.tex index 12fccc2..250d170 100644 --- a/buch/papers/parzyl/main.tex +++ b/buch/papers/parzyl/main.tex @@ -1,9 +1,9 @@ % -% main.tex -- Paper zum Thema +% main.tex -- Paper zum Thema % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:nlwave}} +\chapter{Thema\label{chapter:parzyl}} \lhead{Thema} \begin{refsection} \chapterauthor{Hans Muster} @@ -27,10 +27,10 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. \end{itemize} -\input{papers/nlwave/teil0.tex} -\input{papers/nlwave/teil1.tex} -\input{papers/nlwave/teil2.tex} -\input{papers/nlwave/teil3.tex} +\input{papers/parzyl/teil0.tex} +\input{papers/parzyl/teil1.tex} +\input{papers/parzyl/teil2.tex} +\input{papers/parzyl/teil3.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/parzyl/packages.tex b/buch/papers/parzyl/packages.tex index 7f5be16..f6844a6 100644 --- a/buch/papers/parzyl/packages.tex +++ b/buch/papers/parzyl/packages.tex @@ -1,5 +1,5 @@ % -% packages.tex -- packages required by the paper nlwave +% packages.tex -- packages required by the paper parzyl % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % diff --git a/buch/papers/parzyl/references.bib b/buch/papers/parzyl/references.bib index 976d046..494ff7c 100644 --- a/buch/papers/parzyl/references.bib +++ b/buch/papers/parzyl/references.bib @@ -1,10 +1,10 @@ % -% references.bib -- Bibliography file for the paper nlwave +% references.bib -- Bibliography file for the paper parzyl % % (c) 2020 Autor, Hochschule Rapperswil % -@online{nlwave:bibtex, +@online{parzyl:bibtex, title = {BibTeX}, url = {https://de.wikipedia.org/wiki/BibTeX}, date = {2020-02-06}, @@ -13,7 +13,7 @@ day = {6} } -@book{nlwave:numerical-analysis, +@book{parzyl:numerical-analysis, title = {Numerical Analysis}, author = {David Kincaid and Ward Cheney}, publisher = {American Mathematical Society}, @@ -23,7 +23,7 @@ volume = {2} } -@article{nlwave:mendezmueller, +@article{parzyl:mendezmueller, author = { Tabea Méndez and Andreas Müller }, title = { Noncommutative harmonic analysis and image registration }, journal = { Appl. Comput. Harmon. Anal.}, diff --git a/buch/papers/parzyl/teil0.tex b/buch/papers/parzyl/teil0.tex index cbbf458..09b4024 100644 --- a/buch/papers/parzyl/teil0.tex +++ b/buch/papers/parzyl/teil0.tex @@ -3,11 +3,11 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 0\label{nlwave:section:teil0}} +\section{Teil 0\label{parzyl:section:teil0}} \rhead{Teil 0} Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{nlwave:bibtex}. +erat, sed diam voluptua \cite{parzyl:bibtex}. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. diff --git a/buch/papers/parzyl/teil1.tex b/buch/papers/parzyl/teil1.tex index f64aee9..9ea60e2 100644 --- a/buch/papers/parzyl/teil1.tex +++ b/buch/papers/parzyl/teil1.tex @@ -4,7 +4,7 @@ % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % \section{Teil 1 -\label{nlwave:section:teil1}} +\label{parzyl:section:teil1}} \rhead{Problemstellung} Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium, totam rem aperiam, eaque ipsa @@ -19,7 +19,7 @@ voluptatem sequi nesciunt \left[ \frac13 x^3 \right]_a^b = \frac{b^3-a^3}3. -\label{nlwave:equation1} +\label{parzyl:equation1} \end{equation} Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci velit, sed quia non numquam eius modi tempora @@ -32,7 +32,7 @@ esse quam nihil molestiae consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla pariatur? \subsection{De finibus bonorum et malorum -\label{nlwave:subsection:finibus}} +\label{parzyl:subsection:finibus}} At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non @@ -40,11 +40,11 @@ provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. Et harum quidem rerum facilis est et expedita distinctio -\ref{nlwave:section:loesung}. +\ref{parzyl:section:loesung}. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus -\ref{nlwave:section:folgerung}. +\ref{parzyl:section:folgerung}. Temporibus autem quibusdam et aut officiis debitis aut rerum necessitatibus saepe eveniet ut et voluptates repudiandae sint et molestiae non recusandae. diff --git a/buch/papers/parzyl/teil2.tex b/buch/papers/parzyl/teil2.tex index b93d8b4..75ba259 100644 --- a/buch/papers/parzyl/teil2.tex +++ b/buch/papers/parzyl/teil2.tex @@ -4,7 +4,7 @@ % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % \section{Teil 2 -\label{nlwave:section:teil2}} +\label{parzyl:section:teil2}} \rhead{Teil 2} Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium, totam rem aperiam, eaque ipsa @@ -22,7 +22,7 @@ consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla pariatur? \subsection{De finibus bonorum et malorum -\label{nlwave:subsection:bonorum}} +\label{parzyl:subsection:bonorum}} At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non diff --git a/buch/papers/parzyl/teil3.tex b/buch/papers/parzyl/teil3.tex index c28424a..72c23ca 100644 --- a/buch/papers/parzyl/teil3.tex +++ b/buch/papers/parzyl/teil3.tex @@ -4,7 +4,7 @@ % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % \section{Teil 3 -\label{nlwave:section:teil3}} +\label{parzyl:section:teil3}} \rhead{Teil 3} Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium, totam rem aperiam, eaque ipsa @@ -22,7 +22,7 @@ consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla pariatur? \subsection{De finibus bonorum et malorum -\label{nlwave:subsection:malorum}} +\label{parzyl:subsection:malorum}} At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non -- cgit v1.2.1 From c33458fa86e17af60065df9d973e17e07dba1039 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 8 Mar 2022 07:46:14 +0100 Subject: Korrekturen Teilnehmerliste --- buch/common/teilnehmer.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'buch') diff --git a/buch/common/teilnehmer.tex b/buch/common/teilnehmer.tex index ec6e915..c14790a 100644 --- a/buch/common/teilnehmer.tex +++ b/buch/common/teilnehmer.tex @@ -7,12 +7,12 @@ Joshua Bär, % E %Selvin Blöchlinger, % E Marc Benz, % MSE Manuel Cattaneo, % MSE -Fabian Dünki, % E +Fabian Dünki%, % E \\ %Robin Eberle, % E Enez Erdem, % B Nilakshan Eswararajah, % B -Réde Hadouche%, % E +Réda Haddouche%, % E \\ David Hugentobler, % E Alain Keller, % E -- cgit v1.2.1 From ed0a70c80e7a8c9915f53edbfeb4daf19e030dd8 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 8 Mar 2022 16:27:37 +0100 Subject: add some theory --- buch/papers/common/Makefile.inc | 7 + buch/papers/common/addbibresources.tex | 1 + buch/papers/common/addpackages.tex | 1 + buch/papers/common/addpapers.tex | 1 + buch/papers/common/includes.inc | 2 + buch/papers/common/paperlist | 1 + buch/papers/dreieck/Makefile | 9 ++ buch/papers/dreieck/Makefile.inc | 14 ++ buch/papers/dreieck/main.tex | 26 ++++ buch/papers/dreieck/packages.tex | 10 ++ buch/papers/dreieck/references.bib | 35 +++++ buch/papers/dreieck/teil0.tex | 9 ++ buch/papers/dreieck/teil1.tex | 261 +++++++++++++++++++++++++++++++++ buch/papers/dreieck/teil2.tex | 9 ++ buch/papers/dreieck/teil3.tex | 10 ++ 15 files changed, 396 insertions(+) create mode 100644 buch/papers/dreieck/Makefile create mode 100644 buch/papers/dreieck/Makefile.inc create mode 100644 buch/papers/dreieck/main.tex create mode 100644 buch/papers/dreieck/packages.tex create mode 100644 buch/papers/dreieck/references.bib create mode 100644 buch/papers/dreieck/teil0.tex create mode 100644 buch/papers/dreieck/teil1.tex create mode 100644 buch/papers/dreieck/teil2.tex create mode 100644 buch/papers/dreieck/teil3.tex (limited to 'buch') diff --git a/buch/papers/common/Makefile.inc b/buch/papers/common/Makefile.inc index 1e699cc..eb8b8a7 100644 --- a/buch/papers/common/Makefile.inc +++ b/buch/papers/common/Makefile.inc @@ -24,6 +24,7 @@ PAPERFILES = \ papers/kugel/main.tex \ papers/hermite/main.tex \ papers/ellfilter/main.tex \ + papers/dreieck/main.tex \ buch1-blx.bbl: buch1-blx.aux bibtex buch1-blx @@ -76,6 +77,9 @@ buch16-blx.bbl: buch16-blx.aux buch17-blx.bbl: buch17-blx.aux bibtex buch17-blx +buch18-blx.bbl: buch18-blx.aux + bibtex buch18-blx + BLXFILES = buch.bbl \ buch1-blx.bbl \ buch2-blx.bbl \ @@ -94,6 +98,7 @@ BLXFILES = buch.bbl \ buch15-blx.bbl \ buch16-blx.bbl \ buch17-blx.bbl \ + buch18-blx.bbl \ PAPER_DIRECTORIES = \ 000template \ @@ -112,6 +117,7 @@ PAPER_DIRECTORIES = \ kugel \ hermite \ ellfilter \ + dreieck \ PAPER_MAKEFILEINC = \ papers/000template/Makefile.inc \ @@ -130,4 +136,5 @@ PAPER_MAKEFILEINC = \ papers/kugel/Makefile.inc \ papers/hermite/Makefile.inc \ papers/ellfilter/Makefile.inc \ + papers/dreieck/Makefile.inc \ diff --git a/buch/papers/common/addbibresources.tex b/buch/papers/common/addbibresources.tex index 0d60231..6e354b5 100644 --- a/buch/papers/common/addbibresources.tex +++ b/buch/papers/common/addbibresources.tex @@ -19,3 +19,4 @@ \addbibresource{papers/kugel/references.bib} \addbibresource{papers/hermite/references.bib} \addbibresource{papers/ellfilter/references.bib} +\addbibresource{papers/dreieck/references.bib} diff --git a/buch/papers/common/addpackages.tex b/buch/papers/common/addpackages.tex index c97ce85..31f7455 100644 --- a/buch/papers/common/addpackages.tex +++ b/buch/papers/common/addpackages.tex @@ -19,3 +19,4 @@ \input{papers/kugel/packages.tex} \input{papers/hermite/packages.tex} \input{papers/ellfilter/packages.tex} +\input{papers/dreieck/packages.tex} diff --git a/buch/papers/common/addpapers.tex b/buch/papers/common/addpapers.tex index 9e53036..dd2b07a 100644 --- a/buch/papers/common/addpapers.tex +++ b/buch/papers/common/addpapers.tex @@ -19,3 +19,4 @@ \input{papers/kugel/main.tex} \input{papers/hermite/main.tex} \input{papers/ellfilter/main.tex} +\input{papers/dreieck/main.tex} diff --git a/buch/papers/common/includes.inc b/buch/papers/common/includes.inc index ad8af23..3e064d9 100644 --- a/buch/papers/common/includes.inc +++ b/buch/papers/common/includes.inc @@ -14,6 +14,7 @@ include papers/kra/Makefile.inc include papers/kugel/Makefile.inc include papers/hermite/Makefile.inc include papers/ellfilter/Makefile.inc +include papers/dreieck/Makefile.inc TEXFILES = \ $(dependencies-000template) \ @@ -32,4 +33,5 @@ TEXFILES = \ $(dependencies-kugel) \ $(dependencies-hermite) \ $(dependencies-ellfilter) \ + $(dependencies-dreieck) \ diff --git a/buch/papers/common/paperlist b/buch/papers/common/paperlist index 6eab61d..d4e5c20 100644 --- a/buch/papers/common/paperlist +++ b/buch/papers/common/paperlist @@ -14,3 +14,4 @@ kra kugel hermite ellfilter +dreieck diff --git a/buch/papers/dreieck/Makefile b/buch/papers/dreieck/Makefile new file mode 100644 index 0000000..f0cb602 --- /dev/null +++ b/buch/papers/dreieck/Makefile @@ -0,0 +1,9 @@ +# +# Makefile -- make file for the paper dreieck +# +# (c) 2020 Prof Dr Andreas Mueller +# + +images: + @echo "no images to be created in dreieck" + diff --git a/buch/papers/dreieck/Makefile.inc b/buch/papers/dreieck/Makefile.inc new file mode 100644 index 0000000..843da8d --- /dev/null +++ b/buch/papers/dreieck/Makefile.inc @@ -0,0 +1,14 @@ +# +# Makefile.inc -- dependencies for this article +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +dependencies-dreieck = \ + papers/dreieck/packages.tex \ + papers/dreieck/main.tex \ + papers/dreieck/references.bib \ + papers/dreieck/teil0.tex \ + papers/dreieck/teil1.tex \ + papers/dreieck/teil2.tex \ + papers/dreieck/teil3.tex + diff --git a/buch/papers/dreieck/main.tex b/buch/papers/dreieck/main.tex new file mode 100644 index 0000000..75ba410 --- /dev/null +++ b/buch/papers/dreieck/main.tex @@ -0,0 +1,26 @@ +% +% main.tex -- Paper zum Thema +% +% (c) 2020 Hochschule Rapperswil +% +\chapter{Dreieckstest und Beta-Funktion\label{chapter:dreieck}} +\lhead{Dreieckstest und Beta-Funktion} +\begin{refsection} +\chapterauthor{Andreas Müller} + +\noindent +Mit dem Dreieckstest kann man feststellen, wie gut ein Geruchs- +oder Geschmackstester verschiedene Gerüche oder Geschmäcker +unterscheiden kann. +Seine wahrscheinlichkeitstheoretische Erklärung benötigt die Beta-Funktion, +man kann die Beta-Funktion als durchaus als die mathematische Grundlage +der Weindegustation +bezeichnen. + +\input{papers/dreieck/teil0.tex} +\input{papers/dreieck/teil1.tex} +\input{papers/dreieck/teil2.tex} +\input{papers/dreieck/teil3.tex} + +\printbibliography[heading=subbibliography] +\end{refsection} diff --git a/buch/papers/dreieck/packages.tex b/buch/papers/dreieck/packages.tex new file mode 100644 index 0000000..fd4ebce --- /dev/null +++ b/buch/papers/dreieck/packages.tex @@ -0,0 +1,10 @@ +% +% packages.tex -- packages required by the paper dreieck +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +% if your paper needs special packages, add package commands as in the +% following example +%\usepackage{packagename} + diff --git a/buch/papers/dreieck/references.bib b/buch/papers/dreieck/references.bib new file mode 100644 index 0000000..d2bbe08 --- /dev/null +++ b/buch/papers/dreieck/references.bib @@ -0,0 +1,35 @@ +% +% references.bib -- Bibliography file for the paper dreieck +% +% (c) 2020 Autor, Hochschule Rapperswil +% + +@online{dreieck:bibtex, + title = {BibTeX}, + url = {https://de.wikipedia.org/wiki/BibTeX}, + date = {2020-02-06}, + year = {2020}, + month = {2}, + day = {6} +} + +@book{dreieck:numerical-analysis, + title = {Numerical Analysis}, + author = {David Kincaid and Ward Cheney}, + publisher = {American Mathematical Society}, + year = {2002}, + isbn = {978-8-8218-4788-6}, + inseries = {Pure and applied undegraduate texts}, + volume = {2} +} + +@article{dreieck:mendezmueller, + author = { Tabea Méndez and Andreas Müller }, + title = { Noncommutative harmonic analysis and image registration }, + journal = { Appl. Comput. Harmon. Anal.}, + year = 2019, + volume = 47, + pages = {607--627}, + url = {https://doi.org/10.1016/j.acha.2017.11.004} +} + diff --git a/buch/papers/dreieck/teil0.tex b/buch/papers/dreieck/teil0.tex new file mode 100644 index 0000000..bcf2cf8 --- /dev/null +++ b/buch/papers/dreieck/teil0.tex @@ -0,0 +1,9 @@ +% +% einleitung.tex -- Beispiel-File für die Einleitung +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Testprinzip\label{dreieck:section:testprinzip}} +\rhead{Testprinzip} + + diff --git a/buch/papers/dreieck/teil1.tex b/buch/papers/dreieck/teil1.tex new file mode 100644 index 0000000..255c5d0 --- /dev/null +++ b/buch/papers/dreieck/teil1.tex @@ -0,0 +1,261 @@ +% +% teil1.tex -- Beispiel-File für das Paper +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Ordnungsstatistik und Beta-Funktion +\label{dreieck:section:ordnungsstatistik}} +\rhead{Ordnungsstatistik und Beta-Funktion} +In diesem Abschnitt ist $X$ eine Zufallsvariable mit der Verteilungsfunktion +$F_X(x)$, und $X_i$, $1\le i\le n$ sei ein Stichprobe von unabhängigen +Zufallsvariablen, die wie $X$ verteilt sind. +Ziel ist, die Verteilungsfunktion und die Wahrscheinlichkeitsdichte +des grössten, zweitgrössten, $k$-t-grössten Wertes in der Stichprobe +zu finden. + +\subsection{Verteilung von $\operatorname{max}(X_1,\dots,X_n)$ und +$\operatorname{min}(X_1,\dots,X_n)$ +\label{dreieck:subsection:minmax}} +Die Verteilungsfunktion von $\operatorname{max}(X_1,\dots,X_n)$ hat +den Wert +\begin{align*} +F_{\operatorname{max}(X_1,\dots,X_n)}(x) +&= +P(\operatorname{max}(X_1,\dots,X_n) \le x) +\\ +&= +P(X_1\le x\wedge \dots \wedge X_n\le x) +\\ +&= +P(X_1\le x) \cdot \ldots \cdot P(X_n\le x) +\\ +&= +P(X\le x)^n += +F_X(x)^n. +\end{align*} +Für die Gleichverteilung ist +\[ +F_{\text{equi}}(x) += +\begin{cases} +0&\qquad x< 0 +\\ +x&\qquad 0\le x\le 1 +\\ +1&\qquad 1 X_1\wedge \dots \wedge x > X_n) +\\ +&= +1- +(1-P(x\le X_1)) \cdot\ldots\cdot (1-P(x\le X_n)) +\\ +&= +1-(1-F_X(x))^n, +\end{align*} +Im Speziellen für im Intervall $[0,1]$ gleichverteilte $X_i$ ist die +Verteilungsfunktion des Minimums +\[ +F_{\operatorname{min}(X_1,\dots,X_n)}(x) += +\begin{cases} +0 &\qquad x<0 \\ +1-(1-x)^n&\qquad 0\le x\le 1\\ +1 &\qquad 1 < x +\end{cases} +\] +mit Wahrscheinlichkeitsdichte +\[ +\varphi_{\operatorname{min}(X_1,\dots,X_n)} += +\frac{d}{dx} +F_{\operatorname{min}(X_1,\dots,X_n)} += +\begin{cases} +n(1-x)^{n-1}&\qquad 0\le x\le 1\\ +0 &\qquad \text{sonst} +\end{cases} +\] +und Erwartungswert +\begin{align*} +E(\operatorname{min}(X_1,\dots,X_n) +&= +\int_{-\infty}^\infty x\varphi_{\operatorname{min}(X_1,\dots,X_n)}(x)\,dx += +\int_0^1 x\cdot n(1-x)^{n-1}\,dx +\\ +&= +\bigl[ -x(1-x)^n \bigr]_0^1 + \int_0^1 (1-x)^n\,dx += +\biggl[ +- +\frac{1}{n+1} +(1-x)^{n+1} +\biggr]_0^1 += +\frac{1}{n+1}. +\end{align*} +Es ergibt sich daraus als natürlich Verallgemeinerung die Frage nach +der Verteilung des zweitegrössten oder zweitkleinsten Wertes unter den +Werten $X_i$. + +\subsection{Der $k$-t-grösste Wert} +Sie wieder $X_i$ eine Stichprobe von $n$ unabhängigen wie $X$ verteilten +Zufallsvariablen. +Diese werden jetzt der Grösse nach sortiert, die sortierten Werte werden +mit +\[ +X_{1:n} \le X_{2:n} \le \dots \le X_{(n-1):n} \le X_{n:n} +\] +bezeichnet. +Die Grössen $X_{k:n}$ sind Zufallsvariablen, sie heissen die $k$-ten +Ordnungsstatistiken. +Die in Abschnitt~\ref{dreieck:subsection:minmax} behandelten Zufallsvariablen +$\operatorname{min}(X_1,\dots,X_n)$ +und +$\operatorname{max}(X_1,\dots,X_n)$ +sind die Fälle +\begin{align*} +X_{1:n} &= \operatorname{min}(X_1,\dots,X_n) \\ +X_{n:n} &= \operatorname{max}(X_1,\dots,X_n). +\end{align*} + +Um den Wert der Verteilungsfunktion von $X_{k:n}$ zu berechnen, müssen wir +die Wahrscheinlichkeit bestimmen, dass $k$ der $n$ Werte $X_i$ $x$ nicht +übersteigen. +Es muss also eine Partition von $[n]=\{1,\dots,n\}$ in eine +$k$-elementige $I=\{i_1,\dots,i_k\}$ Teilmenge und ihre +$(n-k)$-elementige Komplementmenge $[n]\setminus I$ geben +derart, dass die $X_{i} \le x$ sind für $i\in I$ und $X_{j}> x$ für +$j\in [n]\setminus I$. +Daraus kann man ablesen, dass +\begin{align*} +F_{X_{k:n}}(x) +&= +P\biggl( +\bigvee_{I\subset[n]\wedge |I|=k} +\bigwedge_{i\in I} (X_i\le x) +\wedge +\bigwedge_{j\in [n]\setminus I} (X_i > x) +\biggr). +\intertext{Da die verschiedenen $k$-elementigen Teilmengen $I\subset[n]$ +zu disjunkten Ereignissen gehören, ist die Wahrscheinlichkeit eine Summe} +&= +\sum_{I\subset[n]\wedge |I|=k} +P\biggl( +\bigwedge_{i\in I} (X_i\le x) +\wedge +\bigwedge_{j\in [n]\setminus I} (X_i > x) +\biggr) +\\ +&= +\sum_{I\subset[n]\wedge |I|=k} +\prod_{i\in I} +P(X_i\le x) +\cdot +\prod_{j\in [n]\setminus I} +P(X_j > x) +\\ +&= +\sum_{I\subset[n]\wedge |I|=k} +F_X(x)^k +(1-F_X(x))^{n-k}. +\intertext{Die Anzahl solcher Teilmengen $I$ ist gegeben durch den +Binomialkoeffizienten gebeben, die Verteilungsfunktion ist daher} +F_{X_{k:n}}(x) +&= +\binom{n}{k} +F_X(x)^k +(1-F_X(x))^{n-k}. +\end{align*} +Für im Intervall $[0,1]$ gleichverteilte $X_i$ ist die Verteilungsfunktion +der $k$-ten Ordnungsstatistik +\[ +F_{X_{k:n}}(x) += +\binom{n}{k} x^k(1-x)^{n-k}. +\] +Ihre Ableitung nach $x$ ist die Wahrscheinlichkeitsdichte und damit +wird es jetzt auch möglich, den Erwartungswert zu ermitteln: +\begin{align*} +E(X_{k:n}) +&= +\int_{0}^1 +\underbrace{x\llap{\phantom{\bigg|}}\mathstrut}_{\downarrow} +\underbrace{\frac{d}{dx}\binom{n}{k}x^k(1-x)^{n-k}}_{\uparrow} +\,dx += +\biggl[ +x\binom{n}{k}x^k(1-x)^{n-k} +\biggr]_0^1 +- +\int_0^1 +\binom{n}{k}x^k(1-x)^{n-k} +\,dx +\\ +&= +\binom{n}{k} +\biggl( +0^{n-k} +- +\int_0^1 x^k(1-x)^{n-k}\,dx +\biggr) +\end{align*} + + + + + diff --git a/buch/papers/dreieck/teil2.tex b/buch/papers/dreieck/teil2.tex new file mode 100644 index 0000000..83ea3cb --- /dev/null +++ b/buch/papers/dreieck/teil2.tex @@ -0,0 +1,9 @@ +% +% teil2.tex -- Beispiel-File für teil2 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Wahrscheinlichkeiten im Dreieckstest +\label{dreieck:section:wahrscheinlichkeiten}} +\rhead{Wahrscheinlichkeiten} + diff --git a/buch/papers/dreieck/teil3.tex b/buch/papers/dreieck/teil3.tex new file mode 100644 index 0000000..e2dfd6b --- /dev/null +++ b/buch/papers/dreieck/teil3.tex @@ -0,0 +1,10 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Erweiterungen +\label{dreieck:section:erweiterungen}} +\rhead{Erweiterungen} + + -- cgit v1.2.1 From 3157b81b70673659b27edbd680af7ef5a4485a22 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 8 Mar 2022 16:53:48 +0100 Subject: add new files --- buch/papers/dreieck/images/Makefile | 8 ++++++++ buch/papers/dreieck/images/order.tex | 34 ++++++++++++++++++++++++++++++++++ 2 files changed, 42 insertions(+) create mode 100644 buch/papers/dreieck/images/Makefile create mode 100644 buch/papers/dreieck/images/order.tex (limited to 'buch') diff --git a/buch/papers/dreieck/images/Makefile b/buch/papers/dreieck/images/Makefile new file mode 100644 index 0000000..02be1bb --- /dev/null +++ b/buch/papers/dreieck/images/Makefile @@ -0,0 +1,8 @@ +# +# Makefile +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +order.pdf: order.tex + pdflatex order.tex + diff --git a/buch/papers/dreieck/images/order.tex b/buch/papers/dreieck/images/order.tex new file mode 100644 index 0000000..826f48c --- /dev/null +++ b/buch/papers/dreieck/images/order.tex @@ -0,0 +1,34 @@ +% +% order.tex -- Verteilungsfunktion für Ordnungsstatistik +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{8} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\draw[color=red,line width=1.4pt] ({-0.1/\skala},0) + -- + plot[domain=0:1,samples=100] ({\x},{0.5*\x*\x*\x*\x*\x*\x}) + -- + ({1+0.1/\skala},0.5); + +\draw[color=red,line width=1.4pt] ({-0.1/\skala},0) + -- + plot[domain=0:1,samples=100] ({\x},{0.5*(\x*\x*\x*\x)}) + -- + ({1+0.1/\skala},0.5); + +\draw[->] ({-0.1/\skala},0) -- (1.1,0) coordinate[label={$1$}]; +\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={left:$F(X)$}]; + +\end{tikzpicture} +\end{document} + -- cgit v1.2.1 From 100498089783148753f2862c4dbfba04f110727f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 9 Mar 2022 09:42:50 +0100 Subject: add order statistics graph --- buch/papers/dreieck/images/Makefile | 4 +- buch/papers/dreieck/images/order.m | 79 ++++++++++++++++++++++++++++++++++ buch/papers/dreieck/images/order.pdf | Bin 0 -> 31044 bytes buch/papers/dreieck/images/order.tex | 81 +++++++++++++++++++++++++++++++---- 4 files changed, 155 insertions(+), 9 deletions(-) create mode 100644 buch/papers/dreieck/images/order.m create mode 100644 buch/papers/dreieck/images/order.pdf (limited to 'buch') diff --git a/buch/papers/dreieck/images/Makefile b/buch/papers/dreieck/images/Makefile index 02be1bb..3907d13 100644 --- a/buch/papers/dreieck/images/Makefile +++ b/buch/papers/dreieck/images/Makefile @@ -3,6 +3,8 @@ # # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -order.pdf: order.tex +order.pdf: order.tex orderpath.tex pdflatex order.tex +orderpath.tex: order.m + octave order.m diff --git a/buch/papers/dreieck/images/order.m b/buch/papers/dreieck/images/order.m new file mode 100644 index 0000000..d37a258 --- /dev/null +++ b/buch/papers/dreieck/images/order.m @@ -0,0 +1,79 @@ +# +# order.m +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global N; +N = 10; +global subdivisions; +subdivisions = 100; +global P; +P = 0.5 + +function retval = orderF(p, n, k) + retval = 0; + for i = (k:n) + retval = retval + nchoosek(n,i) * p^i * (1-p)^(n-i); + end +end + +function retval = orderd(p, n, k) + retval = 0; + for i = (k:n) + s = i * p^(i-1) * (1-p)^(n-i); + s = s - p^i * (n-i) * (1-p)^(n-i-1); + retval = retval + nchoosek(n,i) * s; + end +end + +function orderpath(fn, k, name) + fprintf(fn, "\\def\\order%s{\n\t(0,0)", name); + global N; + global subdivisions; + for i = (0:subdivisions) + p = i/subdivisions; + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", + p, orderF(p, N, k)); + end + fprintf(fn, "\n}\n"); +end + +function orderdpath(fn, k, name) + fprintf(fn, "\\def\\orderd%s{\n\t(0,0)", name); + global N; + global subdivisions; + for i = (1:subdivisions-1) + p = i/subdivisions; + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", + p, orderd(p, N, k)); + end + fprintf(fn, "\n\t-- ({1*\\dx},0)"); + fprintf(fn, "\n}\n"); +end + +fn = fopen("orderpath.tex", "w"); +orderpath(fn, 0, "zero"); +orderdpath(fn, 0, "zero"); +orderpath(fn, 1, "one"); +orderdpath(fn, 1, "one"); +orderpath(fn, 2, "two"); +orderdpath(fn, 2, "two"); +orderpath(fn, 3, "three"); +orderdpath(fn, 3, "three"); +orderpath(fn, 4, "four"); +orderdpath(fn, 4, "four"); +orderpath(fn, 5, "five"); +orderdpath(fn, 5, "five"); +orderpath(fn, 6, "six"); +orderdpath(fn, 6, "six"); +orderpath(fn, 7, "seven"); +orderdpath(fn, 7, "seven"); +orderpath(fn, 8, "eight"); +orderdpath(fn, 8, "eight"); +orderpath(fn, 9, "nine"); +orderdpath(fn, 9, "nine"); +orderpath(fn, 10, "ten"); +orderdpath(fn, 10, "ten"); +fclose(fn); + + diff --git a/buch/papers/dreieck/images/order.pdf b/buch/papers/dreieck/images/order.pdf new file mode 100644 index 0000000..6d9c8c0 Binary files /dev/null and b/buch/papers/dreieck/images/order.pdf differ diff --git a/buch/papers/dreieck/images/order.tex b/buch/papers/dreieck/images/order.tex index 826f48c..083f014 100644 --- a/buch/papers/dreieck/images/order.tex +++ b/buch/papers/dreieck/images/order.tex @@ -12,22 +12,87 @@ \usetikzlibrary{arrows,intersections,math} \begin{document} \def\skala{8} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\input{orderpath.tex} \begin{tikzpicture}[>=latex,thick,scale=\skala] -\draw[color=red,line width=1.4pt] ({-0.1/\skala},0) - -- - plot[domain=0:1,samples=100] ({\x},{0.5*\x*\x*\x*\x*\x*\x}) - -- - ({1+0.1/\skala},0.5); +\def\dx{1} +\def\dy{0.5} -\draw[color=red,line width=1.4pt] ({-0.1/\skala},0) +\def\pfad#1#2{ +\draw[color=#2,line width=1.4pt] ({-0.1/\skala},0) -- - plot[domain=0:1,samples=100] ({\x},{0.5*(\x*\x*\x*\x)}) + #1 -- ({1+0.1/\skala},0.5); +} -\draw[->] ({-0.1/\skala},0) -- (1.1,0) coordinate[label={$1$}]; +\pfad{\orderzero}{darkgreen!20} +\pfad{\orderone}{darkgreen!20} +\pfad{\ordertwo}{darkgreen!20} +\pfad{\orderthree}{darkgreen!20} +\pfad{\orderfour}{darkgreen!20} +\pfad{\orderfive}{darkgreen!20} +\pfad{\ordersix}{darkgreen!20} +\pfad{\ordereight}{darkgreen!20} +\pfad{\ordernine}{darkgreen!20} +\pfad{\orderten}{darkgreen!20} +\pfad{\orderseven}{darkgreen} + +\draw[->] ({-0.1/\skala},0) -- (1.1,0) coordinate[label={$x$}]; \draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={left:$F(X)$}]; +\foreach \x in {0,0.2,0.4,0.6,0.8,1}{ + \draw (\x,{-0.1/\skala}) -- (\x,{0.1/\skala}); + \node at (\x,{-0.1/\skala}) [below] {$\x$}; +} +\foreach \y in {0.5,1}{ + \draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy}); + \node at ({-0.1/\skala},{\y*\dy}) [left] {$\y$}; +} + +\node[color=darkgreen] at (0.65,{0.5*\dy}) [above,rotate=55] {$k=7$}; + +\begin{scope}[yshift=-0.7cm] +\def\dy{0.125} + +\def\pfad#1#2{ + \draw[color=#2,line width=1.4pt] ({-0.1/\skala},0) + -- + #1 + -- + ({1+0.1/\skala},0.0); +} + +\begin{scope} +\clip ({-0.1/\skala},{-0.1/\skala}) + rectangle ({1+0.1/\skala},{0.56+0.1/\skala}); +\pfad{\orderdzero}{red!20} +\pfad{\orderdone}{red!20} +\pfad{\orderdtwo}{red!20} +\pfad{\orderdthree}{red!20} +\pfad{\orderdfour}{red!20} +\pfad{\orderdfive}{red!20} +\pfad{\orderdsix}{red!20} +\pfad{\orderdeight}{red!20} +\pfad{\orderdnine}{red!20} +\pfad{\orderdten}{red!20} +\pfad{\orderdseven}{red} +\end{scope} + +\draw[->] ({-0.1/\skala},0) -- (1.1,0) coordinate[label={$x$}]; +\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={left:$\varphi(X)$}]; +\foreach \x in {0,0.2,0.4,0.6,0.8,1}{ + \draw (\x,{-0.1/\skala}) -- (\x,{0.1/\skala}); + \node at (\x,{-0.1/\skala}) [below] {$\x$}; +} +\foreach \y in {1,2,3,4}{ + \draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy}); + \node at ({-0.1/\skala},{\y*\dy}) [left] {$\y$}; +} + +\node[color=red] at (0.67,{2.7*\dy}) [above] {$k=7$}; + +\end{scope} \end{tikzpicture} \end{document} -- cgit v1.2.1 From b3a9273c682cd6eafd7bf84ddb88adcdc7631a9e Mon Sep 17 00:00:00 2001 From: Alain Date: Wed, 9 Mar 2022 12:46:49 +0100 Subject: Name und Titel --- buch/papers/parzyl/main.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) (limited to 'buch') diff --git a/buch/papers/parzyl/main.tex b/buch/papers/parzyl/main.tex index 250d170..ff21c9f 100644 --- a/buch/papers/parzyl/main.tex +++ b/buch/papers/parzyl/main.tex @@ -3,10 +3,10 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:parzyl}} -\lhead{Thema} +\chapter{Parabolische Zylinderfunktionen\label{chapter:parzyl}} +\lhead{Parabolische Zylinderfunktionen} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Thierry Schwaller, Alain Keller} Ein paar Hinweise für die korrekte Formatierung des Textes \begin{itemize} -- cgit v1.2.1 From ced982f32f430b7e3b82b3cc062411b8130b0bfd Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 11 Mar 2022 22:34:32 +0100 Subject: Bohr-Mollerup und Eindeutigkeit der Gamma-Funktion --- buch/chapters/040-rekursion/Makefile.inc | 2 + buch/chapters/040-rekursion/bohrmollerup.tex | 196 +++++++++++++++++++++++++++ buch/chapters/040-rekursion/gamma.tex | 2 + buch/chapters/040-rekursion/integral.tex | 103 ++++++++++++++ 4 files changed, 303 insertions(+) create mode 100644 buch/chapters/040-rekursion/bohrmollerup.tex create mode 100644 buch/chapters/040-rekursion/integral.tex (limited to 'buch') diff --git a/buch/chapters/040-rekursion/Makefile.inc b/buch/chapters/040-rekursion/Makefile.inc index c5887f7..ed8fd51 100644 --- a/buch/chapters/040-rekursion/Makefile.inc +++ b/buch/chapters/040-rekursion/Makefile.inc @@ -6,6 +6,8 @@ CHAPTERFILES = $(CHAPTERFILES) \ chapters/040-rekursion/gamma.tex \ + chapters/040-rekursion/bohrmollerup.tex \ + chapters/040-rekursion/integral.tex \ chapters/040-rekursion/beta.tex \ chapters/040-rekursion/linear.tex \ chapters/040-rekursion/hypergeometrisch.tex \ diff --git a/buch/chapters/040-rekursion/bohrmollerup.tex b/buch/chapters/040-rekursion/bohrmollerup.tex new file mode 100644 index 0000000..96897be --- /dev/null +++ b/buch/chapters/040-rekursion/bohrmollerup.tex @@ -0,0 +1,196 @@ +% +% bohrmollerup.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\subsection{Der Satz von Bohr-Mollerup +\label{buch:rekursion:subsection:bohr-mollerup}} +Die Integralformel und die Grenzwertdefinition für die Gamma-Funktion +zeigen beide, dass das Problem der Ausdehnung der Fakultät zu einer +Funktion $\mathbb{C}\to\mathbb{C}$ eine Lösung hat, aber es ist noch +nicht klar, in welchem Sinn dies die einzig mögliche Lösung ist. +Der Satz von Bohr-Mollerup gibt darauf eine Antwort. + +\begin{satz} +\label{buch:satz:bohr-mollerup} +Eine Funktion $f\colon \mathbb{R}^+\to\mathbb{R}$ mit den Eigenschaften +\begin{enumerate}[i)] +\item $f(1)=1$, +\item $f(x+1)=xf(x)$ für alle $x\in\mathbb{R}^+$ und +\item die Funktion $\log f(t)$ ist konvex +\end{enumerate} +ist die Gamma-Funktion: $f(t)=\Gamma(t)$. +\end{satz} + +Für den Beweis verwenden wir die folgende Eigenschaft einer konvexen +Funktion $g(x)$. +Sei +\begin{equation} +S(y,x) = \frac{g(y)-g(x)}{y-x} +\qquad\text{für $y-x$} +\end{equation} +die Steigung der Sekante zwischen den Punkten $(x,g(x))$ und $(y,g(y))$ +des Graphen von $g$. +Da $g$ konvex ist, ist $S(y,x)$ eine monoton wachsende Funktion +der beiden Variablen $x$ und $y$, solange $y>x$. + +\begin{proof}[Beweis] +Wir halten zunächst fest, dass die Bedingungen i) und ii) zur Folge haben, +dass $f(n+1)=n!$ ist für alle positiven natürlichen Zahlen. +Für die Steigung einer Sekante der Funktion $g(x)=\log f(x)$ kann damit +für natürliche Argumente bereits berechnet werden, es ist +\[ +S(n,n+1) += +\frac{\log n! - \log (n-1)!}{n+1-n} += +\frac{\log n + \log (n-1)! - \log(n-1)!}{1} += +\log n +\] +und entsprechend auch $S(n-1,n) = \log(n-1)$. + +\begin{figure} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\draw (-6,0) -- (6,0); + +\node at (-5,0) [above] {$n-1\mathstrut$}; +\node at (0,0) [above] {$n\mathstrut$}; +\node at (3,0) [above] {$n+x\mathstrut$}; +\node at (5,0) [above] {$n+1\mathstrut$}; + +\node[color=blue] at (-5,-2.3) {$S(n-1,n)\mathstrut$}; +\node[color=red] at (-1.666,-2.3) {$S(n-1,n+x)\mathstrut$}; +\node[color=darkgreen] at (1.666,-2.3) {$S(n,n+x)\mathstrut$}; +\node[color=orange] at (5,-2.3) {$S(n,n+1)\mathstrut$}; + +\node at (-3.333,-2.3) {$<\mathstrut$}; +\node at (0,-2.3) {$<\mathstrut$}; +\node at (3.333,-2.3) {$<\mathstrut$}; + +\draw[color=blue] (-5,0) -- (-5,-2) -- (0,0); +\draw[color=red] (-5,0) -- (-1.666,-2) -- (3,0); +\draw[color=darkgreen] (0,0) -- (1.666,-2) -- (3,0); +\draw[color=orange] (0,0) -- (5,-2) -- (5,0); + +\fill (-5,0) circle[radius=0.08]; +\fill (0,0) circle[radius=0.08]; +\fill (3,0) circle[radius=0.08]; +\fill (5,0) circle[radius=0.08]; + +\draw[double,color=blue] (-5,-2.5) -- (-5,-3.0); +\draw[double,color=orange] (5,-2.5) -- (5,-3.0); + +\node[color=blue] at (-5,-3.3) {$\log (n-1)\mathstrut$}; +\node[color=orange] at (5,-3.3) {$\log (n)\mathstrut$}; + +\end{tikzpicture} +\end{center} +\caption{Für den Beweis des Satzes von Bohr-Mollerup wird die +Sekantensteigung $S(x,y)$ für die Argumente $n-1$, $n$, $n+x$ und $n+1$ +verwendet. +\label{buch:rekursion:fig:bohr-mollerup}} +\end{figure} +Wir wenden jetzt die eben erwähnte Tatsache, dass $S(x,y)$ monoton +wachsend ist, auf die Punkte $n-1$, $n$, $n+x$ und $n+1$ wie +in Abbildung~\ref{buch:rekursion:fig:bohr-mollerup} an, wobei +$0 Date: Sun, 13 Mar 2022 11:05:56 +0100 Subject: add beta distribution graphs --- buch/chapters/040-rekursion/bohrmollerup.tex | 2 +- buch/chapters/040-rekursion/gamma.tex | 25 ++- buch/papers/dreieck/images/Makefile | 8 + buch/papers/dreieck/images/beta.pdf | Bin 0 -> 100791 bytes buch/papers/dreieck/images/beta.tex | 214 +++++++++++++++++++++ buch/papers/dreieck/images/betadist.m | 50 +++++ buch/papers/dreieck/images/order.m | 40 ++++ buch/papers/dreieck/images/order.pdf | Bin 31044 -> 32692 bytes buch/papers/dreieck/images/order.tex | 52 +++-- buch/papers/dreieck/teil1.tex | 273 +++++++++++++++++++++------ 10 files changed, 586 insertions(+), 78 deletions(-) create mode 100644 buch/papers/dreieck/images/beta.pdf create mode 100644 buch/papers/dreieck/images/beta.tex create mode 100644 buch/papers/dreieck/images/betadist.m (limited to 'buch') diff --git a/buch/chapters/040-rekursion/bohrmollerup.tex b/buch/chapters/040-rekursion/bohrmollerup.tex index 96897be..cd9cadc 100644 --- a/buch/chapters/040-rekursion/bohrmollerup.tex +++ b/buch/chapters/040-rekursion/bohrmollerup.tex @@ -172,7 +172,7 @@ erhalten wir (x)_n f(x) < n^x (n-1)! -\\ +\intertext{oder nach Division durch $(x)_n$} %\underbrace{ \frac{(n-1)^x (n-1)!}{(x)_n} %}_{\displaystyle\to \Gamma(x)} diff --git a/buch/chapters/040-rekursion/gamma.tex b/buch/chapters/040-rekursion/gamma.tex index af5d572..7d4453b 100644 --- a/buch/chapters/040-rekursion/gamma.tex +++ b/buch/chapters/040-rekursion/gamma.tex @@ -651,8 +651,11 @@ Abschnitt~\ref{buch:funktionentheorie:section:fortsetzung} beschrieben wird, kann die Funktion auf ganz $\mathbb{C}$ ausgedehnt werden, mit Ausnahme einzelner Pole. Die Funktionalgleichung gilt natürlich für alle $z\in\mathbb{C}$, -für die $\Gamma(z)$ definiert ist. -In einer Umgebung von $z=-n$ gilt +für die $\Gamma(z)$ definiert ist, nicht nur für diejenigen $z$, für +die das Integral konvergiert. +Wir können Sie daher verwenden, um das Argument in den Bereich +zu bringen, wo das Integral zur Berechnung verwendet werden kann. +Dazu berechnen wir \[ \Gamma(z) = @@ -665,12 +668,20 @@ In einer Umgebung von $z=-n$ gilt \dots = \frac{\Gamma(z+n)}{z(z+1)(z+2)\cdots(z+n-1)} += +\frac{\Gamma(z+n)}{(z)_n}. \] -Keiner der Faktoren im Nenner verschwindet in der Nähe von $z=-n$, der -Zähler hat aber einen Pol erster Ordnung an dieser Stelle. -Daher hat auch der Quotient einen Pol erster Ordnung. -Abbildung~\ref{buch:rekursion:fig:gamma} zeigt die Pole bei den -nicht negativen ganzen Zahlen. +Dies gilt für jedes natürlich $n$. +Für $n$ gross genug, genauer für +$n\ge |\operatorname{Re}z|$, +ist $\operatorname{Re}(z+n)=\operatorname{Re}z + n>0$ und damit +kann $\Gamma(z+n)$ mit der Integralformel berechnet werden. + +Die Gamma-Funktion hat keine Nullstellen, aber in der Nähe von $z=-n$ +hat der Nenner eine Nullstelle erster Ordnung. +Somit hat $\Gamma(z)$ Pole erster Ordnung bei den negativen +ganzen Zahlen und bei $0$, wie sie in +Abbildung~\ref{buch:rekursion:fig:gamma} gezeigt werden. \subsubsection{Numerische Berechnung} \begin{table} diff --git a/buch/papers/dreieck/images/Makefile b/buch/papers/dreieck/images/Makefile index 3907d13..c979599 100644 --- a/buch/papers/dreieck/images/Makefile +++ b/buch/papers/dreieck/images/Makefile @@ -3,8 +3,16 @@ # # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # +all: order.pdf beta.pdf + order.pdf: order.tex orderpath.tex pdflatex order.tex orderpath.tex: order.m octave order.m + +beta.pdf: beta.tex betapaths.tex + pdflatex beta.tex + +betapaths.tex: betadist.m + octave betadist.m diff --git a/buch/papers/dreieck/images/beta.pdf b/buch/papers/dreieck/images/beta.pdf new file mode 100644 index 0000000..c3ab4f6 Binary files /dev/null and b/buch/papers/dreieck/images/beta.pdf differ diff --git a/buch/papers/dreieck/images/beta.tex b/buch/papers/dreieck/images/beta.tex new file mode 100644 index 0000000..50509ee --- /dev/null +++ b/buch/papers/dreieck/images/beta.tex @@ -0,0 +1,214 @@ +% +% beta.tex -- display some symmetric beta distributions +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\input{betapaths.tex} +\begin{document} +\def\skala{12} +\definecolor{colorone}{rgb}{1.0,0.6,0.0} +\definecolor{colortwo}{rgb}{1.0,0.0,0.0} +\definecolor{colorthree}{rgb}{0.6,0.0,0.6} +\definecolor{colorfour}{rgb}{0.6,0.0,1.0} +\definecolor{colorfive}{rgb}{0.0,0.0,1.0} +\definecolor{colorsix}{rgb}{0.4,0.6,1.0} +\definecolor{colorseven}{rgb}{0.0,0.0,0.0} +\definecolor{coloreight}{rgb}{0.0,0.8,0.8} +\definecolor{colornine}{rgb}{0.0,0.8,0.2} +\definecolor{colorten}{rgb}{0.2,0.4,0.0} +\definecolor{coloreleven}{rgb}{1.0,0.8,0.4} + +\def\achsen{ + \foreach \x in {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9}{ + \draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala}); + \node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$}; + } + \foreach \y in {1,2,3,4}{ + \draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy}); + \node at ({-0.1/\skala},{\y*\dy}) [left] {$\y$}; + } + \def\x{1} + \draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala}); + \node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$}; + \def\x{0} + \node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$}; + + \draw[->] ({-0.1/\skala},0) -- ({1*\dx+0.4/\skala},0) + coordinate[label={$x$}]; + \draw[->] (0,{-0.1/\skala}) -- (0,{\betamax*\dy+0.4/\skala},0) + coordinate[label={right:$\beta(a,b,x)$}]; +} + +\def\farbcoord#1#2{ + ({\dx*(0.7+((#1-1)/4)*0.27)},{\dx*(0.15+((#2-1)/4)*0.27)}) +} +\def\farbviereck{ + \foreach \x in {1,2,3,4,5}{ + \draw[color=gray!30] \farbcoord{\x}{1} -- \farbcoord{\x}{5}; + \draw[color=gray!30] \farbcoord{1}{\x} -- \farbcoord{5}{\x}; + } + \draw[->] \farbcoord{1}{1} -- \farbcoord{5.4}{1} + coordinate[label={$a$}]; + \draw[->] \farbcoord{1}{1} -- \farbcoord{1}{5.4} + coordinate[label={left: $b$}]; + \foreach \x in {1,2,3,4,5}{ + \node[color=gray] at \farbcoord{5}{\x} [right] {\tiny $b=\x$}; + \fill[color=white,opacity=0.7] + \farbcoord{(\x-0.1)}{4.3} + rectangle + \farbcoord{(\x+0.1)}{5}; + \node[color=gray] at \farbcoord{\x}{5} [left,rotate=90] + {\tiny $a=\x$}; + } +} +\def\farbpunkt#1#2#3{ + \fill[color=#3] \farbcoord{#1}{#2} circle[radius={0.1/\skala}]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\def\dx{1} +\def\dy{0.1} +\def\opa{0.1} + +\def\betamax{4.2} + +\fill[color=colorone,opacity=\opa] (0,0) -- \betaaa -- (\dx,0) -- cycle; +\fill[color=colortwo,opacity=\opa] (0,0) -- \betabb -- (\dx,0) -- cycle; +\fill[color=colorthree,opacity=\opa] (0,0) -- \betacc -- (\dx,0) -- cycle; +\fill[color=colorfour,opacity=\opa] (0,0) -- \betadd -- (\dx,0) -- cycle; +\fill[color=colorfive,opacity=\opa] (0,0) -- \betaee -- (\dx,0) -- cycle; +\fill[color=colorsix,opacity=\opa] (0,0) -- \betaff -- (\dx,0) -- cycle; +\fill[color=colorseven,opacity=\opa] (0,0) -- \betagg -- (\dx,0) -- cycle; +\fill[color=coloreight,opacity=\opa] (0,0) -- \betahh -- (\dx,0) -- cycle; +\fill[color=colornine,opacity=\opa] (0,0) -- \betaii -- (\dx,0) -- cycle; +\fill[color=colorten,opacity=\opa] (0,0) -- \betajj -- (\dx,0) -- cycle; +\fill[color=coloreleven,opacity=\opa] (0,0) -- \betakk -- (\dx,0) -- cycle; + +\draw[color=colorone] \betaaa; +\draw[color=colortwo] \betabb; +\draw[color=colorthree] \betacc; +\draw[color=colorfour] \betadd; +\draw[color=colorfive] \betaee; +\draw[color=colorsix] \betaff; +\draw[color=colorseven] \betagg; +\draw[color=coloreight] \betahh; +\draw[color=colornine] \betaii; +\draw[color=colorten] \betajj; +\draw[color=coloreleven] \betakk; + +\achsen + +\farbviereck + +\farbpunkt{\alphaeleven}{\betaeleven}{coloreleven} +\farbpunkt{\alphaten}{\betaten}{colorten} +\farbpunkt{\alphanine}{\betanine}{colornine} +\farbpunkt{\alphaeight}{\betaeight}{coloreight} +\farbpunkt{\alphaseven}{\betaseven}{colorseven} +\farbpunkt{\alphasix}{\betasix}{colorsix} +\farbpunkt{\alphafive}{\betafive}{colorfive} +\farbpunkt{\alphafour}{\betafour}{colorfour} +\farbpunkt{\alphathree}{\betathree}{colorthree} +\farbpunkt{\alphatwo}{\betatwo}{colortwo} +\farbpunkt{\alphaone}{\betaone}{colorone} + + +\def\betamax{4.9} + +\begin{scope}[yshift=-0.6cm] +\fill[color=colorone,opacity=\opa] (0,0) -- \betaaa -- (\dx,0) -- cycle; +\fill[color=colortwo,opacity=\opa] (0,0) -- \betaab -- (\dx,0) -- cycle; +\fill[color=colorthree,opacity=\opa] (0,0) -- \betaac -- (\dx,0) -- cycle; +\fill[color=colorfour,opacity=\opa] (0,0) -- \betaad -- (\dx,0) -- cycle; +\fill[color=colorfive,opacity=\opa] (0,0) -- \betaae -- (\dx,0) -- cycle; +\fill[color=colorsix,opacity=\opa] (0,0) -- \betaaf -- (\dx,0) -- cycle; +\fill[color=colorseven,opacity=\opa] (0,0) -- \betaag -- (\dx,0) -- cycle; +\fill[color=coloreight,opacity=\opa] (0,0) -- \betaah -- (\dx,0) -- cycle; +\fill[color=colornine,opacity=\opa] (0,0) -- \betaai -- (\dx,0) -- cycle; +\fill[color=colorten,opacity=\opa] (0,0) -- \betaaj -- (\dx,0) -- cycle; +\fill[color=coloreleven,opacity=\opa] (0,0) -- \betaak -- (\dx,0) -- cycle; + +\draw[color=colorone] \betaaa; +\draw[color=colortwo] \betaab; +\draw[color=colorthree] \betaac; +\draw[color=colorfour] \betaad; +\draw[color=colorfive] \betaae; +\draw[color=colorsix] \betaaf; +\draw[color=colorseven] \betaag; +\draw[color=coloreight] \betaah; +\draw[color=colornine] \betaai; +\draw[color=colorten] \betaaj; +\draw[color=coloreleven] \betaak; + +\achsen + +\farbviereck + +\farbpunkt{\alphaone}{\betaeleven}{coloreleven} +\farbpunkt{\alphaone}{\betaten}{colorten} +\farbpunkt{\alphaone}{\betanine}{colornine} +\farbpunkt{\alphaone}{\betaeight}{coloreight} +\farbpunkt{\alphaone}{\betaseven}{colorseven} +\farbpunkt{\alphaone}{\betasix}{colorsix} +\farbpunkt{\alphaone}{\betafive}{colorfive} +\farbpunkt{\alphaone}{\betafour}{colorfour} +\farbpunkt{\alphaone}{\betathree}{colorthree} +\farbpunkt{\alphaone}{\betatwo}{colortwo} +\farbpunkt{\alphaone}{\betaone}{colorone} + +\end{scope} + +\begin{scope}[yshift=-1.2cm] +\fill[color=colorone,opacity=\opa] (0,0) -- \betaak -- (\dx,0) -- cycle; +\fill[color=colortwo,opacity=\opa] (0,0) -- \betabk -- (\dx,0) -- cycle; +\fill[color=colorthree,opacity=\opa] (0,0) -- \betack -- (\dx,0) -- cycle; +\fill[color=colorfour,opacity=\opa] (0,0) -- \betadk -- (\dx,0) -- cycle; +\fill[color=colorfive,opacity=\opa] (0,0) -- \betaek -- (\dx,0) -- cycle; +\fill[color=colorsix,opacity=\opa] (0,0) -- \betafk -- (\dx,0) -- cycle; +\fill[color=colorseven,opacity=\opa] (0,0) -- \betagk -- (\dx,0) -- cycle; +\fill[color=coloreight,opacity=\opa] (0,0) -- \betahk -- (\dx,0) -- cycle; +\fill[color=colornine,opacity=\opa] (0,0) -- \betaik -- (\dx,0) -- cycle; +\fill[color=colorten,opacity=\opa] (0,0) -- \betajk -- (\dx,0) -- cycle; +\fill[color=coloreleven,opacity=\opa] (0,0) -- \betakk -- (\dx,0) -- cycle; + +\draw[color=colorone] \betaak; +\draw[color=colortwo] \betabk; +\draw[color=colorthree] \betack; +\draw[color=colorfour] \betadk; +\draw[color=colorfive] \betaek; +\draw[color=colorsix] \betafk; +\draw[color=colorseven] \betagk; +\draw[color=coloreight] \betahk; +\draw[color=colornine] \betaik; +\draw[color=colorten] \betajk; +\draw[color=coloreleven] \betakk; + +\achsen + +\farbviereck + +\farbpunkt{\alphaeleven}{\betaeleven}{coloreleven} +\farbpunkt{\alphaten}{\betaeleven}{colorten} +\farbpunkt{\alphanine}{\betaeleven}{colornine} +\farbpunkt{\alphaeight}{\betaeleven}{coloreight} +\farbpunkt{\alphaseven}{\betaeleven}{colorseven} +\farbpunkt{\alphasix}{\betaeleven}{colorsix} +\farbpunkt{\alphafive}{\betaeleven}{colorfive} +\farbpunkt{\alphafour}{\betaeleven}{colorfour} +\farbpunkt{\alphathree}{\betaeleven}{colorthree} +\farbpunkt{\alphatwo}{\betaeleven}{colortwo} +\farbpunkt{\alphaone}{\betaeleven}{colorone} + +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/dreieck/images/betadist.m b/buch/papers/dreieck/images/betadist.m new file mode 100644 index 0000000..9ff78ed --- /dev/null +++ b/buch/papers/dreieck/images/betadist.m @@ -0,0 +1,50 @@ +# +# betadist.m +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global N; +N = 201; +global n; +n = 11; + +t = (0:n-1) / (n-1) +alpha = 1 + 4 * t.^2 + +#alpha = [ 1, 1.03, 1.05, 1.1, 1.25, 1.5, 2, 2.5, 3, 4, 5 ]; +beta = alpha; +names = [ "one"; "two"; "three"; "four"; "five"; "six"; "seven"; "eight"; + "nine"; "ten"; "eleven" ] + +function retval = Beta(a, b, x) + retval = x^(a-1) * (1-x)^(b-1) / beta(a, b); +end + +function plotbeta(fn, a, b, name) + global N; + fprintf(fn, "\\def\\beta%s{\n", name); + fprintf(fn, "\t({%.4f*\\dx},{%.4f*\\dy})", 0, Beta(a, b, 0)); + for x = (1:N-1)/(N-1) + X = (1-cos(pi * x))/2; + fprintf(fn, "\n\t--({%.4f*\\dx},{%.4f*\\dy})", + X, Beta(a, b, X)); + end + fprintf(fn, "\n}\n"); +end + +fn = fopen("betapaths.tex", "w"); + +for i = (1:n) + fprintf(fn, "\\def\\alpha%s{%f}\n", names(i,:), alpha(i)); + fprintf(fn, "\\def\\beta%s{%f}\n", names(i,:), beta(i)); +end + +for i = (1:n) + for j = (1:n) + printf("working on %d,%d:\n", i, j); + plotbeta(fn, alpha(i), beta(j), + char(['a' + i - 1, 'a' + j - 1])); + end +end + +fclose(fn); diff --git a/buch/papers/dreieck/images/order.m b/buch/papers/dreieck/images/order.m index d37a258..762f458 100644 --- a/buch/papers/dreieck/images/order.m +++ b/buch/papers/dreieck/images/order.m @@ -26,6 +26,10 @@ function retval = orderd(p, n, k) end end +function retval = orders(p, n, k) + retval = k * nchoosek(n, k) * p^(k-1) * (1-p)^(n-k); +end + function orderpath(fn, k, name) fprintf(fn, "\\def\\order%s{\n\t(0,0)", name); global N; @@ -51,29 +55,65 @@ function orderdpath(fn, k, name) fprintf(fn, "\n}\n"); end +function orderspath(fn, k, name) + fprintf(fn, "\\def\\orders%s{\n\t(0,0)", name); + global N; + global subdivisions; + for i = (1:subdivisions-1) + p = i/subdivisions; + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", + p, orders(p, N, k)); + end + fprintf(fn, "\n\t-- ({1*\\dx},0)"); + fprintf(fn, "\n}\n"); +end + fn = fopen("orderpath.tex", "w"); + orderpath(fn, 0, "zero"); orderdpath(fn, 0, "zero"); +orderspath(fn, 0, "zero"); + orderpath(fn, 1, "one"); orderdpath(fn, 1, "one"); +orderspath(fn, 1, "one"); + orderpath(fn, 2, "two"); orderdpath(fn, 2, "two"); +orderspath(fn, 2, "two"); + orderpath(fn, 3, "three"); orderdpath(fn, 3, "three"); +orderspath(fn, 3, "three"); + orderpath(fn, 4, "four"); orderdpath(fn, 4, "four"); +orderspath(fn, 4, "four"); + orderpath(fn, 5, "five"); orderdpath(fn, 5, "five"); +orderspath(fn, 5, "five"); + orderpath(fn, 6, "six"); orderdpath(fn, 6, "six"); +orderspath(fn, 6, "six"); + orderpath(fn, 7, "seven"); orderdpath(fn, 7, "seven"); +orderspath(fn, 7, "seven"); + orderpath(fn, 8, "eight"); orderdpath(fn, 8, "eight"); +orderspath(fn, 8, "eight"); + orderpath(fn, 9, "nine"); orderdpath(fn, 9, "nine"); +orderspath(fn, 9, "nine"); + orderpath(fn, 10, "ten"); orderdpath(fn, 10, "ten"); +orderspath(fn, 10, "ten"); + fclose(fn); diff --git a/buch/papers/dreieck/images/order.pdf b/buch/papers/dreieck/images/order.pdf index 6d9c8c0..98a5fbe 100644 Binary files a/buch/papers/dreieck/images/order.pdf and b/buch/papers/dreieck/images/order.pdf differ diff --git a/buch/papers/dreieck/images/order.tex b/buch/papers/dreieck/images/order.tex index 083f014..9a2511c 100644 --- a/buch/papers/dreieck/images/order.tex +++ b/buch/papers/dreieck/images/order.tex @@ -13,10 +13,25 @@ \begin{document} \def\skala{8} \definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\n{10} +\def\E#1#2{ + \draw[color=#2] + ({\dx*#1/(\n+1)},{-0.1/\skala}) -- ({\dx*#1/(\n+1)},{4.4*\dy}); + \node[color=#2] at ({\dx*#1/(\n+1)},{3.2*\dy}) + [rotate=90,above right] {$k=#1$}; +} +\def\var#1#2{ + \pgfmathparse{\dx*sqrt(#1*(\n-#1+1)/((\n+1)*(\n+1)*(\n+2)))} + \xdef\var{\pgfmathresult} + \fill[color=#2,opacity=0.5] + ({\dx*#1/(\n+1)-\var},0) rectangle ({\dx*#1/(\n+1)+\var},{4.4*\dy}); +} + \input{orderpath.tex} \begin{tikzpicture}[>=latex,thick,scale=\skala] -\def\dx{1} +\def\dx{1.6} \def\dy{0.5} \def\pfad#1#2{ @@ -24,7 +39,7 @@ -- #1 -- - ({1+0.1/\skala},0.5); + ({1*\dx+0.1/\skala},0.5); } \pfad{\orderzero}{darkgreen!20} @@ -39,11 +54,11 @@ \pfad{\orderten}{darkgreen!20} \pfad{\orderseven}{darkgreen} -\draw[->] ({-0.1/\skala},0) -- (1.1,0) coordinate[label={$x$}]; -\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={left:$F(X)$}]; +\draw[->] ({-0.1/\skala},0) -- ({1.03*\dx},0) coordinate[label={$x$}]; +\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={right:$F(X)$}]; \foreach \x in {0,0.2,0.4,0.6,0.8,1}{ - \draw (\x,{-0.1/\skala}) -- (\x,{0.1/\skala}); - \node at (\x,{-0.1/\skala}) [below] {$\x$}; + \draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala}); + \node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$}; } \foreach \y in {0.5,1}{ \draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy}); @@ -55,17 +70,25 @@ \begin{scope}[yshift=-0.7cm] \def\dy{0.125} +\foreach \k in {1,2,3,4,5,6,8,9,10}{ + \E{\k}{blue!30} +} +\def\k{7} +\var{\k}{orange!40} +\node[color=blue] at ({\dx*\k/(\n+1)},{4.3*\dy}) [above] {$E(X_{7:n})$}; + \def\pfad#1#2{ \draw[color=#2,line width=1.4pt] ({-0.1/\skala},0) -- #1 -- - ({1+0.1/\skala},0.0); + ({1*\dx+0.1/\skala},0.0); } \begin{scope} \clip ({-0.1/\skala},{-0.1/\skala}) - rectangle ({1+0.1/\skala},{0.56+0.1/\skala}); + rectangle ({1*\dx+0.1/\skala},{0.56+0.1/\skala}); + \pfad{\orderdzero}{red!20} \pfad{\orderdone}{red!20} \pfad{\orderdtwo}{red!20} @@ -76,21 +99,24 @@ \pfad{\orderdeight}{red!20} \pfad{\orderdnine}{red!20} \pfad{\orderdten}{red!20} +\E{\k}{blue} \pfad{\orderdseven}{red} + \end{scope} -\draw[->] ({-0.1/\skala},0) -- (1.1,0) coordinate[label={$x$}]; -\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={left:$\varphi(X)$}]; +\draw[->] ({-0.1/\skala},0) -- ({1.03*\dx},0) coordinate[label={$x$}]; +\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={right:$\varphi(X)$}]; \foreach \x in {0,0.2,0.4,0.6,0.8,1}{ - \draw (\x,{-0.1/\skala}) -- (\x,{0.1/\skala}); - \node at (\x,{-0.1/\skala}) [below] {$\x$}; + \draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala}); + \node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$}; } \foreach \y in {1,2,3,4}{ \draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy}); \node at ({-0.1/\skala},{\y*\dy}) [left] {$\y$}; } -\node[color=red] at (0.67,{2.7*\dy}) [above] {$k=7$}; +\node[color=red] at ({0.67*\dx},{2.7*\dy}) [above] {$k=7$}; + \end{scope} diff --git a/buch/papers/dreieck/teil1.tex b/buch/papers/dreieck/teil1.tex index 255c5d0..5e7090b 100644 --- a/buch/papers/dreieck/teil1.tex +++ b/buch/papers/dreieck/teil1.tex @@ -12,6 +12,8 @@ Zufallsvariablen, die wie $X$ verteilt sind. Ziel ist, die Verteilungsfunktion und die Wahrscheinlichkeitsdichte des grössten, zweitgrössten, $k$-t-grössten Wertes in der Stichprobe zu finden. +Wir schreiben $[n]=\{1,\dots,n\}$ für die Menge der natürlichen +Zahlen von zwischen $1$ und $n$. \subsection{Verteilung von $\operatorname{max}(X_1,\dots,X_n)$ und $\operatorname{min}(X_1,\dots,X_n)$ @@ -176,86 +178,243 @@ X_{n:n} &= \operatorname{max}(X_1,\dots,X_n). Um den Wert der Verteilungsfunktion von $X_{k:n}$ zu berechnen, müssen wir die Wahrscheinlichkeit bestimmen, dass $k$ der $n$ Werte $X_i$ $x$ nicht übersteigen. -Es muss also eine Partition von $[n]=\{1,\dots,n\}$ in eine -$k$-elementige $I=\{i_1,\dots,i_k\}$ Teilmenge und ihre -$(n-k)$-elementige Komplementmenge $[n]\setminus I$ geben -derart, dass die $X_{i} \le x$ sind für $i\in I$ und $X_{j}> x$ für -$j\in [n]\setminus I$. -Daraus kann man ablesen, dass +Der $k$-te Wert $X_{k:n}$ übersteigt genau dann $x$ nicht, wenn +mindestens $k$ der Zufallswerte $X_i$ $x$ nicht übersteigen, also +\[ +P(X_{k:n} \le x) += +P\left( +|\{i\in[n]\,|\, X_i\le x\}| \ge k +\right). +\] + +Das Ereignis $\{X_i\le x\}$ ist eine Bernoulli-Experiment, welches mit +Wahrscheinlichkeit $F_X(x)$ eintritt. +Die Anzahl der Zufallsvariablen $X_i$, die $x$ übertreffen, ist also +Binomialverteilt mit $p=F_X(x)$. +Damit haben wir gefunden, dass mit Wahrscheinlichkeit +\begin{equation} +F_{X_{k:n}}(x) += +P(X_{k:n}\le x) += +\sum_{i=k}^n \binom{n}{i}F_X(x)^i (1-F_X(x))^{n-i} +\label{dreieck:eqn:FXkn} +\end{equation} +mindestens $k$ der Zufallsvariablen den Wert $x$ überschreiten. + +\subsubsection{Wahrscheinlichkeitsdichte der Ordnungsstatistik} +Die Wahrscheinlichkeitsdichte der Ordnungsstatistik kann durch Ableitung +von \eqref{dreieck:eqn:FXkn} gefunden, werden, sie ist \begin{align*} +\varphi_{X_{k:n}}(x) +&= +\frac{d}{dx} F_{X_{k:n}}(x) +\\ &= -P\biggl( -\bigvee_{I\subset[n]\wedge |I|=k} -\bigwedge_{i\in I} (X_i\le x) -\wedge -\bigwedge_{j\in [n]\setminus I} (X_i > x) -\biggr). -\intertext{Da die verschiedenen $k$-elementigen Teilmengen $I\subset[n]$ -zu disjunkten Ereignissen gehören, ist die Wahrscheinlichkeit eine Summe} +\sum_{i=k}^n +\binom{n}{i} +\bigl( +iF_X(x)^{i-1}\varphi_X(x) (1-F_X(x))^{n-i} +- +F_X(x)^k +(n-i) +(1-F_X(x))^{n-i-1} +\varphi_X(x) +\bigr) +\\ &= -\sum_{I\subset[n]\wedge |I|=k} -P\biggl( -\bigwedge_{i\in I} (X_i\le x) -\wedge -\bigwedge_{j\in [n]\setminus I} (X_i > x) +\sum_{i=k}^n +\binom{n}{i} +\varphi_X(x) +F_X(x)^{i-1}(1-F_X(x))^{n-i-1} +\bigl( +iF_X(x)-(n-i)(1-F_X(x)) +\bigr) +\\ +&= +\varphi_X(x) +\biggl( +\sum_{i=k}^n i\binom{n}{i} F_X(x)^{i-1}(1-F_X(x))^{n-i} +- +\sum_{j=k}^n (n-j)\binom{n}{j} F_X(x)^{j}(1-F_X(x))^{n-j-1} \biggr) \\ &= -\sum_{I\subset[n]\wedge |I|=k} -\prod_{i\in I} -P(X_i\le x) -\cdot -\prod_{j\in [n]\setminus I} -P(X_j > x) +\varphi_X(x) +\biggl( +\sum_{i=k}^n i\binom{n}{i} F_X(x)^{i-1}(1-F_X(x))^{n-i} +- +\sum_{i=k+1}^{n+1} (n-i+1)\binom{n}{i-1} F_X(x)^{i-1}(1-F_X(x))^{n-i} +\biggr) \\ &= -\sum_{I\subset[n]\wedge |I|=k} -F_X(x)^k -(1-F_X(x))^{n-k}. -\intertext{Die Anzahl solcher Teilmengen $I$ ist gegeben durch den -Binomialkoeffizienten gebeben, die Verteilungsfunktion ist daher} -F_{X_{k:n}}(x) +\varphi_X(x) +\biggl( +k\binom{n}{k}F_X(x)^{k-1}(1-F_X(x))^{n-k} ++ +\sum_{i=k+1}^{n+1} +\left( +i\binom{n}{i} +- +(n-i+1)\binom{n}{i-1} +\right) +F_X(x)^{i-1}(1-F_X(x))^{n-i} +\biggr) +\end{align*} +Mit den wohlbekannten Identitäten für die Binomialkoeffizienten +\begin{align*} +i\binom{n}{i} +- +(n-i+1)\binom{n}{i-1} &= -\binom{n}{k} -F_X(x)^k -(1-F_X(x))^{n-k}. +n\binom{n-1}{i-1} +- +n +\binom{n-1}{i-1} += +0 +\end{align*} +folgt jetzt +\begin{align*} +\varphi_{X_{k:n}}(x) +&= +\varphi_X(x)k\binom{n}{k} F_X(x)^{k-1}(1-F_X(x))^{n-k}(x). +\intertext{Im Speziellen für gleichverteilte Zufallsvariablen $X_i$ ist +} +\varphi_{X_{k:n}}(x) +&= +k\binom{n}{k} x^{k-1}(1-x)^{n-k}. \end{align*} -Für im Intervall $[0,1]$ gleichverteilte $X_i$ ist die Verteilungsfunktion -der $k$-ten Ordnungsstatistik +Dies ist die Wahrscheinlichkeitsdichte einer Betaverteilung \[ -F_{X_{k:n}}(x) +\beta(k,n-k+1)(x) += +\frac{1}{B(k,n-k+1)} +x^{k-1}(1-x)^{n-k}. +\] +Tatsächlich ist die Normierungskonstante +\begin{align} +\frac{1}{B(k,n-k+1)} +&= +\frac{\Gamma(n+1)}{\Gamma(k)\Gamma(n-k+1)} += +\frac{n!}{(k-1)!(n-k)!}. +\label{dreieck:betaverteilung:normierung1} +\end{align} +Andererseits ist +\[ +k\binom{n}{k} += +k\frac{n!}{k!(n-k)!} = -\binom{n}{k} x^k(1-x)^{n-k}. +\frac{n!}{(k-1)!(n-k)!}, \] -Ihre Ableitung nach $x$ ist die Wahrscheinlichkeitsdichte und damit -wird es jetzt auch möglich, den Erwartungswert zu ermitteln: +in Übereinstimmung mit~\eqref{dreieck:betaverteilung:normierung1}. +Die Verteilungsfunktion und die Wahrscheinlichkeitsdichte der +Ordnungsstatistik sind in Abbildung~\ref{dreieck:fig:order} dargestellt. + +\begin{figure} +\centering +\includegraphics{papers/dreieck/images/order.pdf} +\caption{Verteilungsfunktion und Wahrscheinlichkeitsdichte der +Ordnungsstatistiken $X_{k:n}$ einer gleichverteilung Zuvallsvariable +mit $n=10$. +\label{dreieck:fig:order}} +\end{figure} + +\subsubsection{Erwartungswert} +Mit der Wahrscheinlichkeitsdichte kann man jetzt auch den Erwartungswerte +der $k$-ten Ordnungsstatistik bestimmen. +Die Rechnung ergibt: \begin{align*} E(X_{k:n}) &= -\int_{0}^1 -\underbrace{x\llap{\phantom{\bigg|}}\mathstrut}_{\downarrow} -\underbrace{\frac{d}{dx}\binom{n}{k}x^k(1-x)^{n-k}}_{\uparrow} -\,dx +\int_0^1 x\cdot k\binom{n}{k} x^{k-1}(1-x)^{n-k}\,dx = -\biggl[ -x\binom{n}{k}x^k(1-x)^{n-k} -\biggr]_0^1 -- +k +\binom{n}{k} \int_0^1 -\binom{n}{k}x^k(1-x)^{n-k} -\,dx -\\ +x^{k}(1-x)^{n-k}\,dx. +\intertext{Dies ist das Beta-Integral} &= -\binom{n}{k} -\biggl( -0^{n-k} -- -\int_0^1 x^k(1-x)^{n-k}\,dx -\biggr) +k\binom{n}{k} +B(k+1,n-k+1) +\intertext{welches man durch Gamma-Funktionen bzw.~durch Fakultäten wie in} +&= +k\frac{n!}{k!(n-k)!} +\frac{\Gamma(k+1)\Gamma(n-k+1)}{n+2} += +k\frac{n!}{k!(n-k)!} +\frac{k!(n-k)!}{(n+1)!} += +\frac{k}{n+1} \end{align*} +ausdrücken kann. +Die Erwartungswerte haben also regelmässige Abstände, sie sind in +Abbildung~\ref{dreieck:fig:order} als blaue vertikale Linien eingezeichnet. +\subsubsection{Varianz} +Auch die Varianz lässt sich einfach berechnen, dazu muss zunächst +der Erwartungswert von $X_{k:n}^2$ bestimmt werden. +Er ist +\begin{align*} +E(X_{k:n}^2) +&= +\int_0^1 x^2\cdot k\binom{n}{k} x^{k-1}(1-x)^{n-k}\,dx += +k +\binom{n}{k} +\int_0^1 +x^{k+1}(1-x)^{n-k}\,dx. +\intertext{Auch dies ist ein Beta-Integral, nämlich} +&= +k\binom{n}{k} +B(k+2,n-k+1) += +k\frac{n!}{k!(n-k)!} +\frac{(k+1)!(n-k)!}{(n+2)!} += +\frac{k(k+1)}{(n+1)(n+2)}. +\end{align*} +Die Varianz wird damit +\begin{align} +\operatorname{var}(X_{k:n}) +&= +E(X_{k:n}^2) - E(X_{k:n})^2 +\notag +\\ +& += +\frac{k(k+1)}{(n+1)(n+2)}-\frac{k^2}{(n+1)^2} += +\frac{k(k+1)(n+1)-k^2(n+2)}{(n+1)^2(n+2)} += +\frac{k(n-k+1)}{(n+1)^2(n+2)}. +\label{dreieck:eqn:ordnungsstatistik:varianz} +\end{align} +In Abbildung~\ref{dreieck:fig:order} ist die Varianz der +Ordnungsstatistik $X_{k:n}$ für $k=7$ und $n=10$ als oranges +Rechteck dargestellt. +\begin{figure} +\centering +\includegraphics[width=0.84\textwidth]{papers/dreieck/images/beta.pdf} +\caption{Wahrscheinlichkeitsdichte der Beta-Verteilung +$\beta(a,b,x)$ +für verschiedene Werte der Parameter $a$ und $b$. +Die Werte des Parameters für einen Graphen einer Beta-Verteilung +sind als Punkt im kleinen Quadrat rechts +im Graphen als Punkt mit der gleichen Farbe dargestellt. +\label{dreieck:fig:betaverteilungn}} +\end{figure} +Die Formel~\eqref{dreieck:eqn:ordnungsstatistik:varianz} +besagt auch, dass die Varianz der proportional ist zu $k((n+1)-k)$. +Dieser Ausdruck ist am grössten für $k=(n+1)/2$, die Varianz ist +also grösser für die ``mittleren'' Ordnungstatistiken als für die +extremen $X_{1:n}=\operatorname{min}(X_1,\dots,X_n)$ und +$X_{n:n}=\operatorname{max}(X_1,\dots,X_n)$. -- cgit v1.2.1 From f5047d4d780e996a8b8f7738c1ac7c884a07f135 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 13 Mar 2022 23:26:58 +0100 Subject: new stuff about beta, test2 --- buch/aufgaben2.tex | 4 + buch/chapters/040-rekursion/Makefile.inc | 1 + buch/chapters/040-rekursion/beta.tex | 104 ++------- buch/chapters/040-rekursion/images/Makefile | 16 +- buch/chapters/040-rekursion/images/beta.pdf | Bin 0 -> 109772 bytes buch/chapters/040-rekursion/images/beta.tex | 236 +++++++++++++++++++++ buch/chapters/040-rekursion/images/betadist.m | 58 +++++ buch/chapters/040-rekursion/images/order.m | 119 +++++++++++ buch/chapters/040-rekursion/images/order.pdf | Bin 0 -> 32692 bytes buch/chapters/040-rekursion/images/order.tex | 125 +++++++++++ buch/chapters/070-orthogonalitaet/Makefile.inc | 1 + buch/chapters/070-orthogonalitaet/chapter.tex | 2 +- .../070-orthogonalitaet/uebungsaufgaben/701.tex | 137 ++++++++++++ buch/chapters/090-pde/Makefile.inc | 1 + buch/chapters/090-pde/chapter.tex | 12 +- buch/chapters/090-pde/kreis.tex | 2 +- buch/chapters/090-pde/uebungsaufgaben/901.tex | 82 +++++++ buch/papers/dreieck/images/beta.pdf | Bin 100791 -> 109717 bytes buch/papers/dreieck/images/beta.tex | 208 ++++++++++-------- buch/papers/dreieck/images/betadist.m | 24 ++- 20 files changed, 938 insertions(+), 194 deletions(-) create mode 100644 buch/chapters/040-rekursion/images/beta.pdf create mode 100644 buch/chapters/040-rekursion/images/beta.tex create mode 100644 buch/chapters/040-rekursion/images/betadist.m create mode 100644 buch/chapters/040-rekursion/images/order.m create mode 100644 buch/chapters/040-rekursion/images/order.pdf create mode 100644 buch/chapters/040-rekursion/images/order.tex create mode 100644 buch/chapters/070-orthogonalitaet/uebungsaufgaben/701.tex create mode 100644 buch/chapters/090-pde/uebungsaufgaben/901.tex (limited to 'buch') diff --git a/buch/aufgaben2.tex b/buch/aufgaben2.tex index bed14fb..f98562e 100644 --- a/buch/aufgaben2.tex +++ b/buch/aufgaben2.tex @@ -8,4 +8,8 @@ %\input chapters/40-eigenwerte/uebungsaufgaben/4004.tex %\item %\input chapters/40-eigenwerte/uebungsaufgaben/4005.tex +\item +\input{chapters/090-pde/uebungsaufgaben/901.tex} +\item +\input{chapters/070-orthogonalitaet/uebungsaufgaben/701.tex} diff --git a/buch/chapters/040-rekursion/Makefile.inc b/buch/chapters/040-rekursion/Makefile.inc index ed8fd51..a222b1c 100644 --- a/buch/chapters/040-rekursion/Makefile.inc +++ b/buch/chapters/040-rekursion/Makefile.inc @@ -9,6 +9,7 @@ CHAPTERFILES = $(CHAPTERFILES) \ chapters/040-rekursion/bohrmollerup.tex \ chapters/040-rekursion/integral.tex \ chapters/040-rekursion/beta.tex \ + chapters/040-rekursion/betaverteilung.tex \ chapters/040-rekursion/linear.tex \ chapters/040-rekursion/hypergeometrisch.tex \ chapters/040-rekursion/uebungsaufgaben/401.tex \ diff --git a/buch/chapters/040-rekursion/beta.tex b/buch/chapters/040-rekursion/beta.tex index ea847bc..ff59bad 100644 --- a/buch/chapters/040-rekursion/beta.tex +++ b/buch/chapters/040-rekursion/beta.tex @@ -3,11 +3,17 @@ % % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % -\subsection{Die Beta-Funktion -\label{buch:rekursion:gamma:subsection:beta}} +\section{Die Beta-Funktion +\label{buch:rekursion:gamma:section:beta}} Die Eulersche Integralformel für die Gamma-Funktion in -Definition~\ref{buch:rekursion:def:gamma} wurde bisher nicht -gerechtfertigt. +Definition~\ref{buch:rekursion:def:gamma} wurde in +Abschnitt~\ref{buch:subsection:integral-eindeutig} +mit dem Satz von Mollerup gerechtfertigt. +Man kann Sie aber auch als Grenzfall der Beta-Funktion verstehen, +die in diesem Abschnitt dargestellt wird. + + +\subsection{Beta-Integral} In diesem Abschnitt wird das Beta-Integral eingeführt, eine Funktion von zwei Variablen, welches eine Integral-Definition mit einer reichaltigen Menge von Rekursionsbeziehungen hat, die sich direkt auf @@ -233,6 +239,16 @@ B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} berechnet werden. \end{satz} +% +% Info über die Beta-Verteilung +% +\input{chapters/040-rekursion/betaverteilung.tex} + +\subsection{Weitere Eigenschaften der Gamma-Funktion} +Die nahe Verwandtschaft der Gamma- mit der Beta-Funktion ermöglicht +nun, weitere Eigenschaften der Gamma-Funktion mit Hilfe der Beta-Funktion +herzuleiten. + \subsubsection{Nochmals der Wert von $\Gamma(\frac12)$?} Der Wert von $\Gamma(\frac12)=\sqrt{\pi}$ wurde bereits in \eqref{buch:rekursion:gamma:wert12} @@ -484,83 +500,3 @@ Setzt man $x=\frac12$ in die Verdoppelungsformel ein, erhält man in Übereinstimmung mit dem aus \eqref{buch:rekursion:gamma:gamma12} bereits bekannten Wert. -\subsubsection{Beta-Funktion und Binomialkoeffizienten} -Die Binomialkoeffizienten können mit Hilfe der Fakultät als -\begin{align*} -\binom{n}{k} -&= -\frac{n!}{(n-k)!\,k!} -\intertext{geschrieben werden. -Drückt man die Fakultäten durch die Gamma-Funktion aus, erhält man} -&= -\frac{\Gamma(n+1)}{\Gamma(n-k+1)\Gamma(k+1)}. -\intertext{Schreibt man $x=k-1$ und $y=n-k+1$, wird daraus -wegen $x+y=k+1+n-k+1=n+2=(n+1)+1$} -&= -\frac{\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}. -\intertext{Die Rekursionsformel für die Gamma-Funktion erlaubt, -den Zähler umzuwandeln in $\Gamma(x+y-1)=\Gamma(x+y)/(x+y-1)$, so dass -der Binomialkoeffizient schliesslich} -&= -\frac{\Gamma(x+y)}{(x+y-1)\Gamma(x)\Gamma(y)} -= -\frac{1}{(n-1)B(n-k+1,k+1)} -\label{buch:rekursion:gamma:binombeta} -\end{align*} -geschrieben werden kann. -Die Rekursionsbeziehung -\[ -\binom{n+1}{k} = \binom{n}{k-1} + \binom{n}{k} -\] -der Binomialkoeffizienten erzeugt das vertraute Pascal-Dreieck, -die Formel \eqref{buch:rekursion:gamma:binombeta} für die -Binomialkoeffizienten macht daraus -\[ -\frac{n-1}{B(n-k,k-1)} -= -\frac{n-2}{B(n-k,k-2)} -+ -\frac{n-2}{B(n-k-1,k-1)}, -\] -die für ganzzahlige Argumente gilt. -Wir wollen nachrechnen, dass dies für beliebige Argumente gilt. -\begin{align*} -\frac{(n-1)\Gamma(n-1)}{\Gamma(n-k)\Gamma(k-1)} -&= -\frac{(n-2)\Gamma(n-2)}{\Gamma(n-k)\Gamma(k-2)} -+ -\frac{(n-2)\Gamma(n-2)}{\Gamma(n-k-1)\Gamma(k-1)} -\\ -\frac{\Gamma(n)}{\Gamma(n-k)\Gamma(k-1)} -&= -\frac{\Gamma(n-1)}{\Gamma(n-k)\Gamma(k-2)} -+ -\frac{\Gamma(n-1)}{\Gamma(n-k-1)\Gamma(k-1)} -\intertext{Durch Zusammenfassen der Faktoren im Zähler mit Hilfe -der Rekursionsformel für die Gamma-Funktion und Multiplizieren -mit dem gemeinsamen Nenner -$\Gamma(n-k)\Gamma(k-1)=(n-k-1)\Gamma(n-k-1)(k-2)\Gamma(k-2)$ wird daraus} -\Gamma(n) -&= -(k-2) -\Gamma(n-1) -+ -(n-k-1) -\Gamma(n-1) -\intertext{Indem wir die Rekursionsformel für die Gamma-Funktion auf -die rechte Seite anwenden können wir erreichen, dass in allen Termen -ein Faktor -$\Gamma(n-1)$ auftritt:} -(n-1)\Gamma(n-1) -&= -(k-2)\Gamma(n-1) -+ -(n+k-1)\Gamma(n-1) -\\ -n-1 -&= -k-2 -+ -n-k-1 -\end{align*} - diff --git a/buch/chapters/040-rekursion/images/Makefile b/buch/chapters/040-rekursion/images/Makefile index 9608a94..86dfa1e 100644 --- a/buch/chapters/040-rekursion/images/Makefile +++ b/buch/chapters/040-rekursion/images/Makefile @@ -3,7 +3,7 @@ # # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -all: gammaplot.pdf fibonacci.pdf +all: gammaplot.pdf fibonacci.pdf order.pdf beta.pdf gammaplot.pdf: gammaplot.tex gammapaths.tex pdflatex gammaplot.tex @@ -16,3 +16,17 @@ fibonaccigrid.tex: fibonacci.m fibonacci.pdf: fibonacci.tex fibonaccigrid.tex pdflatex fibonacci.tex + +order.pdf: order.tex orderpath.tex + pdflatex order.tex + +orderpath.tex: order.m + octave order.m + +beta.pdf: beta.tex betapaths.tex + pdflatex beta.tex + +betapaths.tex: betadist.m + octave betadist.m + + diff --git a/buch/chapters/040-rekursion/images/beta.pdf b/buch/chapters/040-rekursion/images/beta.pdf new file mode 100644 index 0000000..0e6567b Binary files /dev/null and b/buch/chapters/040-rekursion/images/beta.pdf differ diff --git a/buch/chapters/040-rekursion/images/beta.tex b/buch/chapters/040-rekursion/images/beta.tex new file mode 100644 index 0000000..1e1a1b3 --- /dev/null +++ b/buch/chapters/040-rekursion/images/beta.tex @@ -0,0 +1,236 @@ +% +% beta.tex -- display some symmetric beta distributions +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\input{betapaths.tex} +\begin{document} +\def\skala{12} +\definecolor{colorone}{rgb}{1.0,0.6,0.0} +\definecolor{colortwo}{rgb}{1.0,0.0,0.0} +\definecolor{colorthree}{rgb}{0.6,0.0,0.6} +\definecolor{colorfour}{rgb}{0.6,0.0,1.0} +\definecolor{colorfive}{rgb}{0.0,0.0,1.0} +\definecolor{colorsix}{rgb}{0.4,0.6,1.0} +\definecolor{colorseven}{rgb}{0.0,0.0,0.0} +\definecolor{coloreight}{rgb}{0.0,0.8,0.8} +\definecolor{colornine}{rgb}{0.0,0.8,0.2} +\definecolor{colorten}{rgb}{0.2,0.4,0.0} +\definecolor{coloreleven}{rgb}{0.6,1.0,0.0} +\definecolor{colortwelve}{rgb}{1.0,0.8,0.4} + +\def\achsen{ + \foreach \x in {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9}{ + \draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala}); + \node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$}; + } + \foreach \y in {1,2,3,4}{ + \draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy}); + \node at ({-0.1/\skala},{\y*\dy}) [left] {$\y$}; + } + \def\x{1} + \draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala}); + \node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$}; + \def\x{0} + \node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$}; + + \draw[->] ({-0.1/\skala},0) -- ({1*\dx+0.4/\skala},0) + coordinate[label={$x$}]; + \draw[->] (0,{-0.1/\skala}) -- (0,{\betamax*\dy+0.4/\skala},0) + coordinate[label={right:$\beta(a,b,x)$}]; +} + +\def\farbcoord#1#2{ + ({\dx*(0.63+((#1)/5)*0.27)},{\dx*(0.18+((#2)/5)*0.27)}) +} +\def\farbviereck{ + \foreach \x in {1,2,3,4}{ + \draw[color=gray!30] \farbcoord{\x}{0} -- \farbcoord{\x}{4}; + \draw[color=gray!30] \farbcoord{0}{\x} -- \farbcoord{4}{\x}; + } + \draw[->] \farbcoord{0}{0} -- \farbcoord{4.4}{0} + coordinate[label={$a$}]; + \draw[->] \farbcoord{0}{0} -- \farbcoord{0}{4.4} + coordinate[label={left: $b$}]; + \foreach \x in {1,2,3,4}{ + \node[color=gray] at \farbcoord{4}{\x} [right] {\tiny $b=\x$}; + %\fill[color=white,opacity=0.7] + % \farbcoord{(\x-0.1)}{3.3} + % rectangle + % \farbcoord{(\x+0.1)}{4}; + \node[color=gray] at \farbcoord{\x}{4} [right,rotate=90] + {\tiny $a=\x$}; + } +} +\def\farbpunkt#1#2#3{ + \fill[color=#3] \farbcoord{#1}{#2} circle[radius={0.1/\skala}]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\def\dx{1.15} +\def\dy{0.1} +\def\opa{0.1} + +\def\betamax{4.9} + +\begin{scope} +\clip (0,0) rectangle ({1*\dx},{\betamax*\dy}); +\fill[color=colorone,opacity=\opa] (0,0) -- \betaaa -- (\dx,0) -- cycle; +\fill[color=colortwo,opacity=\opa] (0,0) -- \betabb -- (\dx,0) -- cycle; +\fill[color=colorthree,opacity=\opa] (0,0) -- \betacc -- (\dx,0) -- cycle; +\fill[color=colorfour,opacity=\opa] (0,0) -- \betadd -- (\dx,0) -- cycle; +\fill[color=colorfive,opacity=\opa] (0,0) -- \betaee -- (\dx,0) -- cycle; +\fill[color=colorsix,opacity=\opa] (0,0) -- \betaff -- (\dx,0) -- cycle; +\fill[color=colorseven,opacity=\opa] (0,0) -- \betagg -- (\dx,0) -- cycle; +\fill[color=coloreight,opacity=\opa] (0,0) -- \betahh -- (\dx,0) -- cycle; +\fill[color=colornine,opacity=\opa] (0,0) -- \betaii -- (\dx,0) -- cycle; +\fill[color=colorten,opacity=\opa] (0,0) -- \betajj -- (\dx,0) -- cycle; +\fill[color=coloreleven,opacity=\opa] (0,0) -- \betakk -- (\dx,0) -- cycle; +\fill[color=colortwelve,opacity=\opa] (0,0) -- \betall -- (\dx,0) -- cycle; + +\draw[color=colorone] \betaaa; +\draw[color=colortwo] \betabb; +\draw[color=colorthree] \betacc; +\draw[color=colorfour] \betadd; +\draw[color=colorfive] \betaee; +\draw[color=colorsix] \betaff; +\draw[color=colorseven] \betagg; +\draw[color=coloreight] \betahh; +\draw[color=colornine] \betaii; +\draw[color=colorten] \betajj; +\draw[color=coloreleven] \betakk; +\draw[color=colortwelve] \betall; + +\end{scope} + +\achsen + +\farbviereck + +\farbpunkt{\alphatwelve}{\betatwelve}{colortwelve} +\farbpunkt{\alphaeleven}{\betaeleven}{coloreleven} +\farbpunkt{\alphaten}{\betaten}{colorten} +\farbpunkt{\alphanine}{\betanine}{colornine} +\farbpunkt{\alphaeight}{\betaeight}{coloreight} +\farbpunkt{\alphaseven}{\betaseven}{colorseven} +\farbpunkt{\alphasix}{\betasix}{colorsix} +\farbpunkt{\alphafive}{\betafive}{colorfive} +\farbpunkt{\alphafour}{\betafour}{colorfour} +\farbpunkt{\alphathree}{\betathree}{colorthree} +\farbpunkt{\alphatwo}{\betatwo}{colortwo} +\farbpunkt{\alphaone}{\betaone}{colorone} + + +\def\betamax{4.9} + +\begin{scope}[yshift=-0.6cm] + +\begin{scope} +\clip (0,0) rectangle ({1*\dx},{\betamax*\dy}); +\fill[color=colorone,opacity=\opa] (0,0) -- \betaea -- (\dx,0) -- cycle; +\fill[color=colortwo,opacity=\opa] (0,0) -- \betaeb -- (\dx,0) -- cycle; +\fill[color=colorthree,opacity=\opa] (0,0) -- \betaec -- (\dx,0) -- cycle; +\fill[color=colorfour,opacity=\opa] (0,0) -- \betaed -- (\dx,0) -- cycle; +\fill[color=colorfive,opacity=\opa] (0,0) -- \betaee -- (\dx,0) -- cycle; +\fill[color=colorsix,opacity=\opa] (0,0) -- \betaef -- (\dx,0) -- cycle; +\fill[color=colorseven,opacity=\opa] (0,0) -- \betaeg -- (\dx,0) -- cycle; +\fill[color=coloreight,opacity=\opa] (0,0) -- \betaeh -- (\dx,0) -- cycle; +\fill[color=colornine,opacity=\opa] (0,0) -- \betaei -- (\dx,0) -- cycle; +\fill[color=colorten,opacity=\opa] (0,0) -- \betaej -- (\dx,0) -- cycle; +\fill[color=coloreleven,opacity=\opa] (0,0) -- \betaek -- (\dx,0) -- cycle; +\fill[color=colortwelve,opacity=\opa] (0,0) -- \betael -- (\dx,0) -- cycle; + +\draw[color=colorone] \betaea; +\draw[color=colortwo] \betaeb; +\draw[color=colorthree] \betaec; +\draw[color=colorfour] \betaed; +\draw[color=colorfive] \betaee; +\draw[color=colorsix] \betaef; +\draw[color=colorseven] \betaeg; +\draw[color=coloreight] \betaeh; +\draw[color=colornine] \betaei; +\draw[color=colorten] \betaej; +\draw[color=coloreleven] \betaek; +\draw[color=colortwelve] \betael; +\end{scope} + +\achsen + +\farbviereck + +\farbpunkt{\alphafive}{\betatwelve}{colortwelve} +\farbpunkt{\alphafive}{\betaeleven}{coloreleven} +\farbpunkt{\alphafive}{\betaten}{colorten} +\farbpunkt{\alphafive}{\betanine}{colornine} +\farbpunkt{\alphafive}{\betaeight}{coloreight} +\farbpunkt{\alphafive}{\betaseven}{colorseven} +\farbpunkt{\alphafive}{\betasix}{colorsix} +\farbpunkt{\alphafive}{\betafive}{colorfive} +\farbpunkt{\alphafive}{\betafour}{colorfour} +\farbpunkt{\alphafive}{\betathree}{colorthree} +\farbpunkt{\alphafive}{\betatwo}{colortwo} +\farbpunkt{\alphafive}{\betaone}{colorone} + +\end{scope} + +\begin{scope}[yshift=-1.2cm] + +\begin{scope} +\clip (0,0) rectangle ({1*\dx},{\betamax*\dy}); +\fill[color=colorone,opacity=\opa] (0,0) -- \betaal -- (\dx,0) -- cycle; +\fill[color=colortwo,opacity=\opa] (0,0) -- \betabl -- (\dx,0) -- cycle; +\fill[color=colorthree,opacity=\opa] (0,0) -- \betacl -- (\dx,0) -- cycle; +\fill[color=colorfour,opacity=\opa] (0,0) -- \betadl -- (\dx,0) -- cycle; +\fill[color=colorfive,opacity=\opa] (0,0) -- \betael -- (\dx,0) -- cycle; +\fill[color=colorsix,opacity=\opa] (0,0) -- \betafl -- (\dx,0) -- cycle; +\fill[color=colorseven,opacity=\opa] (0,0) -- \betagl -- (\dx,0) -- cycle; +\fill[color=coloreight,opacity=\opa] (0,0) -- \betahl -- (\dx,0) -- cycle; +\fill[color=colornine,opacity=\opa] (0,0) -- \betail -- (\dx,0) -- cycle; +\fill[color=colorten,opacity=\opa] (0,0) -- \betajl -- (\dx,0) -- cycle; +\fill[color=coloreleven,opacity=\opa] (0,0) -- \betakl -- (\dx,0) -- cycle; +\fill[color=colortwelve,opacity=\opa] (0,0) -- \betall -- (\dx,0) -- cycle; + +\draw[color=colorone] \betaal; +\draw[color=colortwo] \betabl; +\draw[color=colorthree] \betacl; +\draw[color=colorfour] \betadl; +\draw[color=colorfive] \betael; +\draw[color=colorsix] \betafl; +\draw[color=colorseven] \betagl; +\draw[color=coloreight] \betahl; +\draw[color=colornine] \betail; +\draw[color=colorten] \betajl; +\draw[color=coloreleven] \betakl; +\draw[color=colortwelve] \betall; +\end{scope} + +\achsen + +\farbviereck + +\farbpunkt{\alphatwelve}{\betatwelve}{colortwelve} +\farbpunkt{\alphaeleven}{\betatwelve}{coloreleven} +\farbpunkt{\alphaten}{\betatwelve}{colorten} +\farbpunkt{\alphanine}{\betatwelve}{colornine} +\farbpunkt{\alphaeight}{\betatwelve}{coloreight} +\farbpunkt{\alphaseven}{\betatwelve}{colorseven} +\farbpunkt{\alphasix}{\betatwelve}{colorsix} +\farbpunkt{\alphafive}{\betatwelve}{colorfive} +\farbpunkt{\alphafour}{\betatwelve}{colorfour} +\farbpunkt{\alphathree}{\betatwelve}{colorthree} +\farbpunkt{\alphatwo}{\betatwelve}{colortwo} +\farbpunkt{\alphaone}{\betatwelve}{colorone} + +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/040-rekursion/images/betadist.m b/buch/chapters/040-rekursion/images/betadist.m new file mode 100644 index 0000000..5b466a6 --- /dev/null +++ b/buch/chapters/040-rekursion/images/betadist.m @@ -0,0 +1,58 @@ +# +# betadist.m +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global N; +N = 201; +global nmin; +global nmax; +nmin = -4; +nmax = 7; +n = nmax - nmin + 1 +A = 3; + +t = (nmin:nmax) / nmax; +alpha = 1 + A * t .* abs(t) +#alpha(1) = 0.01; + +#alpha = [ 1, 1.03, 1.05, 1.1, 1.25, 1.5, 2, 2.5, 3, 4, 5 ]; +beta = alpha; +names = [ "one"; "two"; "three"; "four"; "five"; "six"; "seven"; "eight"; + "nine"; "ten"; "eleven"; "twelve" ] + +function retval = Beta(a, b, x) + retval = x^(a-1) * (1-x)^(b-1) / beta(a, b); + if (retval > 100) + retval = 100 + end +end + +function plotbeta(fn, a, b, name) + global N; + fprintf(fn, "\\def\\beta%s{\n", strtrim(name)); + fprintf(fn, "\t({%.4f*\\dx},{%.4f*\\dy})", 0, Beta(a, b, 0)); + for x = (1:N-1)/(N-1) + X = (1-cos(pi * x))/2; + fprintf(fn, "\n\t--({%.4f*\\dx},{%.4f*\\dy})", + X, Beta(a, b, X)); + end + fprintf(fn, "\n}\n"); +end + +fn = fopen("betapaths.tex", "w"); + +for i = (1:n) + fprintf(fn, "\\def\\alpha%s{%f}\n", strtrim(names(i,:)), alpha(i)); + fprintf(fn, "\\def\\beta%s{%f}\n", strtrim(names(i,:)), beta(i)); +end + +for i = (1:n) + for j = (1:n) + printf("working on %d,%d:\n", i, j); + plotbeta(fn, alpha(i), beta(j), + char(['a' + i - 1, 'a' + j - 1])); + end +end + +fclose(fn); diff --git a/buch/chapters/040-rekursion/images/order.m b/buch/chapters/040-rekursion/images/order.m new file mode 100644 index 0000000..762f458 --- /dev/null +++ b/buch/chapters/040-rekursion/images/order.m @@ -0,0 +1,119 @@ +# +# order.m +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global N; +N = 10; +global subdivisions; +subdivisions = 100; +global P; +P = 0.5 + +function retval = orderF(p, n, k) + retval = 0; + for i = (k:n) + retval = retval + nchoosek(n,i) * p^i * (1-p)^(n-i); + end +end + +function retval = orderd(p, n, k) + retval = 0; + for i = (k:n) + s = i * p^(i-1) * (1-p)^(n-i); + s = s - p^i * (n-i) * (1-p)^(n-i-1); + retval = retval + nchoosek(n,i) * s; + end +end + +function retval = orders(p, n, k) + retval = k * nchoosek(n, k) * p^(k-1) * (1-p)^(n-k); +end + +function orderpath(fn, k, name) + fprintf(fn, "\\def\\order%s{\n\t(0,0)", name); + global N; + global subdivisions; + for i = (0:subdivisions) + p = i/subdivisions; + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", + p, orderF(p, N, k)); + end + fprintf(fn, "\n}\n"); +end + +function orderdpath(fn, k, name) + fprintf(fn, "\\def\\orderd%s{\n\t(0,0)", name); + global N; + global subdivisions; + for i = (1:subdivisions-1) + p = i/subdivisions; + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", + p, orderd(p, N, k)); + end + fprintf(fn, "\n\t-- ({1*\\dx},0)"); + fprintf(fn, "\n}\n"); +end + +function orderspath(fn, k, name) + fprintf(fn, "\\def\\orders%s{\n\t(0,0)", name); + global N; + global subdivisions; + for i = (1:subdivisions-1) + p = i/subdivisions; + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", + p, orders(p, N, k)); + end + fprintf(fn, "\n\t-- ({1*\\dx},0)"); + fprintf(fn, "\n}\n"); +end + +fn = fopen("orderpath.tex", "w"); + +orderpath(fn, 0, "zero"); +orderdpath(fn, 0, "zero"); +orderspath(fn, 0, "zero"); + +orderpath(fn, 1, "one"); +orderdpath(fn, 1, "one"); +orderspath(fn, 1, "one"); + +orderpath(fn, 2, "two"); +orderdpath(fn, 2, "two"); +orderspath(fn, 2, "two"); + +orderpath(fn, 3, "three"); +orderdpath(fn, 3, "three"); +orderspath(fn, 3, "three"); + +orderpath(fn, 4, "four"); +orderdpath(fn, 4, "four"); +orderspath(fn, 4, "four"); + +orderpath(fn, 5, "five"); +orderdpath(fn, 5, "five"); +orderspath(fn, 5, "five"); + +orderpath(fn, 6, "six"); +orderdpath(fn, 6, "six"); +orderspath(fn, 6, "six"); + +orderpath(fn, 7, "seven"); +orderdpath(fn, 7, "seven"); +orderspath(fn, 7, "seven"); + +orderpath(fn, 8, "eight"); +orderdpath(fn, 8, "eight"); +orderspath(fn, 8, "eight"); + +orderpath(fn, 9, "nine"); +orderdpath(fn, 9, "nine"); +orderspath(fn, 9, "nine"); + +orderpath(fn, 10, "ten"); +orderdpath(fn, 10, "ten"); +orderspath(fn, 10, "ten"); + +fclose(fn); + + diff --git a/buch/chapters/040-rekursion/images/order.pdf b/buch/chapters/040-rekursion/images/order.pdf new file mode 100644 index 0000000..cc175a9 Binary files /dev/null and b/buch/chapters/040-rekursion/images/order.pdf differ diff --git a/buch/chapters/040-rekursion/images/order.tex b/buch/chapters/040-rekursion/images/order.tex new file mode 100644 index 0000000..9a2511c --- /dev/null +++ b/buch/chapters/040-rekursion/images/order.tex @@ -0,0 +1,125 @@ +% +% order.tex -- Verteilungsfunktion für Ordnungsstatistik +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{8} +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\n{10} +\def\E#1#2{ + \draw[color=#2] + ({\dx*#1/(\n+1)},{-0.1/\skala}) -- ({\dx*#1/(\n+1)},{4.4*\dy}); + \node[color=#2] at ({\dx*#1/(\n+1)},{3.2*\dy}) + [rotate=90,above right] {$k=#1$}; +} +\def\var#1#2{ + \pgfmathparse{\dx*sqrt(#1*(\n-#1+1)/((\n+1)*(\n+1)*(\n+2)))} + \xdef\var{\pgfmathresult} + \fill[color=#2,opacity=0.5] + ({\dx*#1/(\n+1)-\var},0) rectangle ({\dx*#1/(\n+1)+\var},{4.4*\dy}); +} + +\input{orderpath.tex} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\def\dx{1.6} +\def\dy{0.5} + +\def\pfad#1#2{ +\draw[color=#2,line width=1.4pt] ({-0.1/\skala},0) + -- + #1 + -- + ({1*\dx+0.1/\skala},0.5); +} + +\pfad{\orderzero}{darkgreen!20} +\pfad{\orderone}{darkgreen!20} +\pfad{\ordertwo}{darkgreen!20} +\pfad{\orderthree}{darkgreen!20} +\pfad{\orderfour}{darkgreen!20} +\pfad{\orderfive}{darkgreen!20} +\pfad{\ordersix}{darkgreen!20} +\pfad{\ordereight}{darkgreen!20} +\pfad{\ordernine}{darkgreen!20} +\pfad{\orderten}{darkgreen!20} +\pfad{\orderseven}{darkgreen} + +\draw[->] ({-0.1/\skala},0) -- ({1.03*\dx},0) coordinate[label={$x$}]; +\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={right:$F(X)$}]; +\foreach \x in {0,0.2,0.4,0.6,0.8,1}{ + \draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala}); + \node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$}; +} +\foreach \y in {0.5,1}{ + \draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy}); + \node at ({-0.1/\skala},{\y*\dy}) [left] {$\y$}; +} + +\node[color=darkgreen] at (0.65,{0.5*\dy}) [above,rotate=55] {$k=7$}; + +\begin{scope}[yshift=-0.7cm] +\def\dy{0.125} + +\foreach \k in {1,2,3,4,5,6,8,9,10}{ + \E{\k}{blue!30} +} +\def\k{7} +\var{\k}{orange!40} +\node[color=blue] at ({\dx*\k/(\n+1)},{4.3*\dy}) [above] {$E(X_{7:n})$}; + +\def\pfad#1#2{ + \draw[color=#2,line width=1.4pt] ({-0.1/\skala},0) + -- + #1 + -- + ({1*\dx+0.1/\skala},0.0); +} + +\begin{scope} +\clip ({-0.1/\skala},{-0.1/\skala}) + rectangle ({1*\dx+0.1/\skala},{0.56+0.1/\skala}); + +\pfad{\orderdzero}{red!20} +\pfad{\orderdone}{red!20} +\pfad{\orderdtwo}{red!20} +\pfad{\orderdthree}{red!20} +\pfad{\orderdfour}{red!20} +\pfad{\orderdfive}{red!20} +\pfad{\orderdsix}{red!20} +\pfad{\orderdeight}{red!20} +\pfad{\orderdnine}{red!20} +\pfad{\orderdten}{red!20} +\E{\k}{blue} +\pfad{\orderdseven}{red} + +\end{scope} + +\draw[->] ({-0.1/\skala},0) -- ({1.03*\dx},0) coordinate[label={$x$}]; +\draw[->] (0,{-0.1/\skala}) -- (0,0.6) coordinate[label={right:$\varphi(X)$}]; +\foreach \x in {0,0.2,0.4,0.6,0.8,1}{ + \draw ({\x*\dx},{-0.1/\skala}) -- ({\x*\dx},{0.1/\skala}); + \node at ({\x*\dx},{-0.1/\skala}) [below] {$\x$}; +} +\foreach \y in {1,2,3,4}{ + \draw ({-0.1/\skala},{\y*\dy}) -- ({0.1/\skala},{\y*\dy}); + \node at ({-0.1/\skala},{\y*\dy}) [left] {$\y$}; +} + +\node[color=red] at ({0.67*\dx},{2.7*\dy}) [above] {$k=7$}; + + +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/070-orthogonalitaet/Makefile.inc b/buch/chapters/070-orthogonalitaet/Makefile.inc index 48e5356..286ab2e 100644 --- a/buch/chapters/070-orthogonalitaet/Makefile.inc +++ b/buch/chapters/070-orthogonalitaet/Makefile.inc @@ -13,4 +13,5 @@ CHAPTERFILES = $(CHAPTERFILES) \ chapters/070-orthogonalitaet/jacobi.tex \ chapters/070-orthogonalitaet/sturm.tex \ chapters/070-orthogonalitaet/gaussquadratur.tex \ + chapters/070-orthogonalitaet/uebungsaufgaben/701.tex \ chapters/070-orthogonalitaet/chapter.tex diff --git a/buch/chapters/070-orthogonalitaet/chapter.tex b/buch/chapters/070-orthogonalitaet/chapter.tex index 5ebb795..4756844 100644 --- a/buch/chapters/070-orthogonalitaet/chapter.tex +++ b/buch/chapters/070-orthogonalitaet/chapter.tex @@ -25,7 +25,7 @@ \rhead{Übungsaufgaben} \aufgabetoplevel{chapters/070-orthogonalitaet/uebungsaufgaben} \begin{uebungsaufgaben} -%\uebungsaufgabe{0} +\uebungsaufgabe{701} %\uebungsaufgabe{1} \end{uebungsaufgaben} diff --git a/buch/chapters/070-orthogonalitaet/uebungsaufgaben/701.tex b/buch/chapters/070-orthogonalitaet/uebungsaufgaben/701.tex new file mode 100644 index 0000000..dad489f --- /dev/null +++ b/buch/chapters/070-orthogonalitaet/uebungsaufgaben/701.tex @@ -0,0 +1,137 @@ +Für Funktionen auf dem Interval $(-\frac{\pi}2,\frac{\pi}2)$ ist +\[ +\langle f,g\rangle += +\frac12\int_{-\frac{\pi}2}^{\frac{\pi}2} f(x)g(x)\cos x\,dx +\] +ein Skalarprodukt. +Bestimmen Sie bezüglich dieses Skalarproduktes orthogonale Polynome +bis zum Grad $2$. + +\begin{hinweis} +Verwenden Sie +\begin{align*} +\int_{-\frac{\pi}2}^{\frac{\pi}2} 1\cos x\,dx +&= +1, +& +\int_{-\frac{\pi}2}^{\frac{\pi}2} x^2\cos x\,dx +&= +\frac{\pi^2-8}{2}, +& +\int_{-\frac{\pi}2}^{\frac{\pi}2} x^4\cos x\,dx +&= +\frac{\pi^4-48\pi^2+384}{8}. +\end{align*} +\end{hinweis} + +\begin{loesung} +Wir müssen den Gram-Schmidt-Orthogonalisierungsprozess für die +Polynome $f_0(x)=1$, $f_1(x)=x$ und $f_2(x)=x^2$ durchführen. +Zunächst halten wir fest, dass +\[ +\langle f_0,f_0\rangle += +\frac12 +\int_{-\frac{\pi}2}^{\frac{\pi}2} \cos x\,dx += +1, +\] +das Polynom $g_0(x)=f_0(x)$ ist hat also Norm $1$. + +Ein dazu orthogonales Polynom ist +\( +f_1(x) - \langle g_0,f_1\rangle g_0(x), +\) +wir müssen also das Skalarprodukt +\[ +\langle g_0,f_1\rangle += +\frac{1}{2} +\int_{-\frac{\pi}2}^{\frac{\pi}2} +x\cos x\,dx +\] +bestimmen. +Es verschwindet, weil die Funktion $x\cos x$ ungerade ist. +Somit ist die Funktion $f_1(x)=x$ orthogonal zu $f_0(x)=1$, um sie auch zu +normieren berechnen wir das Integral +\[ +\| f_1\|^2 += +\frac12\int_{-\frac{\pi}2}^{\frac{\pi}2} x^2\cos x\,dx += +\frac{\pi^2-8}{4}, +\] +und +\[ +g_1(x) += +\frac{2}{\sqrt{\pi^2-8}} x. +\] + +Zur Berechnung von $g_2$ müssen wir die Skalarprodukte +\begin{align*} +\langle g_0,f_2\rangle +&= +\frac{1}{2} +\int_{-\frac{\pi}2}^{\frac{\pi}2} +x^2 +\cos x +\,dx += +\frac{\pi^2-8}{4} +\\ +\langle g_1,f_2\rangle +&= +\frac{1}{2} +\int_{-\frac{\pi}2}^{\frac{\pi}2} +\frac{2}{\sqrt{\pi^2-8}} +x +\cdot x^2 +\cos x +\,dx += +0 +\end{align*} +bestimmen. +Damit wird das dritte Polynom +\[ +f_2(x) +- g_0(x)\langle g_0,f_2\rangle +- g_1(x)\langle g_1,f_2\rangle += +x^2 - \frac{\pi^2-8}{4}, +\] +welches bereits orthogonal ist zu $g_0$ und $g_1$. +Wir können auch noch erreichen, obwohl das nicht verlangt war, +dass es normiert ist, indem wir die Norm berechnen: +\[ +\left\| x^2-\frac{\pi^2-8}{4} \right\|^2 += +\frac12 +\int_{-\frac{\pi}2}^{\frac{\pi}2} +\biggl(x^2-\frac{\pi^2-8}{4}\biggr)^2 +\cos x\,dx += +20-2\pi^2 +\] +woraus sich +\[ +g_2(x) += +\frac{1}{\sqrt{20-2\pi^2}} +\biggl( +x^2 - \frac{\pi^2-8}{4} +\biggr). +\] +Damit haben wir die ersten drei bezüglich des obigen Skalarproduktes +orthogonalen Polynome +\begin{align*} +g_0(x)&=1, +& +g_1(x)&=\frac{2x}{\sqrt{\pi^2-8}}, +& +g_2(x)&=\frac{1}{\sqrt{20-2\pi^2}}\biggl(x^2-\frac{\pi^2-8}{4}\biggr) +\end{align*} +gefunden. +\end{loesung} diff --git a/buch/chapters/090-pde/Makefile.inc b/buch/chapters/090-pde/Makefile.inc index a9ef74a..c64af06 100644 --- a/buch/chapters/090-pde/Makefile.inc +++ b/buch/chapters/090-pde/Makefile.inc @@ -10,4 +10,5 @@ CHAPTERFILES = $(CHAPTERFILES) \ chapters/090-pde/rechteck.tex \ chapters/090-pde/kreis.tex \ chapters/090-pde/kugel.tex \ + chapters/090-pde/uebungsaufgaben/901.tex \ chapters/090-pde/chapter.tex diff --git a/buch/chapters/090-pde/chapter.tex b/buch/chapters/090-pde/chapter.tex index db909ee..a393da5 100644 --- a/buch/chapters/090-pde/chapter.tex +++ b/buch/chapters/090-pde/chapter.tex @@ -21,11 +21,11 @@ deren Lösungen spezielle Funktionen sind. \input{chapters/090-pde/kreis.tex} \input{chapters/090-pde/kugel.tex} -%\section*{Übungsaufgaben} -%\rhead{Übungsaufgaben} -%\aufgabetoplevel{chapters/020-exponential/uebungsaufgaben} -%\begin{uebungsaufgaben} -%\uebungsaufgabe{0} +\section*{Übungsaufgaben} +\rhead{Übungsaufgaben} +\aufgabetoplevel{chapters/090-pde/uebungsaufgaben} +\begin{uebungsaufgaben} +\uebungsaufgabe{901} %\uebungsaufgabe{1} -%\end{uebungsaufgaben} +\end{uebungsaufgaben} diff --git a/buch/chapters/090-pde/kreis.tex b/buch/chapters/090-pde/kreis.tex index a24b6bb..b4ce8d7 100644 --- a/buch/chapters/090-pde/kreis.tex +++ b/buch/chapters/090-pde/kreis.tex @@ -120,7 +120,7 @@ für $\Phi(\varphi)$. Die Gleichung für $\Phi$ hat für $\mu\ne 0$ die Lösungen \begin{align*} \Phi(\varphi) &= \cos\mu\varphi -\text{und}\qquad +&&\text{und}& \Phi(\varphi) &= \sin\mu\varphi. \end{align*} Die Lösung muss aber auch stetig sein, d.~h.~es muss $\Phi(0)=\Phi(2\pi)$ diff --git a/buch/chapters/090-pde/uebungsaufgaben/901.tex b/buch/chapters/090-pde/uebungsaufgaben/901.tex new file mode 100644 index 0000000..67fa8e5 --- /dev/null +++ b/buch/chapters/090-pde/uebungsaufgaben/901.tex @@ -0,0 +1,82 @@ +Die Differentialgleichung +\begin{equation} +\frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\partial x^2} +\qquad +\text{im Gebiet} +\qquad +(t,x)\in \Omega=\mathbb{R}^+\times (0,l) +\label{505:waermeleitungsgleichung} +\end{equation} +beschreibt die Änderung der Temperatur eines Stabes der Länge $l$. +Die homogene Randbedingung +\begin{equation} +u(t,0)= +u(t,l)=0 +\label{505:homogene-randbedingung} +\end{equation} +besagt, dass der Stab an seinen Enden auf Temperatur $0$ gehalten. +Zur Lösung dieser Differentialgleichung muss auch die Temperatur +zur Zeit $t=0$ in Form einer Randbedingung +\[ +u(0,x) = T_0(x) +\] +gegeben sein. +Führen Sie Separation für die +Differentialgleichung~\eqref{505:waermeleitungsgleichung} +durch und bestimmen Sie die zulässigen Werte der Separationskonstanten. + +\begin{loesung} +Man verwendet den Ansatz $u(t,x)= T(t)\cdot X(x)$ und setzt diesen +in die Differentialgleichung ein, die dadurch zu +\[ +T'(t)X(x) = \kappa T(t) X''(x) +\] +wird. +Division durch $T(t)X(x)$ wird dies zu +\[ +\frac{T'(t)}{T(t)} += +\kappa +\frac{X''(x)}{X(x)}. +\] +Da die linke Seite nur von $t$ abhängt, die rechte aber nur von $x$, müssen +beide Seiten konstant sein. +Wir bezeichnen die Konstante mit $-\lambda^2$, so dass wir die beiden +gewöhnlichen Differentialgleichungen +\begin{align*} +\frac{1}{\kappa} +\frac{T'(t)}{T(t)}&=-\lambda^2 +& +\frac{X''(x)}{X(x)}&=-\lambda^2 +\\ +T'(t)&=-\lambda^2\kappa T(t) +& +X''(x) &= -\lambda^2 X(x) +\intertext{welche die Lösungen} +T(t)&=Ce^{-\lambda^2\kappa t} +& +X(x)&= A\cos\lambda x + B\sin\lambda x +\end{align*} +haben. +Die Lösung $X(x)$ muss aber auch die homogene Randbedingung +\eqref{505:homogene-randbedingung} erfüllen. +Setzt man $x=0$ und $x=l$ ein, folgt +\begin{align*} +0 = X(0)&=A\cos 0 + B\sin 0 = A +& +0 = X(l)&=B\sin \lambda l, +\end{align*} +woraus man schliessen kann, dass $\lambda l$ ein ganzzahliges +Vielfaches von $\pi$ ist, wir schreiben $\lambda l = k\pi$ oder +\[ +\lambda = \frac{k\pi}{l}. +\] +Damit sind die möglichen Werte $\lambda$ bestimmt und man kann jetzt +auch die möglichen Lösungen aufschreiben, sie sind +\[ +u(t,x) += +\sum_{k=1}^\infty b_k e^{-k^2\pi^2\kappa t/l^2}\sin\frac{k\pi x}{l}. +\qedhere +\] +\end{loesung} diff --git a/buch/papers/dreieck/images/beta.pdf b/buch/papers/dreieck/images/beta.pdf index c3ab4f6..cd5ed80 100644 Binary files a/buch/papers/dreieck/images/beta.pdf and b/buch/papers/dreieck/images/beta.pdf differ diff --git a/buch/papers/dreieck/images/beta.tex b/buch/papers/dreieck/images/beta.tex index 50509ee..f0ffdf0 100644 --- a/buch/papers/dreieck/images/beta.tex +++ b/buch/papers/dreieck/images/beta.tex @@ -23,7 +23,8 @@ \definecolor{coloreight}{rgb}{0.0,0.8,0.8} \definecolor{colornine}{rgb}{0.0,0.8,0.2} \definecolor{colorten}{rgb}{0.2,0.4,0.0} -\definecolor{coloreleven}{rgb}{1.0,0.8,0.4} +\definecolor{coloreleven}{rgb}{0.6,1.0,0.0} +\definecolor{colortwelve}{rgb}{1.0,0.8,0.4} \def\achsen{ \foreach \x in {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9}{ @@ -47,24 +48,24 @@ } \def\farbcoord#1#2{ - ({\dx*(0.7+((#1-1)/4)*0.27)},{\dx*(0.15+((#2-1)/4)*0.27)}) + ({\dx*(0.63+((#1)/5)*0.27)},{\dx*(0.18+((#2)/5)*0.27)}) } \def\farbviereck{ - \foreach \x in {1,2,3,4,5}{ - \draw[color=gray!30] \farbcoord{\x}{1} -- \farbcoord{\x}{5}; - \draw[color=gray!30] \farbcoord{1}{\x} -- \farbcoord{5}{\x}; + \foreach \x in {1,2,3,4}{ + \draw[color=gray!30] \farbcoord{\x}{0} -- \farbcoord{\x}{4}; + \draw[color=gray!30] \farbcoord{0}{\x} -- \farbcoord{4}{\x}; } - \draw[->] \farbcoord{1}{1} -- \farbcoord{5.4}{1} + \draw[->] \farbcoord{0}{0} -- \farbcoord{4.4}{0} coordinate[label={$a$}]; - \draw[->] \farbcoord{1}{1} -- \farbcoord{1}{5.4} + \draw[->] \farbcoord{0}{0} -- \farbcoord{0}{4.4} coordinate[label={left: $b$}]; - \foreach \x in {1,2,3,4,5}{ - \node[color=gray] at \farbcoord{5}{\x} [right] {\tiny $b=\x$}; - \fill[color=white,opacity=0.7] - \farbcoord{(\x-0.1)}{4.3} - rectangle - \farbcoord{(\x+0.1)}{5}; - \node[color=gray] at \farbcoord{\x}{5} [left,rotate=90] + \foreach \x in {1,2,3,4}{ + \node[color=gray] at \farbcoord{4}{\x} [right] {\tiny $b=\x$}; + %\fill[color=white,opacity=0.7] + % \farbcoord{(\x-0.1)}{3.3} + % rectangle + % \farbcoord{(\x+0.1)}{4}; + \node[color=gray] at \farbcoord{\x}{4} [right,rotate=90] {\tiny $a=\x$}; } } @@ -74,23 +75,26 @@ \begin{tikzpicture}[>=latex,thick,scale=\skala] -\def\dx{1} +\def\dx{1.1} \def\dy{0.1} \def\opa{0.1} -\def\betamax{4.2} - -\fill[color=colorone,opacity=\opa] (0,0) -- \betaaa -- (\dx,0) -- cycle; -\fill[color=colortwo,opacity=\opa] (0,0) -- \betabb -- (\dx,0) -- cycle; -\fill[color=colorthree,opacity=\opa] (0,0) -- \betacc -- (\dx,0) -- cycle; -\fill[color=colorfour,opacity=\opa] (0,0) -- \betadd -- (\dx,0) -- cycle; -\fill[color=colorfive,opacity=\opa] (0,0) -- \betaee -- (\dx,0) -- cycle; -\fill[color=colorsix,opacity=\opa] (0,0) -- \betaff -- (\dx,0) -- cycle; -\fill[color=colorseven,opacity=\opa] (0,0) -- \betagg -- (\dx,0) -- cycle; -\fill[color=coloreight,opacity=\opa] (0,0) -- \betahh -- (\dx,0) -- cycle; -\fill[color=colornine,opacity=\opa] (0,0) -- \betaii -- (\dx,0) -- cycle; -\fill[color=colorten,opacity=\opa] (0,0) -- \betajj -- (\dx,0) -- cycle; +\def\betamax{4.9} + +\begin{scope} +\clip (0,0) rectangle ({1*\dx},{\betamax*\dy}); +\fill[color=colorone,opacity=\opa] (0,0) -- \betaaa -- (\dx,0) -- cycle; +\fill[color=colortwo,opacity=\opa] (0,0) -- \betabb -- (\dx,0) -- cycle; +\fill[color=colorthree,opacity=\opa] (0,0) -- \betacc -- (\dx,0) -- cycle; +\fill[color=colorfour,opacity=\opa] (0,0) -- \betadd -- (\dx,0) -- cycle; +\fill[color=colorfive,opacity=\opa] (0,0) -- \betaee -- (\dx,0) -- cycle; +\fill[color=colorsix,opacity=\opa] (0,0) -- \betaff -- (\dx,0) -- cycle; +\fill[color=colorseven,opacity=\opa] (0,0) -- \betagg -- (\dx,0) -- cycle; +\fill[color=coloreight,opacity=\opa] (0,0) -- \betahh -- (\dx,0) -- cycle; +\fill[color=colornine,opacity=\opa] (0,0) -- \betaii -- (\dx,0) -- cycle; +\fill[color=colorten,opacity=\opa] (0,0) -- \betajj -- (\dx,0) -- cycle; \fill[color=coloreleven,opacity=\opa] (0,0) -- \betakk -- (\dx,0) -- cycle; +\fill[color=colortwelve,opacity=\opa] (0,0) -- \betall -- (\dx,0) -- cycle; \draw[color=colorone] \betaaa; \draw[color=colortwo] \betabb; @@ -103,11 +107,15 @@ \draw[color=colornine] \betaii; \draw[color=colorten] \betajj; \draw[color=coloreleven] \betakk; +\draw[color=colortwelve] \betall; + +\end{scope} \achsen \farbviereck +\farbpunkt{\alphatwelve}{\betatwelve}{colortwelve} \farbpunkt{\alphaeleven}{\betaeleven}{coloreleven} \farbpunkt{\alphaten}{\betaten}{colorten} \farbpunkt{\alphanine}{\betanine}{colornine} @@ -124,88 +132,102 @@ \def\betamax{4.9} \begin{scope}[yshift=-0.6cm] -\fill[color=colorone,opacity=\opa] (0,0) -- \betaaa -- (\dx,0) -- cycle; -\fill[color=colortwo,opacity=\opa] (0,0) -- \betaab -- (\dx,0) -- cycle; -\fill[color=colorthree,opacity=\opa] (0,0) -- \betaac -- (\dx,0) -- cycle; -\fill[color=colorfour,opacity=\opa] (0,0) -- \betaad -- (\dx,0) -- cycle; -\fill[color=colorfive,opacity=\opa] (0,0) -- \betaae -- (\dx,0) -- cycle; -\fill[color=colorsix,opacity=\opa] (0,0) -- \betaaf -- (\dx,0) -- cycle; -\fill[color=colorseven,opacity=\opa] (0,0) -- \betaag -- (\dx,0) -- cycle; -\fill[color=coloreight,opacity=\opa] (0,0) -- \betaah -- (\dx,0) -- cycle; -\fill[color=colornine,opacity=\opa] (0,0) -- \betaai -- (\dx,0) -- cycle; -\fill[color=colorten,opacity=\opa] (0,0) -- \betaaj -- (\dx,0) -- cycle; -\fill[color=coloreleven,opacity=\opa] (0,0) -- \betaak -- (\dx,0) -- cycle; -\draw[color=colorone] \betaaa; -\draw[color=colortwo] \betaab; -\draw[color=colorthree] \betaac; -\draw[color=colorfour] \betaad; -\draw[color=colorfive] \betaae; -\draw[color=colorsix] \betaaf; -\draw[color=colorseven] \betaag; -\draw[color=coloreight] \betaah; -\draw[color=colornine] \betaai; -\draw[color=colorten] \betaaj; -\draw[color=coloreleven] \betaak; +\begin{scope} +\clip (0,0) rectangle ({1*\dx},{\betamax*\dy}); +\fill[color=colorone,opacity=\opa] (0,0) -- \betaea -- (\dx,0) -- cycle; +\fill[color=colortwo,opacity=\opa] (0,0) -- \betaeb -- (\dx,0) -- cycle; +\fill[color=colorthree,opacity=\opa] (0,0) -- \betaec -- (\dx,0) -- cycle; +\fill[color=colorfour,opacity=\opa] (0,0) -- \betaed -- (\dx,0) -- cycle; +\fill[color=colorfive,opacity=\opa] (0,0) -- \betaee -- (\dx,0) -- cycle; +\fill[color=colorsix,opacity=\opa] (0,0) -- \betaef -- (\dx,0) -- cycle; +\fill[color=colorseven,opacity=\opa] (0,0) -- \betaeg -- (\dx,0) -- cycle; +\fill[color=coloreight,opacity=\opa] (0,0) -- \betaeh -- (\dx,0) -- cycle; +\fill[color=colornine,opacity=\opa] (0,0) -- \betaei -- (\dx,0) -- cycle; +\fill[color=colorten,opacity=\opa] (0,0) -- \betaej -- (\dx,0) -- cycle; +\fill[color=coloreleven,opacity=\opa] (0,0) -- \betaek -- (\dx,0) -- cycle; +\fill[color=colortwelve,opacity=\opa] (0,0) -- \betael -- (\dx,0) -- cycle; + +\draw[color=colorone] \betaea; +\draw[color=colortwo] \betaeb; +\draw[color=colorthree] \betaec; +\draw[color=colorfour] \betaed; +\draw[color=colorfive] \betaee; +\draw[color=colorsix] \betaef; +\draw[color=colorseven] \betaeg; +\draw[color=coloreight] \betaeh; +\draw[color=colornine] \betaei; +\draw[color=colorten] \betaej; +\draw[color=coloreleven] \betaek; +\draw[color=colortwelve] \betael; +\end{scope} \achsen \farbviereck -\farbpunkt{\alphaone}{\betaeleven}{coloreleven} -\farbpunkt{\alphaone}{\betaten}{colorten} -\farbpunkt{\alphaone}{\betanine}{colornine} -\farbpunkt{\alphaone}{\betaeight}{coloreight} -\farbpunkt{\alphaone}{\betaseven}{colorseven} -\farbpunkt{\alphaone}{\betasix}{colorsix} -\farbpunkt{\alphaone}{\betafive}{colorfive} -\farbpunkt{\alphaone}{\betafour}{colorfour} -\farbpunkt{\alphaone}{\betathree}{colorthree} -\farbpunkt{\alphaone}{\betatwo}{colortwo} -\farbpunkt{\alphaone}{\betaone}{colorone} +\farbpunkt{\alphafive}{\betatwelve}{colortwelve} +\farbpunkt{\alphafive}{\betaeleven}{coloreleven} +\farbpunkt{\alphafive}{\betaten}{colorten} +\farbpunkt{\alphafive}{\betanine}{colornine} +\farbpunkt{\alphafive}{\betaeight}{coloreight} +\farbpunkt{\alphafive}{\betaseven}{colorseven} +\farbpunkt{\alphafive}{\betasix}{colorsix} +\farbpunkt{\alphafive}{\betafive}{colorfive} +\farbpunkt{\alphafive}{\betafour}{colorfour} +\farbpunkt{\alphafive}{\betathree}{colorthree} +\farbpunkt{\alphafive}{\betatwo}{colortwo} +\farbpunkt{\alphafive}{\betaone}{colorone} \end{scope} \begin{scope}[yshift=-1.2cm] -\fill[color=colorone,opacity=\opa] (0,0) -- \betaak -- (\dx,0) -- cycle; -\fill[color=colortwo,opacity=\opa] (0,0) -- \betabk -- (\dx,0) -- cycle; -\fill[color=colorthree,opacity=\opa] (0,0) -- \betack -- (\dx,0) -- cycle; -\fill[color=colorfour,opacity=\opa] (0,0) -- \betadk -- (\dx,0) -- cycle; -\fill[color=colorfive,opacity=\opa] (0,0) -- \betaek -- (\dx,0) -- cycle; -\fill[color=colorsix,opacity=\opa] (0,0) -- \betafk -- (\dx,0) -- cycle; -\fill[color=colorseven,opacity=\opa] (0,0) -- \betagk -- (\dx,0) -- cycle; -\fill[color=coloreight,opacity=\opa] (0,0) -- \betahk -- (\dx,0) -- cycle; -\fill[color=colornine,opacity=\opa] (0,0) -- \betaik -- (\dx,0) -- cycle; -\fill[color=colorten,opacity=\opa] (0,0) -- \betajk -- (\dx,0) -- cycle; -\fill[color=coloreleven,opacity=\opa] (0,0) -- \betakk -- (\dx,0) -- cycle; -\draw[color=colorone] \betaak; -\draw[color=colortwo] \betabk; -\draw[color=colorthree] \betack; -\draw[color=colorfour] \betadk; -\draw[color=colorfive] \betaek; -\draw[color=colorsix] \betafk; -\draw[color=colorseven] \betagk; -\draw[color=coloreight] \betahk; -\draw[color=colornine] \betaik; -\draw[color=colorten] \betajk; -\draw[color=coloreleven] \betakk; +\begin{scope} +\clip (0,0) rectangle ({1*\dx},{\betamax*\dy}); +\fill[color=colorone,opacity=\opa] (0,0) -- \betaal -- (\dx,0) -- cycle; +\fill[color=colortwo,opacity=\opa] (0,0) -- \betabl -- (\dx,0) -- cycle; +\fill[color=colorthree,opacity=\opa] (0,0) -- \betacl -- (\dx,0) -- cycle; +\fill[color=colorfour,opacity=\opa] (0,0) -- \betadl -- (\dx,0) -- cycle; +\fill[color=colorfive,opacity=\opa] (0,0) -- \betael -- (\dx,0) -- cycle; +\fill[color=colorsix,opacity=\opa] (0,0) -- \betafl -- (\dx,0) -- cycle; +\fill[color=colorseven,opacity=\opa] (0,0) -- \betagl -- (\dx,0) -- cycle; +\fill[color=coloreight,opacity=\opa] (0,0) -- \betahl -- (\dx,0) -- cycle; +\fill[color=colornine,opacity=\opa] (0,0) -- \betail -- (\dx,0) -- cycle; +\fill[color=colorten,opacity=\opa] (0,0) -- \betajl -- (\dx,0) -- cycle; +\fill[color=coloreleven,opacity=\opa] (0,0) -- \betakl -- (\dx,0) -- cycle; +\fill[color=colortwelve,opacity=\opa] (0,0) -- \betall -- (\dx,0) -- cycle; + +\draw[color=colorone] \betaal; +\draw[color=colortwo] \betabl; +\draw[color=colorthree] \betacl; +\draw[color=colorfour] \betadl; +\draw[color=colorfive] \betael; +\draw[color=colorsix] \betafl; +\draw[color=colorseven] \betagl; +\draw[color=coloreight] \betahl; +\draw[color=colornine] \betail; +\draw[color=colorten] \betajl; +\draw[color=coloreleven] \betakl; +\draw[color=colortwelve] \betall; +\end{scope} \achsen \farbviereck -\farbpunkt{\alphaeleven}{\betaeleven}{coloreleven} -\farbpunkt{\alphaten}{\betaeleven}{colorten} -\farbpunkt{\alphanine}{\betaeleven}{colornine} -\farbpunkt{\alphaeight}{\betaeleven}{coloreight} -\farbpunkt{\alphaseven}{\betaeleven}{colorseven} -\farbpunkt{\alphasix}{\betaeleven}{colorsix} -\farbpunkt{\alphafive}{\betaeleven}{colorfive} -\farbpunkt{\alphafour}{\betaeleven}{colorfour} -\farbpunkt{\alphathree}{\betaeleven}{colorthree} -\farbpunkt{\alphatwo}{\betaeleven}{colortwo} -\farbpunkt{\alphaone}{\betaeleven}{colorone} +\farbpunkt{\alphatwelve}{\betatwelve}{colortwelve} +\farbpunkt{\alphaeleven}{\betatwelve}{coloreleven} +\farbpunkt{\alphaten}{\betatwelve}{colorten} +\farbpunkt{\alphanine}{\betatwelve}{colornine} +\farbpunkt{\alphaeight}{\betatwelve}{coloreight} +\farbpunkt{\alphaseven}{\betatwelve}{colorseven} +\farbpunkt{\alphasix}{\betatwelve}{colorsix} +\farbpunkt{\alphafive}{\betatwelve}{colorfive} +\farbpunkt{\alphafour}{\betatwelve}{colorfour} +\farbpunkt{\alphathree}{\betatwelve}{colorthree} +\farbpunkt{\alphatwo}{\betatwelve}{colortwo} +\farbpunkt{\alphaone}{\betatwelve}{colorone} \end{scope} diff --git a/buch/papers/dreieck/images/betadist.m b/buch/papers/dreieck/images/betadist.m index 9ff78ed..5b466a6 100644 --- a/buch/papers/dreieck/images/betadist.m +++ b/buch/papers/dreieck/images/betadist.m @@ -5,24 +5,32 @@ # global N; N = 201; -global n; -n = 11; +global nmin; +global nmax; +nmin = -4; +nmax = 7; +n = nmax - nmin + 1 +A = 3; -t = (0:n-1) / (n-1) -alpha = 1 + 4 * t.^2 +t = (nmin:nmax) / nmax; +alpha = 1 + A * t .* abs(t) +#alpha(1) = 0.01; #alpha = [ 1, 1.03, 1.05, 1.1, 1.25, 1.5, 2, 2.5, 3, 4, 5 ]; beta = alpha; names = [ "one"; "two"; "three"; "four"; "five"; "six"; "seven"; "eight"; - "nine"; "ten"; "eleven" ] + "nine"; "ten"; "eleven"; "twelve" ] function retval = Beta(a, b, x) retval = x^(a-1) * (1-x)^(b-1) / beta(a, b); + if (retval > 100) + retval = 100 + end end function plotbeta(fn, a, b, name) global N; - fprintf(fn, "\\def\\beta%s{\n", name); + fprintf(fn, "\\def\\beta%s{\n", strtrim(name)); fprintf(fn, "\t({%.4f*\\dx},{%.4f*\\dy})", 0, Beta(a, b, 0)); for x = (1:N-1)/(N-1) X = (1-cos(pi * x))/2; @@ -35,8 +43,8 @@ end fn = fopen("betapaths.tex", "w"); for i = (1:n) - fprintf(fn, "\\def\\alpha%s{%f}\n", names(i,:), alpha(i)); - fprintf(fn, "\\def\\beta%s{%f}\n", names(i,:), beta(i)); + fprintf(fn, "\\def\\alpha%s{%f}\n", strtrim(names(i,:)), alpha(i)); + fprintf(fn, "\\def\\beta%s{%f}\n", strtrim(names(i,:)), beta(i)); end for i = (1:n) -- cgit v1.2.1 From 9f0ff73d26b3e096d848abfd20cd124433e2a4a7 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 13 Mar 2022 23:27:43 +0100 Subject: cleanup --- buch/aufgaben2.tex | 4 ---- 1 file changed, 4 deletions(-) (limited to 'buch') diff --git a/buch/aufgaben2.tex b/buch/aufgaben2.tex index f98562e..8073f26 100644 --- a/buch/aufgaben2.tex +++ b/buch/aufgaben2.tex @@ -4,10 +4,6 @@ % (c) 2022 Prof. Dr. Andreas Mueller, OST % -%\item -%\input chapters/40-eigenwerte/uebungsaufgaben/4004.tex -%\item -%\input chapters/40-eigenwerte/uebungsaufgaben/4005.tex \item \input{chapters/090-pde/uebungsaufgaben/901.tex} \item -- cgit v1.2.1 From 18e46179f2da76a3147d3f3b466206c6b5405859 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 14 Mar 2022 08:20:28 +0100 Subject: describe link between Jacobi-weights and Beta-distribution --- buch/chapters/070-orthogonalitaet/jacobi.tex | 22 ++ buch/chapters/070-orthogonalitaet/orthogonal.tex | 51 +++ buch/papers/dreieck/teil1.tex | 411 +---------------------- 3 files changed, 74 insertions(+), 410 deletions(-) (limited to 'buch') diff --git a/buch/chapters/070-orthogonalitaet/jacobi.tex b/buch/chapters/070-orthogonalitaet/jacobi.tex index 042d466..f776c03 100644 --- a/buch/chapters/070-orthogonalitaet/jacobi.tex +++ b/buch/chapters/070-orthogonalitaet/jacobi.tex @@ -189,6 +189,28 @@ rechten Rand haben. \label{buch:orthogonal:fig:jacobi-parameter}} \end{figure} +\subsection{Jacobi-Gewichtsfunktion und Beta-Verteilung +\label{buch:orthogonal:subsection:beta-verteilung}} +Die Jacobi-Gewichtsfunktion entsteht aus der Wahrscheinlichkeitsdichte +der Beta-Verteilung, die in +Abschnitt~\ref{buch:rekursion:subsection:beta-verteilung} +eingeführt wurde mit Hilfe der Variablen-Transformation $x = 2t-1$ +oder $t=(x+1)/2$. +Das Integral mit der Jacobi-Gewichtsfunktion $w^{(\alpha,\beta)}(x)$ +kann damit umgeformt werden in +\[ +\int_{-1}^1 +f(x)\,w^{(\alpha,\beta)}(x)\,dx += +\int_0^1 +f(2t-1) w^{(\alpha,\beta)}(2t-1)\,2\,dt += +\int_0^1 +f(2t-1) +(1-(2t-1))^\alpha (1+(2t-1))^\beta +\,2\,dt +\] + % % % diff --git a/buch/chapters/070-orthogonalitaet/orthogonal.tex b/buch/chapters/070-orthogonalitaet/orthogonal.tex index d06f46e..a84248a 100644 --- a/buch/chapters/070-orthogonalitaet/orthogonal.tex +++ b/buch/chapters/070-orthogonalitaet/orthogonal.tex @@ -737,6 +737,57 @@ rechten Rand haben. \label{buch:orthogonal:fig:jacobi-parameter}} \end{figure} +\subsubsection{Jacobi-Gewichtsfunktion und Beta-Verteilung +\label{buch:orthogonal:subsection:beta-verteilung}} +Die Jacobi-Gewichtsfunktion entsteht aus der Wahrscheinlichkeitsdichte +der Beta-Verteilung, die in +Abschnitt~\ref{buch:rekursion:subsection:beta-verteilung} +eingeführt wurde mit Hilfe der Variablen-Transformation $x = 2t-1$ +oder $t=(x+1)/2$. +Das Integral mit der Jacobi-Gewichtsfunktion $w^{(\alpha,\beta)}(x)$ +kann damit umgeformt werden in +\begin{align*} +\int_{-1}^1 +f(x)\,w^{(\alpha,\beta)}(x)\,dx +&= +\int_0^1 +f(2t-1) w^{(\alpha,\beta)}(2t-1)\,2\,dt +\\ +&= +\int_0^1 +f(2t-1) +(1-(2t-1))^\alpha (1+(2t-1))^\beta +\,2\,dt +\\ +&= +2^{\alpha+\beta+1} +\int_0^1 +f(2t-1) +\, +t^\beta +(1-t)^\alpha +\,dt +\\ +&= +2^{\alpha+\beta+1} +B(\alpha+1,\beta+1) +\int_0^1 +f(2t-1) +\, +\frac{ +t^\beta +(1-t)^\alpha +}{B(\alpha+1,\beta+1)} +\,dt. +\end{align*} +Auf der letzten Zeile steht ein Integral mit der Wahrscheinlichkeitsdichte +der Beta-Verteilung. +Orthogonale Funktionen bezüglich der Jacobischen Gewichtsfunktion +$w^{(\alpha,\beta)}$ werden mit der genannten Substitution also +zu orthogonalen Funktionen bezüglich der Beta-Verteilung mit +Parametern $\beta+1$ und $\alpha+1$. + + % % Tschebyscheff-Gewichtsfunktion % diff --git a/buch/papers/dreieck/teil1.tex b/buch/papers/dreieck/teil1.tex index 5e7090b..4abe2e1 100644 --- a/buch/papers/dreieck/teil1.tex +++ b/buch/papers/dreieck/teil1.tex @@ -5,416 +5,7 @@ % \section{Ordnungsstatistik und Beta-Funktion \label{dreieck:section:ordnungsstatistik}} -\rhead{Ordnungsstatistik und Beta-Funktion} -In diesem Abschnitt ist $X$ eine Zufallsvariable mit der Verteilungsfunktion -$F_X(x)$, und $X_i$, $1\le i\le n$ sei ein Stichprobe von unabhängigen -Zufallsvariablen, die wie $X$ verteilt sind. -Ziel ist, die Verteilungsfunktion und die Wahrscheinlichkeitsdichte -des grössten, zweitgrössten, $k$-t-grössten Wertes in der Stichprobe -zu finden. -Wir schreiben $[n]=\{1,\dots,n\}$ für die Menge der natürlichen -Zahlen von zwischen $1$ und $n$. +\rhead{} -\subsection{Verteilung von $\operatorname{max}(X_1,\dots,X_n)$ und -$\operatorname{min}(X_1,\dots,X_n)$ -\label{dreieck:subsection:minmax}} -Die Verteilungsfunktion von $\operatorname{max}(X_1,\dots,X_n)$ hat -den Wert -\begin{align*} -F_{\operatorname{max}(X_1,\dots,X_n)}(x) -&= -P(\operatorname{max}(X_1,\dots,X_n) \le x) -\\ -&= -P(X_1\le x\wedge \dots \wedge X_n\le x) -\\ -&= -P(X_1\le x) \cdot \ldots \cdot P(X_n\le x) -\\ -&= -P(X\le x)^n -= -F_X(x)^n. -\end{align*} -Für die Gleichverteilung ist -\[ -F_{\text{equi}}(x) -= -\begin{cases} -0&\qquad x< 0 -\\ -x&\qquad 0\le x\le 1 -\\ -1&\qquad 1 X_1\wedge \dots \wedge x > X_n) -\\ -&= -1- -(1-P(x\le X_1)) \cdot\ldots\cdot (1-P(x\le X_n)) -\\ -&= -1-(1-F_X(x))^n, -\end{align*} -Im Speziellen für im Intervall $[0,1]$ gleichverteilte $X_i$ ist die -Verteilungsfunktion des Minimums -\[ -F_{\operatorname{min}(X_1,\dots,X_n)}(x) -= -\begin{cases} -0 &\qquad x<0 \\ -1-(1-x)^n&\qquad 0\le x\le 1\\ -1 &\qquad 1 < x -\end{cases} -\] -mit Wahrscheinlichkeitsdichte -\[ -\varphi_{\operatorname{min}(X_1,\dots,X_n)} -= -\frac{d}{dx} -F_{\operatorname{min}(X_1,\dots,X_n)} -= -\begin{cases} -n(1-x)^{n-1}&\qquad 0\le x\le 1\\ -0 &\qquad \text{sonst} -\end{cases} -\] -und Erwartungswert -\begin{align*} -E(\operatorname{min}(X_1,\dots,X_n) -&= -\int_{-\infty}^\infty x\varphi_{\operatorname{min}(X_1,\dots,X_n)}(x)\,dx -= -\int_0^1 x\cdot n(1-x)^{n-1}\,dx -\\ -&= -\bigl[ -x(1-x)^n \bigr]_0^1 + \int_0^1 (1-x)^n\,dx -= -\biggl[ -- -\frac{1}{n+1} -(1-x)^{n+1} -\biggr]_0^1 -= -\frac{1}{n+1}. -\end{align*} -Es ergibt sich daraus als natürlich Verallgemeinerung die Frage nach -der Verteilung des zweitegrössten oder zweitkleinsten Wertes unter den -Werten $X_i$. - -\subsection{Der $k$-t-grösste Wert} -Sie wieder $X_i$ eine Stichprobe von $n$ unabhängigen wie $X$ verteilten -Zufallsvariablen. -Diese werden jetzt der Grösse nach sortiert, die sortierten Werte werden -mit -\[ -X_{1:n} \le X_{2:n} \le \dots \le X_{(n-1):n} \le X_{n:n} -\] -bezeichnet. -Die Grössen $X_{k:n}$ sind Zufallsvariablen, sie heissen die $k$-ten -Ordnungsstatistiken. -Die in Abschnitt~\ref{dreieck:subsection:minmax} behandelten Zufallsvariablen -$\operatorname{min}(X_1,\dots,X_n)$ -und -$\operatorname{max}(X_1,\dots,X_n)$ -sind die Fälle -\begin{align*} -X_{1:n} &= \operatorname{min}(X_1,\dots,X_n) \\ -X_{n:n} &= \operatorname{max}(X_1,\dots,X_n). -\end{align*} - -Um den Wert der Verteilungsfunktion von $X_{k:n}$ zu berechnen, müssen wir -die Wahrscheinlichkeit bestimmen, dass $k$ der $n$ Werte $X_i$ $x$ nicht -übersteigen. -Der $k$-te Wert $X_{k:n}$ übersteigt genau dann $x$ nicht, wenn -mindestens $k$ der Zufallswerte $X_i$ $x$ nicht übersteigen, also -\[ -P(X_{k:n} \le x) -= -P\left( -|\{i\in[n]\,|\, X_i\le x\}| \ge k -\right). -\] - -Das Ereignis $\{X_i\le x\}$ ist eine Bernoulli-Experiment, welches mit -Wahrscheinlichkeit $F_X(x)$ eintritt. -Die Anzahl der Zufallsvariablen $X_i$, die $x$ übertreffen, ist also -Binomialverteilt mit $p=F_X(x)$. -Damit haben wir gefunden, dass mit Wahrscheinlichkeit -\begin{equation} -F_{X_{k:n}}(x) -= -P(X_{k:n}\le x) -= -\sum_{i=k}^n \binom{n}{i}F_X(x)^i (1-F_X(x))^{n-i} -\label{dreieck:eqn:FXkn} -\end{equation} -mindestens $k$ der Zufallsvariablen den Wert $x$ überschreiten. - -\subsubsection{Wahrscheinlichkeitsdichte der Ordnungsstatistik} -Die Wahrscheinlichkeitsdichte der Ordnungsstatistik kann durch Ableitung -von \eqref{dreieck:eqn:FXkn} gefunden, werden, sie ist -\begin{align*} -\varphi_{X_{k:n}}(x) -&= -\frac{d}{dx} -F_{X_{k:n}}(x) -\\ -&= -\sum_{i=k}^n -\binom{n}{i} -\bigl( -iF_X(x)^{i-1}\varphi_X(x) (1-F_X(x))^{n-i} -- -F_X(x)^k -(n-i) -(1-F_X(x))^{n-i-1} -\varphi_X(x) -\bigr) -\\ -&= -\sum_{i=k}^n -\binom{n}{i} -\varphi_X(x) -F_X(x)^{i-1}(1-F_X(x))^{n-i-1} -\bigl( -iF_X(x)-(n-i)(1-F_X(x)) -\bigr) -\\ -&= -\varphi_X(x) -\biggl( -\sum_{i=k}^n i\binom{n}{i} F_X(x)^{i-1}(1-F_X(x))^{n-i} -- -\sum_{j=k}^n (n-j)\binom{n}{j} F_X(x)^{j}(1-F_X(x))^{n-j-1} -\biggr) -\\ -&= -\varphi_X(x) -\biggl( -\sum_{i=k}^n i\binom{n}{i} F_X(x)^{i-1}(1-F_X(x))^{n-i} -- -\sum_{i=k+1}^{n+1} (n-i+1)\binom{n}{i-1} F_X(x)^{i-1}(1-F_X(x))^{n-i} -\biggr) -\\ -&= -\varphi_X(x) -\biggl( -k\binom{n}{k}F_X(x)^{k-1}(1-F_X(x))^{n-k} -+ -\sum_{i=k+1}^{n+1} -\left( -i\binom{n}{i} -- -(n-i+1)\binom{n}{i-1} -\right) -F_X(x)^{i-1}(1-F_X(x))^{n-i} -\biggr) -\end{align*} -Mit den wohlbekannten Identitäten für die Binomialkoeffizienten -\begin{align*} -i\binom{n}{i} -- -(n-i+1)\binom{n}{i-1} -&= -n\binom{n-1}{i-1} -- -n -\binom{n-1}{i-1} -= -0 -\end{align*} -folgt jetzt -\begin{align*} -\varphi_{X_{k:n}}(x) -&= -\varphi_X(x)k\binom{n}{k} F_X(x)^{k-1}(1-F_X(x))^{n-k}(x). -\intertext{Im Speziellen für gleichverteilte Zufallsvariablen $X_i$ ist -} -\varphi_{X_{k:n}}(x) -&= -k\binom{n}{k} x^{k-1}(1-x)^{n-k}. -\end{align*} -Dies ist die Wahrscheinlichkeitsdichte einer Betaverteilung -\[ -\beta(k,n-k+1)(x) -= -\frac{1}{B(k,n-k+1)} -x^{k-1}(1-x)^{n-k}. -\] -Tatsächlich ist die Normierungskonstante -\begin{align} -\frac{1}{B(k,n-k+1)} -&= -\frac{\Gamma(n+1)}{\Gamma(k)\Gamma(n-k+1)} -= -\frac{n!}{(k-1)!(n-k)!}. -\label{dreieck:betaverteilung:normierung1} -\end{align} -Andererseits ist -\[ -k\binom{n}{k} -= -k\frac{n!}{k!(n-k)!} -= -\frac{n!}{(k-1)!(n-k)!}, -\] -in Übereinstimmung mit~\eqref{dreieck:betaverteilung:normierung1}. -Die Verteilungsfunktion und die Wahrscheinlichkeitsdichte der -Ordnungsstatistik sind in Abbildung~\ref{dreieck:fig:order} dargestellt. - -\begin{figure} -\centering -\includegraphics{papers/dreieck/images/order.pdf} -\caption{Verteilungsfunktion und Wahrscheinlichkeitsdichte der -Ordnungsstatistiken $X_{k:n}$ einer gleichverteilung Zuvallsvariable -mit $n=10$. -\label{dreieck:fig:order}} -\end{figure} - -\subsubsection{Erwartungswert} -Mit der Wahrscheinlichkeitsdichte kann man jetzt auch den Erwartungswerte -der $k$-ten Ordnungsstatistik bestimmen. -Die Rechnung ergibt: -\begin{align*} -E(X_{k:n}) -&= -\int_0^1 x\cdot k\binom{n}{k} x^{k-1}(1-x)^{n-k}\,dx -= -k -\binom{n}{k} -\int_0^1 -x^{k}(1-x)^{n-k}\,dx. -\intertext{Dies ist das Beta-Integral} -&= -k\binom{n}{k} -B(k+1,n-k+1) -\intertext{welches man durch Gamma-Funktionen bzw.~durch Fakultäten wie in} -&= -k\frac{n!}{k!(n-k)!} -\frac{\Gamma(k+1)\Gamma(n-k+1)}{n+2} -= -k\frac{n!}{k!(n-k)!} -\frac{k!(n-k)!}{(n+1)!} -= -\frac{k}{n+1} -\end{align*} -ausdrücken kann. -Die Erwartungswerte haben also regelmässige Abstände, sie sind in -Abbildung~\ref{dreieck:fig:order} als blaue vertikale Linien eingezeichnet. - -\subsubsection{Varianz} -Auch die Varianz lässt sich einfach berechnen, dazu muss zunächst -der Erwartungswert von $X_{k:n}^2$ bestimmt werden. -Er ist -\begin{align*} -E(X_{k:n}^2) -&= -\int_0^1 x^2\cdot k\binom{n}{k} x^{k-1}(1-x)^{n-k}\,dx -= -k -\binom{n}{k} -\int_0^1 -x^{k+1}(1-x)^{n-k}\,dx. -\intertext{Auch dies ist ein Beta-Integral, nämlich} -&= -k\binom{n}{k} -B(k+2,n-k+1) -= -k\frac{n!}{k!(n-k)!} -\frac{(k+1)!(n-k)!}{(n+2)!} -= -\frac{k(k+1)}{(n+1)(n+2)}. -\end{align*} -Die Varianz wird damit -\begin{align} -\operatorname{var}(X_{k:n}) -&= -E(X_{k:n}^2) - E(X_{k:n})^2 -\notag -\\ -& -= -\frac{k(k+1)}{(n+1)(n+2)}-\frac{k^2}{(n+1)^2} -= -\frac{k(k+1)(n+1)-k^2(n+2)}{(n+1)^2(n+2)} -= -\frac{k(n-k+1)}{(n+1)^2(n+2)}. -\label{dreieck:eqn:ordnungsstatistik:varianz} -\end{align} -In Abbildung~\ref{dreieck:fig:order} ist die Varianz der -Ordnungsstatistik $X_{k:n}$ für $k=7$ und $n=10$ als oranges -Rechteck dargestellt. - -\begin{figure} -\centering -\includegraphics[width=0.84\textwidth]{papers/dreieck/images/beta.pdf} -\caption{Wahrscheinlichkeitsdichte der Beta-Verteilung -$\beta(a,b,x)$ -für verschiedene Werte der Parameter $a$ und $b$. -Die Werte des Parameters für einen Graphen einer Beta-Verteilung -sind als Punkt im kleinen Quadrat rechts -im Graphen als Punkt mit der gleichen Farbe dargestellt. -\label{dreieck:fig:betaverteilungn}} -\end{figure} - -Die Formel~\eqref{dreieck:eqn:ordnungsstatistik:varianz} -besagt auch, dass die Varianz der proportional ist zu $k((n+1)-k)$. -Dieser Ausdruck ist am grössten für $k=(n+1)/2$, die Varianz ist -also grösser für die ``mittleren'' Ordnungstatistiken als für die -extremen $X_{1:n}=\operatorname{min}(X_1,\dots,X_n)$ und -$X_{n:n}=\operatorname{max}(X_1,\dots,X_n)$. -- cgit v1.2.1 From 8ecf2cd93564d76aee7dd81736d3cd5908b273cd Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 14 Mar 2022 12:41:22 +0100 Subject: move command configuration to common/Makefile.inc --- buch/Makefile | 28 ++++++++++++++-------------- buch/common/Makefile.inc | 8 +++++++- 2 files changed, 21 insertions(+), 15 deletions(-) (limited to 'buch') diff --git a/buch/Makefile b/buch/Makefile index e2ad4c0..00fcf42 100755 --- a/buch/Makefile +++ b/buch/Makefile @@ -18,15 +18,15 @@ ALLTEXFILES = $(TEXFILES) $(CHAPTERFILES) # Buchblock für Druckerei # buch.pdf: buch.tex $(TEXFILES) buch.ind $(BLXFILES) - pdflatex buch.tex - bibtex buch + $(pdflatex) buch.tex + $(bibtex) buch buch.idx: buch.tex $(TEXFILES) images - touch buch.ind - pdflatex buch.tex + $(touch) buch.ind + $(pdflatex) buch.tex buch.ind: buch.idx - makeindex buch.idx + $(makeindex) buch.idx # # Papers in einzelne PDF-Files separieren für digitales Feedback @@ -39,16 +39,16 @@ separate: buch.aux buch.pdf # SeminarSpezielleFunktionen.pdf: SeminarSpezielleFunktionen.tex $(TEXFILES) \ SeminarSpezielleFunktionen.ind $(BLXFILES) - pdflatex SeminarSpezielleFunktionen.tex - bibtex SeminarSpezielleFunktionen + $(pdflatex) SeminarSpezielleFunktionen.tex + $(bibtex) SeminarSpezielleFunktionen SeminarSpezielleFunktionen.idx: SeminarSpezielleFunktionen.tex $(TEXFILES) \ images - touch SeminarSpezielleFunktionen.ind - pdflatex SeminarSpezielleFunktionen.tex + $(touch) SeminarSpezielleFunktionen.ind + $(pdflatex) SeminarSpezielleFunktionen.tex SeminarSpezielleFunktionen.ind: SeminarSpezielleFunktionen.idx - makeindex SeminarSpezielleFunktionen + $(makeindex) SeminarSpezielleFunktionen # # This Makefile can also construct the short tests @@ -56,17 +56,17 @@ SeminarSpezielleFunktionen.ind: SeminarSpezielleFunktionen.idx tests: test1.pdf test2.pdf test3.pdf test1.pdf: common/test-common.tex common/test1.tex aufgaben1.tex - pdflatex common/test1.tex + $(pdflatex) common/test1.tex test2.pdf: common/test-common.tex common/test1.tex aufgaben2.tex - pdflatex common/test2.tex + $(pdflatex) common/test2.tex test3.pdf: common/test-common.tex common/test1.tex aufgaben3.tex - pdflatex common/test3.tex + $(pdflatex) common/test3.tex # # Errata # errata.pdf: errata.tex - pdflatex errata.tex + $(pdflatex) errata.tex diff --git a/buch/common/Makefile.inc b/buch/common/Makefile.inc index c8b0f6e..b9461e5 100755 --- a/buch/common/Makefile.inc +++ b/buch/common/Makefile.inc @@ -4,9 +4,15 @@ # (c) 2021 Prof Dr Andreas Mueller, OST Ostschweizer Fachhochschule # - SUBDIRECTORIES = chapters +# change the following variables to suit your environment + +pdflatex = pdflatex +bibtex = bibtex +makeindex = makeindex +touch = touch + .PHONY: images images: -- cgit v1.2.1 From bfaa02edc586b384691399c81aad54d92b5de986 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Tue, 15 Mar 2022 09:48:35 +0100 Subject: kugel: Add paper title and notes --- buch/papers/kugel/Makefile.inc | 8 ++---- buch/papers/kugel/main.tex | 57 ++++++++++++++++++++++-------------------- buch/papers/kugel/teil0.tex | 22 ---------------- buch/papers/kugel/teil1.tex | 55 ---------------------------------------- buch/papers/kugel/teil2.tex | 40 ----------------------------- buch/papers/kugel/teil3.tex | 40 ----------------------------- 6 files changed, 32 insertions(+), 190 deletions(-) delete mode 100644 buch/papers/kugel/teil0.tex delete mode 100644 buch/papers/kugel/teil1.tex delete mode 100644 buch/papers/kugel/teil2.tex delete mode 100644 buch/papers/kugel/teil3.tex (limited to 'buch') diff --git a/buch/papers/kugel/Makefile.inc b/buch/papers/kugel/Makefile.inc index d926229..50d6825 100644 --- a/buch/papers/kugel/Makefile.inc +++ b/buch/papers/kugel/Makefile.inc @@ -4,11 +4,7 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # dependencies-kugel = \ - papers/kugel/packages.tex \ + papers/kugel/packages.tex \ papers/kugel/main.tex \ - papers/kugel/references.bib \ - papers/kugel/teil0.tex \ - papers/kugel/teil1.tex \ - papers/kugel/teil2.tex \ - papers/kugel/teil3.tex + papers/kugel/references.bib diff --git a/buch/papers/kugel/main.tex b/buch/papers/kugel/main.tex index 0e632ec..06368af 100644 --- a/buch/papers/kugel/main.tex +++ b/buch/papers/kugel/main.tex @@ -1,36 +1,39 @@ % + % main.tex -- Paper zum Thema % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:kugel}} -\lhead{Thema} +\chapter{Recurrence Relations for Spherical Harmonics in Quantum Mechanics\label{chapter:kugel}} +\lhead{Recurrence Relations in Quantum Mechanics} \begin{refsection} -\chapterauthor{Hans Muster} - -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} - -\input{papers/kugel/teil0.tex} -\input{papers/kugel/teil1.tex} -\input{papers/kugel/teil2.tex} -\input{papers/kugel/teil3.tex} +\chapterauthor{Manuel Cattaneo, Naoki Pross} + +\begin{verbatim} + +Ideas and current research goals +-------------------------------- + +- Recurrence relations for spherical harmonics +- Associated Legendre polynomials +- Rodrigues' type formula aka Rodrigues' formula +- Applications: + * Quantization of angular momentum + * Gravitational field measurements (NASA ebb and flow, ESA goce) + * Literally anything that needs basis functions on the surface of a sphere + +Literature +---------- + +- Nichtkommutative Bildverarbeitung, T. Mendez, p57+ +- Linear Algebra Done Right, S. Axler, p212,221,231,237 +- Introduction to Quantum Mechanics, D. J. Griffith, p201+ +- Seminar Quantenmechanik, A. Müller, p101,106,114,121 +- Introduction to Partial Differential Equations, J. Oliver, p510+ +- Partial Differential Equations in Engineering Problems, K. Miller, p175,190 + +\end{verbatim} + \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/kugel/teil0.tex b/buch/papers/kugel/teil0.tex deleted file mode 100644 index f921a82..0000000 --- a/buch/papers/kugel/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{kugel:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{kugel:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/kugel/teil1.tex b/buch/papers/kugel/teil1.tex deleted file mode 100644 index e56bb18..0000000 --- a/buch/papers/kugel/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{kugel:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{kugel:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{kugel:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{kugel:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{kugel:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/kugel/teil2.tex b/buch/papers/kugel/teil2.tex deleted file mode 100644 index cb9e427..0000000 --- a/buch/papers/kugel/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{kugel:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{kugel:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/kugel/teil3.tex b/buch/papers/kugel/teil3.tex deleted file mode 100644 index 734fff9..0000000 --- a/buch/papers/kugel/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{kugel:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{kugel:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - -- cgit v1.2.1 From 9f542be91a61e6eb1fcb3c29e8a908eb9e5b8949 Mon Sep 17 00:00:00 2001 From: Marc Benz Date: Tue, 15 Mar 2022 14:00:47 +0100 Subject: initial commit from Marc Benz --- buch/papers/transfer/main.tex | 23 ++--------------------- 1 file changed, 2 insertions(+), 21 deletions(-) (limited to 'buch') diff --git a/buch/papers/transfer/main.tex b/buch/papers/transfer/main.tex index 2aae635..ed16998 100644 --- a/buch/papers/transfer/main.tex +++ b/buch/papers/transfer/main.tex @@ -3,29 +3,10 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:transfer}} +\chapter{Transferfunktionen\label{chapter:transfer}} \lhead{Thema} \begin{refsection} -\chapterauthor{Hans Muster} - -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} +\chapterauthor{Marc Benz} \input{papers/transfer/teil0.tex} \input{papers/transfer/teil1.tex} -- cgit v1.2.1 From 3381bfe1e6ebcd66249cd4c6f49bdd820643a5be Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 15 Mar 2022 17:35:00 +0100 Subject: add some stuff about separation --- buch/chapters/090-pde/kreis.tex | 2 +- buch/chapters/090-pde/kugel.tex | 146 ++++++++++++++++++++++++++++++++++++++ buch/papers/000template/main.tex | 5 +- buch/papers/000template/teil0.tex | 3 + buch/papers/000template/teil1.tex | 3 + buch/papers/000template/teil2.tex | 3 + buch/papers/000template/teil3.tex | 3 + 7 files changed, 163 insertions(+), 2 deletions(-) (limited to 'buch') diff --git a/buch/chapters/090-pde/kreis.tex b/buch/chapters/090-pde/kreis.tex index b4ce8d7..c60fd44 100644 --- a/buch/chapters/090-pde/kreis.tex +++ b/buch/chapters/090-pde/kreis.tex @@ -32,7 +32,7 @@ Der Laplace-Operator hat in Polarkoordinaten die Form \frac1r \frac{\partial}{\partial r} + -\frac{1}{r 2} +\frac{1}{r^2} \frac{\partial^2}{\partial\varphi^2}. \label{buch:pde:kreis:laplace} \end{equation} diff --git a/buch/chapters/090-pde/kugel.tex b/buch/chapters/090-pde/kugel.tex index 0e3524f..c081029 100644 --- a/buch/chapters/090-pde/kugel.tex +++ b/buch/chapters/090-pde/kugel.tex @@ -5,4 +5,150 @@ % \section{Kugelfunktionen \label{buch:pde:section:kugel}} +Kugelsymmetrische Probleme können oft vorteilhaft in Kugelkoordinaten +beschrieben werden. +Die Separationsmethode kann auf partielle Differentialgleichungen +mit dem Laplace-Operator angewendet werden. +Die daraus resultierenden gewöhnlichen Differentialgleichungen führen +einerseits auf die Laguerre-Differentialgleichung für den radialen +Anteil sowie auf Kugelfunktionen für die Koordinaten der +geographischen Länge und Breite. + +\subsection{Kugelkoordinaten} +Wir verwenden Kugelkoordinaten $(r,\vartheta,\varphi)$, wobei $r$ +der Radius ist, $\vartheta$ die geographische Breite gemessen vom +Nordpol der Kugel und $\varphi$ die geographische Breite. +Der Definitionsbereich für Kugelkoordinaten ist +\[ +\Omega += +\{(r,\vartheta,\varphi) +\;|\; +r\ge 0\wedge +0\le \vartheta\le \pi\wedge +0\le \varphi< 2\pi +\}. +\] +Die Entfernung eines Punktes von der $z$-Achse ist $r\sin\vartheta$. +Daraus lassen sich die karteischen Koordinaten eines Punktes mit Hilfe +von +\[ +\begin{pmatrix}x\\y\\z\end{pmatrix} += +\begin{pmatrix} +r\cos\vartheta\\ +r\sin\vartheta\cos\varphi\\ +r\sin\vartheta\sin\varphi +\end{pmatrix}. +\] +Man beachte, dass die Punkte auf der $z$-Achse keine eindeutigen +Kugelkoordinaten haben. +Sie sind charakterisiert durch $r\sin\vartheta=0$, was $\cos\vartheta=\pm1$ +impliziert. +Entsprechend führen alle Werte von $\varphi$ auf den gleichen Punkt +$(0,0,\pm r)$. + +\subsection{Der Laplace-Operator in Kugelkoordinaten} +Der Laplace-Operator in Kugelkoordinaten lautet +\begin{align} +\Delta +&= +\frac{1}{r^2} \frac{\partial}{\partial r}r^2\frac{\partial}{\partial r} ++ +\frac{1}{r^2\sin\vartheta}\frac{\partial}{\partial\vartheta} +\sin\vartheta\frac{\partial}{\partial\vartheta} ++ +\frac{1}{r^2\sin^2\vartheta}\frac{\partial^2}{\partial\varphi^2}. +\label{buch:pde:kugel:laplace1} +\intertext{Dies kann auch geschrieben werden als} +&= +\frac{\partial^2}{\partial r^2} ++ +\frac{2}{r}\frac{\partial}{\partial r} ++ +\frac{1}{r^2\sin\vartheta}\frac{\partial}{\partial\vartheta} +\sin\vartheta\frac{\partial}{\partial\vartheta} ++ +\frac{1}{r^2\sin^2\vartheta}\frac{\partial^2}{\partial\varphi^2} +\label{buch:pde:kugel:laplace2} +\intertext{oder} +&= +\frac{1}{r} +\frac{\partial^2}{\partial r^2} r ++ +\frac{1}{r^2\sin\vartheta}\frac{\partial}{\partial\vartheta} +\sin\vartheta\frac{\partial}{\partial\vartheta} ++ +\frac{1}{r^2\sin^2\vartheta}\frac{\partial^2}{\partial\varphi^2}. +\label{buch:pde:kugel:laplace3} +\end{align} +Dabei ist zu berücksichtigen, dass mit der Notation gemeint ist, +dass ein Ableitungsoperator auf alles wirkt, was rechts im gleichen +Term steht. +Der Operator +\[ +\frac{1}{r} +\frac{\partial^2}{\partial r^2}r +\quad\text{wirkt daher als}\quad +\frac{1}{r} +\frac{\partial^2}{\partial r^2}rf += +\frac{1}{r} +\frac{\partial}{\partial r}\biggl(f + r\frac{\partial f}{\partial r}\biggr) += +\frac{1}{r} +\frac{\partial f}{\partial r} ++ +\frac{1}{r} +\frac{\partial f}{\partial r} ++ +\frac{\partial^2f}{\partial r^2}. += +\frac{2}{r}\frac{\partial f}{\partial r} ++ +\frac{\partial^2f}{\partial r^2}, +\] +was die Äquivalenz der beiden Formen +\eqref{buch:pde:kugel:laplace2} +und +\eqref{buch:pde:kugel:laplace3} +rechtfertigt. +Auch die Äquivalenz mit +\eqref{buch:pde:kugel:laplace1} +kann auf ähnliche Weise verstanden werden. + +Die Herleitung dieser Formel ist ziemlich aufwendig und soll hier +nicht dargestellt werden. +Es sei aber darauf hingewiesen, dass sich für $\vartheta=\frac{\pi}2$ +wegen $\sin\vartheta=\sin\frac{\pi}2=1$ +der eingeschränkte Operator +\[ +\Delta += +\frac{1}{r^2}\frac{\partial }{\partial r} r^2\frac{\partial}{\partial r} ++ +\frac{1}{r^2}\frac{\partial^2}{\partial\varphi^2} +\] +ergibt. +Wendet man wie oben die Produktregel auf den ersten Term an, entsteht die +Form +\[ +\frac{\partial^2}{\partial r^2} ++ +\frac{2}{r} +\frac{\partial}{\partial r} ++ +\frac{1}{r^2}\frac{\partial^2}{\partial\varphi^2} +\] +die {\em nicht} übereinstimmt mit dem Laplace-Operator in +Polarkoordinaten~\eqref{buch:pde:kreis:laplace}. +Der Unterschied rührt daher, dass der Laplace-Operator die Krümmung +der Koordinatenlinien berücksichtigt, in diesem Fall der Meridiane. + + +\subsection{Separation} + + + + diff --git a/buch/papers/000template/main.tex b/buch/papers/000template/main.tex index 87a5685..91b6d6e 100644 --- a/buch/papers/000template/main.tex +++ b/buch/papers/000template/main.tex @@ -1,7 +1,10 @@ % % main.tex -- Paper zum Thema <000template> % -% (c) 2020 Hochschule Rapperswil +% (c) 2020 Autor, OST Ostschweizer Fachhochschule +% +% !TEX root = ../../buch.tex +% !TEX encoding = UTF-8 % \chapter{Thema\label{chapter:000template}} \lhead{Thema} diff --git a/buch/papers/000template/teil0.tex b/buch/papers/000template/teil0.tex index 7b9f088..65d7ae1 100644 --- a/buch/papers/000template/teil0.tex +++ b/buch/papers/000template/teil0.tex @@ -3,6 +3,9 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % +% !TEX root = ../../buch.tex +% !TEX encoding = UTF-8 +% \section{Teil 0\label{000template:section:teil0}} \rhead{Teil 0} Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam diff --git a/buch/papers/000template/teil1.tex b/buch/papers/000template/teil1.tex index 00d3058..0f8dfae 100644 --- a/buch/papers/000template/teil1.tex +++ b/buch/papers/000template/teil1.tex @@ -3,6 +3,9 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % +% !TEX root = ../../buch.tex +% !TEX encoding = UTF-8 +% \section{Teil 1 \label{000template:section:teil1}} \rhead{Problemstellung} diff --git a/buch/papers/000template/teil2.tex b/buch/papers/000template/teil2.tex index 471adae..496557f 100644 --- a/buch/papers/000template/teil2.tex +++ b/buch/papers/000template/teil2.tex @@ -3,6 +3,9 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % +% !TEX root = ../../buch.tex +% !TEX encoding = UTF-8 +% \section{Teil 2 \label{000template:section:teil2}} \rhead{Teil 2} diff --git a/buch/papers/000template/teil3.tex b/buch/papers/000template/teil3.tex index 4697813..ef2aa75 100644 --- a/buch/papers/000template/teil3.tex +++ b/buch/papers/000template/teil3.tex @@ -3,6 +3,9 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % +% !TEX root = ../../buch.tex +% !TEX encoding = UTF-8 +% \section{Teil 3 \label{000template:section:teil3}} \rhead{Teil 3} -- cgit v1.2.1 From 61202dbd7a7762ceeee673cf27da26e47d72b966 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 16 Mar 2022 09:42:00 +0100 Subject: Kugelfunktionen --- buch/chapters/090-pde/kugel.tex | 237 +++++++++++++++++++++++++++++++++++++++- 1 file changed, 236 insertions(+), 1 deletion(-) (limited to 'buch') diff --git a/buch/chapters/090-pde/kugel.tex b/buch/chapters/090-pde/kugel.tex index c081029..d466e26 100644 --- a/buch/chapters/090-pde/kugel.tex +++ b/buch/chapters/090-pde/kugel.tex @@ -145,8 +145,243 @@ Polarkoordinaten~\eqref{buch:pde:kreis:laplace}. Der Unterschied rührt daher, dass der Laplace-Operator die Krümmung der Koordinatenlinien berücksichtigt, in diesem Fall der Meridiane. - \subsection{Separation} +In Abschnitt~\ref{buch:pde:subsection:eigenwertproblem} +wurde bereits gzeigt, wie die Wellengleichung +\[ +\frac{1}{c^2} +\frac{\partial^2 U}{\partial t^2} +-\Delta U += +0 +\] +durch Separation der Zeit auf ein Eigenwertproblem für eine +Funktion $u$ reduziert werden kann, die nur von den Ortskoordinaten +abhängt. +Es geht also nur noch darum, dass Eigenwertproblem +\[ +\Delta u = -\lambda^2 u +\] +mit geeigneten Randbedingungen zu lösen. +Dazu gehören einerseits eventuelle Gebietsränder, die im Moment +nicht interessieren. +Andererseits muss sichergestellt sein, dass die Lösungsfunktionen +stetig und differentierbar sind an den Orten, wo das Koordinatensystem +singulär ist. +So müssen $u(r,\vartheta,\varphi)$ $2\pi$-periodisch in $\varphi$ sein. +% XXX Ableitungen + +\subsubsection{Separation des radialen Anteils} +Für das Eigenwertproblem verwenden wir den Ansatz +\[ +u(r,\vartheta,\varphi) += +R(r) \Theta(\vartheta) \Phi(\varphi), +\] +den wir in die Differentialgleichung einsetzen. +So erhalten wir +\[ +\biggl(\frac{1}{r^2}R''(r)+\frac{2}{r}R'(r) \biggr) +\Theta(\vartheta)\Phi(\varphi) ++ +R(r) +\frac{1}{r^2\sin\vartheta} +\frac{\partial}{\partial\vartheta}(\sin\vartheta \Theta'(\vartheta)) +\Phi(\varphi) ++ +R(r)\Theta(\vartheta) +\frac{1}{r^2\sin\vartheta} \Phi''(\varphi) += +-\lambda^2 R(r)\Theta(\vartheta)\Phi(\varphi). +\] +Die Gleichung lässt sich nach Multiplikation mit $r^2$ und +Division durch $u$ separieren in +\begin{equation} +\frac{R''(r)+2rR'(r)+\lambda^2r^2}{R(r)} ++ +\frac{1}{\Theta(\vartheta) \sin\vartheta} +\frac{\partial}{\partial\vartheta}\sin\vartheta\Theta'(\vartheta) ++ +\frac{1}{\sin^2\vartheta}\frac{\Phi''(\varphi)}{\Phi(\varphi)} += +0 +\label{buch:pde:kugel:separiert2} +\end{equation} +Der erste Term hängt nur von $r$ ab, die anderen nur von $\vartheta$ und +$\varphi$, daher muss der erste Term konstant sein. +Damit ergbit sich für den Radialanteil die gewöhnliche Differentialgleichung +\[ +R''(r) + 2rR'(r) +\lambda^2 r^2 = \mu^2 R(r), +\] +die zum Beispiel mit der Potenzreihenmethode gelöst werden kann. +Sie kann aber durch eine geeignete Substition nochmals auf die +Laguerre-Differentialgleichung reduziert werden, wie in +Kapitel~\ref{chapter:laguerre} dargelegt wird. + +\subsubsection{Kugelflächenanteil} +Für die Separation der verbleibenden winkelabhängigen Teile muss die +Gleichung +\[ +\frac{1}{\Theta(\vartheta) \sin\vartheta} +\frac{\partial}{\partial\vartheta}\sin\vartheta\Theta'(\vartheta) ++ +\frac{1}{\sin^2\vartheta}\frac{\Phi''(\varphi)}{\Phi(\varphi)} += +-\mu^2 +\] +mit $\sin^2\vartheta$ multipliziert werden, was auf +\[ +\frac{\sin\vartheta}{\Theta(\vartheta)} +\frac{\partial}{\partial\vartheta}\sin\vartheta\Theta'(\vartheta) ++ +\frac{\Phi''(\varphi)}{\Phi(\varphi)} += +-\mu^2\sin^2\vartheta +\quad\Rightarrow\quad +\frac{\sin\vartheta}{\Theta(\vartheta)} +\frac{\partial}{\partial\vartheta}\sin\vartheta\Theta'(\vartheta) ++ +\mu^2\sin^2\vartheta += +- +\frac{\Phi''(\varphi)}{\Phi(\varphi)} +\] +führt. +Die linke Seite der letzten Gleichung hängt nur von $\vartheta$ +ab, die rechte nur von $\varphi$, beide Seiten müssen daher +konstant sein, wir bezeichnen diese Konstante mit $\alpha^2$. +So ergibt sich die Differentialgleichung +\[ +\alpha^2 += +-\frac{\Phi''(\varphi)}{\Phi(\varphi)} +\] +für die Abhängigkeit von $\varphi$, mit der allgemeinen Lösung +\[ +\Phi(\varphi) += +A\cos\alpha \varphi ++ +B\sin\alpha \varphi. +\] +Die Randbedingungen verlangen, dass $\Phi(\varphi)$ eine $2\pi$-periodische +Funktion ist, was genau dann möglich ist, wenn $\alpha=m$ ganzzahlig ist. +Damit ergibt sich für die $\vartheta$-Abhängigkeit die Differentialgleichung +\begin{equation} +\frac{\sin\vartheta}{\Theta(\vartheta)} +\frac{\partial}{\partial\vartheta}\sin\vartheta\Theta'(\vartheta) ++ +\mu^2\sin^2\vartheta += +m^2. +\label{buch:pde:kugel:eqn:thetaanteil} +\end{equation} + +\subsubsection{Abhängigkeit von $\vartheta$} +Die Differentialgleichung~\eqref{buch:pde:kugel:eqn:thetaanteil} +ist etwas unhandlich, daher verwenden wir die Substitution $z=\cos\vartheta$, +um die trigonometrischen Funktionen los zu werden. +Wegen +\[ +\frac{dz}{d\vartheta} = -\sin\vartheta =-\sqrt{1-z^2} +\] +können die Ableitungen nach $\vartheta$ auch durch Ableitungen nach $z$ +ausgedrückt werden. +Wir schreiben dazu $Z(z)=\Theta(\vartheta)$ und berechnen +\[ +\Theta'(\vartheta) += +\frac{d\Theta}{d\vartheta} += +\frac{dZ}{dz}\frac{dz}{d\vartheta} += +- +\sqrt{1-z^2} +Z'(z). +\] +Dies bedeutet auch, dass +\[ +\sin\vartheta\frac{d}{d\vartheta} += +- +(1-z^2)\frac{d}{dz}, +\] +damit lässt sich die Differentialgleichung für $\Theta(\vartheta)$ umschreiben +in eine Differentialgleichung für $Z(z)$, nämlich +\[ +(1-z^2)\frac{d}{dz}(1-z^2)\frac{d}{dz} Z(z) ++ +\mu^2 +(1-z^2) +Z(z) += +m^2 +Z(z). +\] +Indem man die Ableitung im ersten Term mit Hilfe der Produktregel +ausführt, kann man die Gleichung +\[ +(1-z^2)\biggl( +-2zZ'(z) + (1-z^2)Z''(z) +\biggr) ++ +\mu^2(1-z^2)Z(z) += +-m^2 Z(z) +\] +bekommen. +Division durch $1-z^2$ ergibt die +{\em Legendre-Differentialgleichung} +\begin{equation} +(1-z^2)Z''(z) +-2zZ'(z) ++ +\biggl( +\mu^2 - \frac{m^2}{1-z^2} +\biggr) +Z(z) += +0. +\label{buch:pde:kugel:eqn:legendre-dgl} +\end{equation} +Eine Diskussion der Lösungen dieser Differentialgleichung erfolgt im +Kapitel~\ref{chapter:kugel}. + +\subsection{Kugelfunktionen} +Die Legendre-Differentialgleichung~\eqref{buch:pde:kugel:eqn:legendre-dgl} +hat Lösungen für Werte von $\mu$ derart, dass $\mu^2=l(l+1)$ für natürliche +Zahlen $l$. +Die Lösungen sind sogar Polynome, die wir mit $P_l^{(m)}(z)$ +bezeichnen, dabei ist $m$ eine ganze Zahl mit $-l\le m\le l$. +Die Funktionen $P_l^{(m)}(\cos\vartheta)e^{im\varphi}$ +sind daher alle Lösungen des von $\vartheta$ und $\varphi$ +abhängigen Teils der Lösungen des Eigenwertproblems. +Mit einer geeigneten Normierung kann man zudem eine Familie von +bezüglich des Skalarproduktes +\[ +\langle f,g\rangle_{S^2} += +\int_{-\pi}^{\pi} +\int_{0}^{\pi} +\overline{f(\vartheta,\varphi)} +g(\vartheta,\varphi) +\sin\vartheta +\,d\vartheta +\,d\varphi +\] +orthonormiete Funktionen auf der Kugeloberfläche erhalten, die +man normalerweise als +\[ +Y_{lm}(\vartheta,\varphi) += +\frac{1}{\sqrt{2\pi}} +\sqrt{ +\frac{2l+1}{2}\cdot +\frac{(l-m)!}{(l+m)!} +} +P_{l}^{(m)}(\cos\vartheta)e^{im\varphi} +\] +bezeichnet. -- cgit v1.2.1 From fd8e4b595f3f2c4245e7ba201a727585e34cfc82 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 16 Mar 2022 14:36:37 +0100 Subject: add missing file --- buch/chapters/040-rekursion/betaverteilung.tex | 487 +++++++++++++++++++++++++ buch/chapters/090-pde/gleichung.tex | 1 + buch/chapters/090-pde/kreis.tex | 1 + buch/chapters/090-pde/kugel.tex | 1 + buch/chapters/090-pde/rechteck.tex | 1 + buch/chapters/090-pde/separation.tex | 1 + 6 files changed, 492 insertions(+) create mode 100644 buch/chapters/040-rekursion/betaverteilung.tex (limited to 'buch') diff --git a/buch/chapters/040-rekursion/betaverteilung.tex b/buch/chapters/040-rekursion/betaverteilung.tex new file mode 100644 index 0000000..979d04c --- /dev/null +++ b/buch/chapters/040-rekursion/betaverteilung.tex @@ -0,0 +1,487 @@ +% +% teil1.tex -- Beispiel-File für das Paper +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\subsection{Ordnungsstatistik und Beta-Funktion +\label{buch:rekursion:ordnung:section:ordnungsstatistik}} +\rhead{Ordnungsstatistik und Beta-Funktion} +In diesem Abschnitt ist $X$ eine Zufallsvariable mit der Verteilungsfunktion +$F_X(x)$, und $X_i$, $1\le i\le n$ sei ein Stichprobe von unabhängigen +Zufallsvariablen, die wie $X$ verteilt sind. +Ziel ist, die Verteilungsfunktion und die Wahrscheinlichkeitsdichte +des grössten, zweitgrössten, $k$-t-grössten Wertes in der Stichprobe +zu finden. +Wir schreiben $[n]=\{1,\dots,n\}$ für die Menge der natürlichen +Zahlen von zwischen $1$ und $n$. + +\subsubsection{Verteilung von $\operatorname{max}(X_1,\dots,X_n)$ und +$\operatorname{min}(X_1,\dots,X_n)$ +\label{buch:rekursion:ordnung:subsection:minmax}} +Die Verteilungsfunktion von $\operatorname{max}(X_1,\dots,X_n)$ hat +den Wert +\begin{align*} +F_{\operatorname{max}(X_1,\dots,X_n)}(x) +&= +P(\operatorname{max}(X_1,\dots,X_n) \le x) +\\ +&= +P(X_1\le x\wedge \dots \wedge X_n\le x) +\\ +&= +P(X_1\le x) \cdot \ldots \cdot P(X_n\le x) +\\ +&= +P(X\le x)^n += +F_X(x)^n. +\end{align*} +Für die Gleichverteilung ist +\[ +F_{\text{equi}}(x) += +\begin{cases} +0&\qquad x< 0 +\\ +x&\qquad 0\le x\le 1 +\\ +1&\qquad 1 X_1\wedge \dots \wedge x > X_n) +\\ +&= +1- +(1-P(x\le X_1)) \cdot\ldots\cdot (1-P(x\le X_n)) +\\ +&= +1-(1-F_X(x))^n, +\end{align*} +Im Speziellen für im Intervall $[0,1]$ gleichverteilte $X_i$ ist die +Verteilungsfunktion des Minimums +\[ +F_{\operatorname{min}(X_1,\dots,X_n)}(x) += +\begin{cases} +0 &\qquad x<0 \\ +1-(1-x)^n&\qquad 0\le x\le 1\\ +1 &\qquad 1 < x +\end{cases} +\] +mit Wahrscheinlichkeitsdichte +\[ +\varphi_{\operatorname{min}(X_1,\dots,X_n)} += +\frac{d}{dx} +F_{\operatorname{min}(X_1,\dots,X_n)} += +\begin{cases} +n(1-x)^{n-1}&\qquad 0\le x\le 1\\ +0 &\qquad \text{sonst} +\end{cases} +\] +und Erwartungswert +\begin{align*} +E(\operatorname{min}(X_1,\dots,X_n) +&= +\int_{-\infty}^\infty x\varphi_{\operatorname{min}(X_1,\dots,X_n)}(x)\,dx += +\int_0^1 x\cdot n(1-x)^{n-1}\,dx +\\ +&= +\bigl[ -x(1-x)^n \bigr]_0^1 + \int_0^1 (1-x)^n\,dx += +\biggl[ +- +\frac{1}{n+1} +(1-x)^{n+1} +\biggr]_0^1 += +\frac{1}{n+1}. +\end{align*} +Es ergibt sich daraus als natürlich Verallgemeinerung die Frage nach +der Verteilung des zweitegrössten oder zweitkleinsten Wertes unter den +Werten $X_i$. + +\subsubsection{Der $k$-t-grösste Wert} +Sie wieder $X_i$ eine Stichprobe von $n$ unabhängigen wie $X$ verteilten +Zufallsvariablen. +Diese werden jetzt der Grösse nach sortiert, die sortierten Werte werden +mit +\[ +X_{1:n} \le X_{2:n} \le \dots \le X_{(n-1):n} \le X_{n:n} +\] +bezeichnet. +Die Grössen $X_{k:n}$ sind Zufallsvariablen, sie heissen die $k$-ten +Ordnungsstatistiken. +Die in Abschnitt~\ref{buch:rekursion:ordnung:subsection:minmax} behandelten Zufallsvariablen +$\operatorname{min}(X_1,\dots,X_n)$ +und +$\operatorname{max}(X_1,\dots,X_n)$ +sind die Fälle +\begin{align*} +X_{1:n} &= \operatorname{min}(X_1,\dots,X_n) \\ +X_{n:n} &= \operatorname{max}(X_1,\dots,X_n). +\end{align*} + +Um den Wert der Verteilungsfunktion von $X_{k:n}$ zu berechnen, müssen wir +die Wahrscheinlichkeit bestimmen, dass $k$ der $n$ Werte $X_i$ $x$ nicht +übersteigen. +Der $k$-te Wert $X_{k:n}$ übersteigt genau dann $x$ nicht, wenn +mindestens $k$ der Zufallswerte $X_i$ $x$ nicht übersteigen, also +\[ +P(X_{k:n} \le x) += +P\left( +|\{i\in[n]\,|\, X_i\le x\}| \ge k +\right). +\] + +Das Ereignis $\{X_i\le x\}$ ist eine Bernoulli-Experiment, welches mit +Wahrscheinlichkeit $F_X(x)$ eintritt. +Die Anzahl der Zufallsvariablen $X_i$, die $x$ übertreffen, ist also +Binomialverteilt mit $p=F_X(x)$. +Damit haben wir gefunden, dass mit Wahrscheinlichkeit +\begin{equation} +F_{X_{k:n}}(x) += +P(X_{k:n}\le x) += +\sum_{i=k}^n \binom{n}{i}F_X(x)^i (1-F_X(x))^{n-i} +\label{buch:rekursion:ordnung:eqn:FXkn} +\end{equation} +mindestens $k$ der Zufallsvariablen den Wert $x$ überschreiten. + +\subsubsection{Wahrscheinlichkeitsdichte der Ordnungsstatistik} +Die Wahrscheinlichkeitsdichte der Ordnungsstatistik kann durch Ableitung +von \eqref{buch:rekursion:ordnung:eqn:FXkn} gefunden, werden, sie ist +\begin{align*} +\varphi_{X_{k:n}}(x) +&= +\frac{d}{dx} +F_{X_{k:n}}(x) +\\ +&= +\sum_{i=k}^n +\binom{n}{i} +\bigl( +iF_X(x)^{i-1}\varphi_X(x) (1-F_X(x))^{n-i} +- +F_X(x)^k +(n-i) +(1-F_X(x))^{n-i-1} +\varphi_X(x) +\bigr) +\\ +&= +\sum_{i=k}^n +\binom{n}{i} +\varphi_X(x) +F_X(x)^{i-1}(1-F_X(x))^{n-i-1} +\bigl( +iF_X(x)-(n-i)(1-F_X(x)) +\bigr) +\\ +&= +\varphi_X(x) +\biggl( +\sum_{i=k}^n i\binom{n}{i} F_X(x)^{i-1}(1-F_X(x))^{n-i} +- +\sum_{j=k}^n (n-j)\binom{n}{j} F_X(x)^{j}(1-F_X(x))^{n-j-1} +\biggr) +\\ +&= +\varphi_X(x) +\biggl( +\sum_{i=k}^n i\binom{n}{i} F_X(x)^{i-1}(1-F_X(x))^{n-i} +- +\sum_{i=k+1}^{n+1} (n-i+1)\binom{n}{i-1} F_X(x)^{i-1}(1-F_X(x))^{n-i} +\biggr) +\\ +&= +\varphi_X(x) +\biggl( +k\binom{n}{k}F_X(x)^{k-1}(1-F_X(x))^{n-k} ++ +\sum_{i=k+1}^{n+1} +\left( +i\binom{n}{i} +- +(n-i+1)\binom{n}{i-1} +\right) +F_X(x)^{i-1}(1-F_X(x))^{n-i} +\biggr) +\end{align*} +Mit den wohlbekannten Identitäten für die Binomialkoeffizienten +\begin{align*} +i\binom{n}{i} +- +(n-i+1)\binom{n}{i-1} +&= +n\binom{n-1}{i-1} +- +n +\binom{n-1}{i-1} += +0 +\end{align*} +folgt jetzt +\begin{align*} +\varphi_{X_{k:n}}(x) +&= +\varphi_X(x)k\binom{n}{k} F_X(x)^{k-1}(1-F_X(x))^{n-k}(x). +\intertext{Im Speziellen für gleichverteilte Zufallsvariablen $X_i$ ist +} +\varphi_{X_{k:n}}(x) +&= +k\binom{n}{k} x^{k-1}(1-x)^{n-k}. +\end{align*} +Dies ist die Wahrscheinlichkeitsdichte einer Betaverteilung +\[ +\beta(k,n-k+1)(x) += +\frac{1}{B(k,n-k+1)} +x^{k-1}(1-x)^{n-k}. +\] +Tatsächlich ist die Normierungskonstante +\begin{align} +\frac{1}{B(k,n-k+1)} +&= +\frac{\Gamma(n+1)}{\Gamma(k)\Gamma(n-k+1)} += +\frac{n!}{(k-1)!(n-k)!}. +\label{buch:rekursion:ordnung:betaverteilung:normierung1} +\end{align} +Andererseits ist +\[ +k\binom{n}{k} += +k\frac{n!}{k!(n-k)!} += +\frac{n!}{(k-1)!(n-k)!}, +\] +in Übereinstimmung mit~\eqref{buch:rekursion:ordnung:betaverteilung:normierung1}. +Die Verteilungsfunktion und die Wahrscheinlichkeitsdichte der +Ordnungsstatistik sind in Abbildung~\ref{buch:rekursion:ordnung:fig:order} dargestellt. + +\begin{figure} +\centering +\includegraphics{chapters/040-rekursion/images/order.pdf} +\caption{Verteilungsfunktion und Wahrscheinlichkeitsdichte der +Ordnungsstatistiken $X_{k:n}$ einer gleichverteilung Zuvallsvariable +mit $n=10$. +\label{buch:rekursion:ordnung:fig:order}} +\end{figure} + +% +% Die Beta-Funktion +% +\subsection{Die Beta-Verteilung +\label{buch:rekursion:subsection:beta-verteilung}} +Die Wahrscheinlichkeitsdichte, die im +Abschnitt~\ref{buch:rekursion:ordnung:section:ordnungsstatistik} +gefunden worden ist, ist nicht nur für ganzzahlige Exponenten +definiert. + +\begin{figure} +\centering +\includegraphics[width=0.92\textwidth]{chapters/040-rekursion/images/beta.pdf} +\caption{Wahrscheinlichkeitsdichte der Beta-Verteilung +$\beta(a,b,x)$ +für verschiedene Werte der Parameter $a$ und $b$. +Die Werte des Parameters für einen Graphen einer Beta-Verteilung +sind im kleinen Quadrat rechts im Graphen +als Punkt mit der gleichen Farbe dargestellt. +\label{buch:rekursion:ordnung:fig:betaverteilungn}} +\end{figure} + +\begin{definition} +Die Beta-Verteilung ist die Verteilung mit der Wahrscheinlichkeitsdichte +\[ +\beta_{a,b}(x) += +\begin{cases} +\displaystyle +\frac{1}{B(a,b)} +x^{a-1}(1-x)^{b-1}&\qquad 0\le x \le 1\\ +0&\qquad\text{sonst.} +\end{cases} +\] +\end{definition} + +Die Beta-Funktion ist also die Normierungskonstante der Beta-Verteilung. +Die wichtigsten Kennzahlen der Beta-Verteilung wie Erwartungswert und +Varianz lassen sich alle ebenfalls als Werte der Beta-Funktion ausdrücken. + +\subsubsection{Erwartungswert} +Mit der Wahrscheinlichkeitsdichte kann man jetzt auch den Erwartungswerte +der $k$-ten Ordnungsstatistik bestimmen. +Die Rechnung ergibt: +\begin{align*} +E(X_{k:n}) +&= +\int_0^1 x\cdot k\binom{n}{k} x^{k-1}(1-x)^{n-k}\,dx += +k +\binom{n}{k} +\int_0^1 +x^{k}(1-x)^{n-k}\,dx. +\intertext{Dies ist das Beta-Integral} +&= +k\binom{n}{k} +B(k+1,n-k+1) +\intertext{welches man durch Gamma-Funktionen bzw.~durch Fakultäten wie in} +&= +k\frac{n!}{k!(n-k)!} +\frac{\Gamma(k+1)\Gamma(n-k+1)}{n+2} += +k\frac{n!}{k!(n-k)!} +\frac{k!(n-k)!}{(n+1)!} += +\frac{k}{n+1} +\end{align*} +ausdrücken kann. +Die Erwartungswerte haben also regelmässige Abstände, sie sind in +Abbildung~\ref{buch:rekursion:ordnung:fig:order} als blaue vertikale Linien eingezeichnet. + +Für die Beta-Verteilung lässt sich die Rechnung noch allgemeiner +durchführen. +Der Erwartungswert einer $\beta_{a,b}$-verteilten Zufallsvariablen $X$ +ist +\begin{align*} +E(X) +&= +\int_0^1 x \beta_{a,b}(x)\,dx += +\frac{1}{B(a,b)} +\int_0^1 x\cdot x^{a-1}(1-x)^{b-1}\,dx += +\frac{B(a+1,b)}{B(a,b)} += +\frac{a}{a+b}. +\end{align*} +Durch Einsetzen von $a=k+1$ und $b=n-k+1$ lassen sich die für die +Ordnungsstatistik berechneten Werte wiederfinden. + +\subsubsection{Varianz} +Auch die Varianz lässt sich einfach berechnen, dazu muss zunächst +der Erwartungswert von $X_{k:n}^2$ bestimmt werden. +Er ist +\begin{align*} +E(X_{k:n}^2) +&= +\int_0^1 x^2\cdot k\binom{n}{k} x^{k-1}(1-x)^{n-k}\,dx += +k +\binom{n}{k} +\int_0^1 +x^{k+1}(1-x)^{n-k}\,dx. +\intertext{Auch dies ist ein Beta-Integral, nämlich} +&= +k\binom{n}{k} +B(k+2,n-k+1) += +k\frac{n!}{k!(n-k)!} +\frac{(k+1)!(n-k)!}{(n+2)!} += +\frac{k(k+1)}{(n+1)(n+2)}. +\end{align*} +Die Varianz wird damit +\begin{align} +\operatorname{var}(X_{k:n}) +&= +E(X_{k:n}^2) - E(X_{k:n})^2 +\notag +\\ +& += +\frac{k(k+1)}{(n+1)(n+2)}-\frac{k^2}{(n+1)^2} += +\frac{k(k+1)(n+1)-k^2(n+2)}{(n+1)^2(n+2)} += +\frac{k(n-k+1)}{(n+1)^2(n+2)}. +\label{buch:rekursion:ordnung:eqn:ordnungsstatistik:varianz} +\end{align} +In Abbildung~\ref{buch:rekursion:ordnung:fig:order} ist die Varianz der +Ordnungsstatistik $X_{k:n}$ für $k=7$ und $n=10$ als oranges +Rechteck dargestellt. + +Auch die Varianz kann ganz allgemein für die Beta-Verteilung +bestimmt werden. +Dazu berechnen wir zunächst +\begin{align*} +E(X^2) +&= +\frac{1}{B(a,b)} +\int_0^1 +x^2\cdot x^{a-1}(1-y)^{b-1}\,dx += +\frac{B(a+2,b)}{B(a,b)}. +\end{align*} +Daraus folgt dann +\[ +\operatorname{var}(X) += +E(X^2)-E(X)^2 += +\frac{B(a+2,b)B(a,b)-B(a+1,b)^2}{B(a,b)^2}. +\] + +Die Formel~\eqref{buch:rekursion:ordnung:eqn:ordnungsstatistik:varianz} +besagt auch, dass die Varianz der proportional ist zu $k((n+1)-k)$. +Dieser Ausdruck ist am grössten für $k=(n+1)/2$, die Varianz ist +also grösser für die ``mittleren'' Ordnungstatistiken als für die +extremen $X_{1:n}=\operatorname{min}(X_1,\dots,X_n)$ und +$X_{n:n}=\operatorname{max}(X_1,\dots,X_n)$. + diff --git a/buch/chapters/090-pde/gleichung.tex b/buch/chapters/090-pde/gleichung.tex index 7f65f06..583895d 100644 --- a/buch/chapters/090-pde/gleichung.tex +++ b/buch/chapters/090-pde/gleichung.tex @@ -5,6 +5,7 @@ % \section{Gleichungen und Randbedingungen \label{buch:pde:section:gleichungen-und-randbedingungen}} +\rhead{Gebiete, Gleichungen und Randbedingungen} \subsection{Gebiete, Differentialoperatoren, Randbedingungen} diff --git a/buch/chapters/090-pde/kreis.tex b/buch/chapters/090-pde/kreis.tex index c60fd44..a8cab3e 100644 --- a/buch/chapters/090-pde/kreis.tex +++ b/buch/chapters/090-pde/kreis.tex @@ -5,6 +5,7 @@ % \section{Kreisförmige Membran \label{buch:pde:section:kreis}} +\rhead{Kreisförmige Membran} In diesem Abschnitt soll die Differentialgleichung einer kreisförmigen Membran mit Hilfe der Separationsmethode gelöst werden. Dabei werden die Bessel-Funktionen als Lösungsfunktionen diff --git a/buch/chapters/090-pde/kugel.tex b/buch/chapters/090-pde/kugel.tex index d466e26..ee56316 100644 --- a/buch/chapters/090-pde/kugel.tex +++ b/buch/chapters/090-pde/kugel.tex @@ -5,6 +5,7 @@ % \section{Kugelfunktionen \label{buch:pde:section:kugel}} +\rhead{Kugelfunktionen} Kugelsymmetrische Probleme können oft vorteilhaft in Kugelkoordinaten beschrieben werden. Die Separationsmethode kann auf partielle Differentialgleichungen diff --git a/buch/chapters/090-pde/rechteck.tex b/buch/chapters/090-pde/rechteck.tex index 72e2806..b7dfe11 100644 --- a/buch/chapters/090-pde/rechteck.tex +++ b/buch/chapters/090-pde/rechteck.tex @@ -5,6 +5,7 @@ % \section{Rechteckige Membran \label{buch:pde:section:rechteck}} +\rhead{Rechteckige Membran} Als Beispiel für die Lösung des in Abschnitt~\ref{buch:pde:subsection:eigenwertproblem} aus der Wellengleichung abgeleiteten Eigenwertproblems diff --git a/buch/chapters/090-pde/separation.tex b/buch/chapters/090-pde/separation.tex index 6faceaa..e5e144a 100644 --- a/buch/chapters/090-pde/separation.tex +++ b/buch/chapters/090-pde/separation.tex @@ -5,6 +5,7 @@ % \section{Separationsmethode \label{buch:pde:section:separation}} +\rhead{Separationsmethode} Die Existenz der Lösung einer gewöhnlichen Differentialgleichung ist unter einigermassen milden Bedingungen in der Nähe der Anfangsbedingung garantiert. -- cgit v1.2.1 From e245386805c7bb6d7515154f2bd50aba3a7713a8 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 18 Mar 2022 08:45:05 +0100 Subject: typos --- buch/chapters/070-orthogonalitaet/gaussquadratur.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'buch') diff --git a/buch/chapters/070-orthogonalitaet/gaussquadratur.tex b/buch/chapters/070-orthogonalitaet/gaussquadratur.tex index 55f9700..b7a5643 100644 --- a/buch/chapters/070-orthogonalitaet/gaussquadratur.tex +++ b/buch/chapters/070-orthogonalitaet/gaussquadratur.tex @@ -135,7 +135,7 @@ p(x)&=x^2\colon& \frac23 &= A_0x_0^2 + A_1x_1^2\\ p(x)&=x^3\colon& 0 &= A_0x_0^3 + A_1x_1^3. \end{aligned} \] -Dividiert man die zweite und vierte Gleichung in der Form +Dividiert man die zweite und dritte Gleichung in der Form \[ \left. \begin{aligned} @@ -155,7 +155,7 @@ x_1=-x_0. \] Indem wir dies in die zweite Gleichung einsetzen, finden wir \[ -0 = A_0x_0 + A_1x_1 = A_0x_1 -A_1x_0 = (A_0-A_1)x_0 +0 = A_0x_0 + A_1x_1 = A_0x_0 -A_1x_0 = (A_0-A_1)x_0 \quad\Rightarrow\quad A_0=A_1. \] -- cgit v1.2.1 From 8ea715ed37adfbaf6369c7fbd889380857b574e0 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 18 Mar 2022 08:56:45 +0100 Subject: fix Gauss quadrature example --- buch/chapters/070-orthogonalitaet/gaussquadratur.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) (limited to 'buch') diff --git a/buch/chapters/070-orthogonalitaet/gaussquadratur.tex b/buch/chapters/070-orthogonalitaet/gaussquadratur.tex index b7a5643..acfdb1a 100644 --- a/buch/chapters/070-orthogonalitaet/gaussquadratur.tex +++ b/buch/chapters/070-orthogonalitaet/gaussquadratur.tex @@ -135,12 +135,12 @@ p(x)&=x^2\colon& \frac23 &= A_0x_0^2 + A_1x_1^2\\ p(x)&=x^3\colon& 0 &= A_0x_0^3 + A_1x_1^3. \end{aligned} \] -Dividiert man die zweite und dritte Gleichung in der Form +Dividiert man die vierte durch die zweite Gleichung in der Form \[ \left. \begin{aligned} -A_0x_0 &= -A_1x_1\\ -A_0x_0^2 &= -A_1x_1^2 +A_0x_0^3 &= -A_1x_1^3 &\qquad&\text{(vierte Gleichung)}\\ +A_0x_0 &= -A_1x_1 &\qquad&\text{(zweite Gleichung)} \end{aligned} \quad \right\} -- cgit v1.2.1 From 9b8e3f91862b5acd367cf7bbf9d52cb17563390d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 26 Mar 2022 22:08:42 +0100 Subject: add new lecture notes --- buch/chapters/110-elliptisch/images/jacobiplots.pdf | Bin 57192 -> 56975 bytes buch/chapters/110-elliptisch/images/jacobiplots.tex | 2 +- 2 files changed, 1 insertion(+), 1 deletion(-) (limited to 'buch') diff --git a/buch/chapters/110-elliptisch/images/jacobiplots.pdf b/buch/chapters/110-elliptisch/images/jacobiplots.pdf index d11bde8..88cf119 100644 Binary files a/buch/chapters/110-elliptisch/images/jacobiplots.pdf and b/buch/chapters/110-elliptisch/images/jacobiplots.pdf differ diff --git a/buch/chapters/110-elliptisch/images/jacobiplots.tex b/buch/chapters/110-elliptisch/images/jacobiplots.tex index 4fc572e..fec04fc 100644 --- a/buch/chapters/110-elliptisch/images/jacobiplots.tex +++ b/buch/chapters/110-elliptisch/images/jacobiplots.tex @@ -31,7 +31,7 @@ \fill[color=gray!50] (-0.2,1.65) rectangle (7.0,2.3); \draw[line width=0.5pt] (-0.2,-6) rectangle (7.0,2.3); \begin{scope}[scale=0.5] -\node at (6.5,{\dy+2}) {$m = #1$}; +\node at (6.5,{\dy+2}) {$k^2 = #1$}; \end{scope} } \def\jacobiplot#1#2#3#4{ -- cgit v1.2.1 From 4a49ccec57384ba582c1c132a33942c938bc1b43 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 28 Mar 2022 13:26:54 +0200 Subject: new stuff about bessel --- buch/chapters/075-fourier/2d.tex | 19 + buch/chapters/075-fourier/Makefile.inc | 2 + buch/chapters/075-fourier/bessel.tex | 620 +++++++++++++++++++++++++++++++++ buch/chapters/075-fourier/chapter.tex | 3 +- 4 files changed, 643 insertions(+), 1 deletion(-) create mode 100644 buch/chapters/075-fourier/2d.tex create mode 100644 buch/chapters/075-fourier/bessel.tex (limited to 'buch') diff --git a/buch/chapters/075-fourier/2d.tex b/buch/chapters/075-fourier/2d.tex new file mode 100644 index 0000000..cc019c7 --- /dev/null +++ b/buch/chapters/075-fourier/2d.tex @@ -0,0 +1,19 @@ +% +% 2d.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\section{Zweidimensionale Fourier-Transformation +\label{buch:fourier:section:2d}} +\rhead{Zweidimensionale Fourier-Transformation} + +\subsection{Fourier-Transformation und partielle Differentialgleichungen} + +\subsection{Fourier-Transformation in kartesischen Koordinaten} + +\subsection{Basisfunktionen in Polarkoordinaten} + + + + + diff --git a/buch/chapters/075-fourier/Makefile.inc b/buch/chapters/075-fourier/Makefile.inc index ee9641c..c153dc4 100644 --- a/buch/chapters/075-fourier/Makefile.inc +++ b/buch/chapters/075-fourier/Makefile.inc @@ -5,4 +5,6 @@ # CHAPTERFILES = $(CHAPTERFILES) \ + chapters/075-fourier/bessel.tex \ + chapters/075-fourier/2d.tex \ chapters/075-fourier/chapter.tex diff --git a/buch/chapters/075-fourier/bessel.tex b/buch/chapters/075-fourier/bessel.tex new file mode 100644 index 0000000..7e978f7 --- /dev/null +++ b/buch/chapters/075-fourier/bessel.tex @@ -0,0 +1,620 @@ +% +% bessel.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\section{Fourier-Transformation und Bessel-Funktionen +\label{buch:fourier:section:fourier-und-bessel}} +\rhead{Fourier-Transformation und Bessel-Funktionen} + +Sei $f\colon \mathbb{R}^2\to\mathbb{C}$ eine auf $\mathbb{R}$ definierte +Funktion. +Die Fourier-Transformation von $f$ ist das Integral +\begin{equation} +(\mathscr{F}f)(u,v) += +F(u,v) += +\frac{1}{2\pi} +\int_{-\infty}^\infty +\int_{-\infty}^\infty +f(x,y) e^{i(xu+yv)} +\,dx\,dy. +\label{buch:fourier:eqn:2dfourier} +\end{equation} +Die Funktionen $e_{u,v}\colon (x,y)\mapsto e^{i(xu+yv)}$ +sind die Eigenfunktionen des Laplace-Operators in kartesischen Koordinaten, +sie erfüllen +\[ +\Delta e_{u,v} = (u^2+v^2) \Delta e_{u,v}. +\] +Die Fourier-Integrale sind die Skalarprodukte +\[ +(\mathscr{F}f)(u,v) += +\langle +e_{u,v}, +f +\rangle, +\] +wobei das Skalarprodukt durch +\[ +\langle f,g\rangle += +\int_{-\infty}^\infty +\int_{-\infty}^\infty +\overline{f(x)} g(x) +\,dx\,dy +\] +definiert ist. + +Jede Funktion in der Ebene kann auch in Polarkoordinaten ausgedrückt werden. +Die kartesischen Koordinaten können mittels +\begin{align*} +x&=r\cos\varphi +y&=r\sin\varphi +\end{align*} +durch die Polarkoordinaten $(r,\varphi)$ ausgedrückt werden. +Wir schreiben +\[ +\tilde{f}(r,\varphi) += +f(r\cos\varphi,r\sin\varphi) +\] +für die Funktion $f$ ausgedrückt in Polarkoordinaten. + +In Polarkoordinaten wird das Skalarprodukt +\[ +\langle f,g\rangle += +\int_0^\infty \int_{0}^{2\pi} e^{in\varphi} +\overline{ +\tilde{f}(r,\varphi) +} +\tilde{g}(r,\varphi) +r\,dr\,d\varphi. +\] +Auch die Fouriertransformation kann jetzt durch Berechnung eines +doppelten Integrals in Polarkoordinaten ermittelt werden. +Ziel dieses Abschnitts ist zu zeigen, dass auch diese Berechnung auf +Bessel-Funktionen führt. +Im Gegenzug werden sich neue Eigenschaften und Darstellungen derselben +ergeben. + + +\subsection{Berechnung der Fourier-Transformation in Polarkoordinaten} +Die Fourier-Transformation $(\mathscr{F}f)(u,v)$ ist eine Funktion +$\mathbb{R}^2\to\mathbb{C}$, die vom Wellenvektor $(u,v)$ abhängt. +Auch dieser Vektor kann in Polarkoordinaten ausgedrückt werden. +Für die Polarkoordinaten in der Wellenvektor-Ebene soll die Bezeichnung +$(R,\vartheta)$ verwendet werden, was auf die Transformationsgleichungen +\begin{align*} +u&=R\cos\vartheta\\ +v&=R\sin\vartheta +\end{align*} +führt. +Im Exponenten der Exponentialfunktion +des Fourier-Integrals~\eqref{buch:fourier:eqn:2dfourier} +steht der Ausdruck +\[ +xu+yv += +r\cos\varphi\cdot R\cos\vartheta ++ +r\sin\varphi\cdot R\sin\vartheta += +rR\cos(\varphi-\vartheta). +\] +Mit diesen Bezeichnungen wird das +Fourier-Integral~\eqref{buch:fourier:eqn:2dfourier} +zu +\begin{align} +\tilde{F}(R,\vartheta) +&= +\frac{1}{2\pi} +\int_{0}^{\infty} +\int_{0}^{2\pi} +f(r\cos\varphi,r\sin\varphi) +e^{irR\cos(\varphi-\vartheta)} +\,d\varphi\,r\, dr +\notag +\\ +&= +\frac{1}{2\pi} +\int_{0}^{\infty} +\int_{0}^{2\pi} +\tilde{f}(r,\varphi) +e^{irR\cos(\varphi-\vartheta)} +\,d\varphi\,r\, dr. +\label{buch:fourier:eqn:fouriertrafopolar} +\end{align} +Die partielle Funktion $\varphi\mapsto \tilde{f}(r,\varphi)$ +ist eine $2\pi$-periodische Funktion, sie lässt sich also als +komplexe Fourier-Reihe +\begin{equation} +\tilde{f}(r,\varphi) += +\sum_{n\in\mathbb{Z}} \hat{f}_n(r) e^{in\varphi} +\label{buch:fourier:eqn:fourierkoef} +\end{equation} +schreiben, die Funktionen $\hat{f}_n(r)$ sind die komplexen +Fourier-Koeffizienten. +Setzt man \eqref{buch:fourier:eqn:fourierkoef} in die Fourier-Transformation +\eqref{buch:fourier:eqn:fouriertrafopolar} ein, erhält man +\begin{align*} +\tilde{F}(R,\vartheta) +&= +\sum_{n\in\mathbb{Z}} +\int_0^\infty +\hat{f}_n(r) +\frac{1}{2\pi} +\int_0^{2\pi} +e^{in\varphi+irR\cos(\varphi-\vartheta)} +\,d\varphi +\, +r\,dr. +\end{align*} +Der Exponent im inneren Integral kann als +\[ +in\varphi+irR\cos(\varphi-\vartheta) += +i(n(\varphi-\vartheta)+rR\cos(\varphi-\vartheta)) ++ +in\vartheta, +\] +oder im Integral als +\[ +\tilde{F}(R,\vartheta) += +\sum_{n\in\mathbb{Z}} +\int_0^\infty +\hat{f}_n(r) +\frac{1}{2\pi} +\int_0^{2\pi} +e^{in(\varphi-\vartheta)+irR\cos(\varphi-\vartheta)} +e^{in\vartheta} +\,d\varphi +\, +r\,dr +\] +geschrieben werden. +Der zweite Exonentialfaktor hängt nicht von $\varphi$ ab und kann daher +aus dem Integral herausgezogen werden. +Der erste Exponentialfaktor hängt nur von $\varphi-\vartheta$ ab. +Da die Exponentialfunktion $2\pi$-periodisch ist, hat die Verschiebung +um $\vartheta$ keinen Einfluss auf den Wert des Integrals. +Die Fourier-Transformation ist daher auch +\[ +\tilde{F}(R,\vartheta) += +\sum_{n\in\mathbb{Z}} +\int_0^\infty +\hat{f}_n(r) +e^{in\vartheta} +\underbrace{ +\frac{1}{2\pi} +\int_0^{2\pi} +e^{in\varphi+irR\cos\varphi} +\,d\varphi +}_{\displaystyle =:F_n(rR)} +\, +r\,dr. +\] +Die Beziehung zu den Besselfunktionen können wir daraus herstellen, +indem wir zunächst $\xi = rR$ abkürzen und dann das innere Integral +\begin{equation} +F_n(\xi) += +\frac{1}{2\pi} +\int_{0}^{2\pi} +e^{in\varphi+i\xi\cos\varphi} +\,d\varphi += +\frac{1}{2\pi} +\int_{0}^{2\pi} +e^{in\varphi}e^{i\xi\cos\varphi} +\,d\varphi +\label{buch:fourier:eqn:Fncosphi} +\end{equation} +auswerten. +Exponentialfunktion als Potenzreihe entwickeln: +\[ +F_n(\xi) += +\frac{1}{2\pi} +\int_0^{2\pi} +e^{in\varphi} +\sum_{k=0}^\infty +\frac{ +i^k\xi^k \cos^k\varphi +}{k!} +\,d\varphi += +\sum_{k=0}^\infty +\frac{i^k\xi^k}{k!} +\underbrace{ +\frac{1}{2\pi} +\int_0^{2\pi} +e^{in\varphi} +\cos^k\varphi +\,d\varphi}_{\displaystyle =c_{n,k}}. +\] +Das Integral auf der rechten Seite ist im Wesentlichen ein +Fourier-Koeffizient der Funktion $\varphi\mapsto \cos^k\varphi$. + +\subsubsection{Berechnung der Fourier-Koeffizienten von $\cos^k\varphi$} +Indem man die Kosinus-Funktion als die Linearkombination +\[ +\cos\varphi += +\frac{e^{i\varphi}+e^{-i\varphi}}2 +\] +von Exponentialfunktionen ausdrückt, kann man auch die $k$-te Potenz +mit Hilfe des binomischen Satzes als +\[ +\cos^k\varphi += +\sum_{m=0}^k +\frac{1}{2^k} +\binom{k}{m} +e^{im\varphi}e^{i(m-k)\varphi} += +\sum_{m=0}^k +\frac{1}{2^k} +\binom{k}{m} +e^{i(2m-k)\varphi} +\] +ausdrücken. +Der Fourier-Koeffizient von $\cos^k\varphi$ ist daher das Integral +\begin{align*} +c_{n,k} +&= +\frac{1}{2\pi} +\int_0^{2\pi} +e^{in\varphi}\cos^k\varphi\,d\varphi +\\ +&= +\frac{1}{2^k} +\sum_{m=0}^k +\binom{k}{m} +\frac{1}{2\pi} +\int_0^{2\pi} +e^{in\varphi}e^{i(2m-k)\varphi} +\,d\varphi +\\ +&= +\frac{1}{2^k} +\sum_{m=0}^k +\binom{k}{m} +\frac{1}{2\pi} +\int_0^{2\pi} +e^{i(2m-k+n)\varphi} +\,d\varphi. +\end{align*} +Für $2m-k+n=0$ ist das Integral ein Integral der Funktion $1$ über +ein Intervall der Länge $2\pi$, zusammen mit dem Faktor $1/2\pi$ hat +es daher den Wert $1$. +Für $2m-k+n\ne 0$ ist das Integral +\[ +\frac{1}{2\pi} +\int_0^{2\pi} +e^{i(2m-k+n)\varphi} +\,d\varphi += +\frac{1}{i} +\biggl[ +\frac{e^{i(2m-k+n)\varphi}}{2m-k+n} +\biggr]_0^{2\pi} += +0 +\] +weil die Exponentialfunktion $2\pi$-periodisch ist. +Nur für $k=2m+n$ ergibt sich ein nicht verschwindender +Fourier-Koeffizient. +Eine Summe über $k\in\mathbb{N}$ kann daher auch als Summe über +$m\in\mathbb{N}$ interpretiert werden, in der $k$ durch die Formel +$k=2m+n$ gegeben wird. +Mit dieser Konvention wird +\[ +c_{n,k} += +c_{n,2m+n} +%= +%\frac{1}{2\pi} +%\int_0^{2\pi} +%e^{-i(2m+n)\varphi} +%\cos^{2m+n}\varphi +%\,d\varphi += +\frac{1}{2^{2m+n}} +\binom{2m+n}{m} +\] +schreiben lässt. + +\subsubsection{Berechnung von $F_n(\xi)$} +Die Reihe für $F_n(\xi)$ lässt sich weiter vereinfachen. +Wir verwenden wieder die Tatsache, dass sich nur für $n=-2m-k$ +ein Beitrag ergibt. +Dies bedeutet, dass $k=2m+n$ sein muss, die Summe kann damit als +Summe über $m$ statt über $k$ geschrieben werden. +Somit ist +\begin{align*} +F_n(\xi) +&= +\sum_{k=0}^\infty +\frac{i^k\xi^k}{k!} +c_{n,k} += +\sum_{m=0}^\infty +\frac{i^{2m+n}\xi^{2m+n}}{(2m+n)!} +c_{n,2m+n} +\\ +&= +\sum_{m=0}^\infty +\frac{1}{2^{2m+n}} +\binom{2m+n}{m} +\frac{i^{2m+n}\xi^{2m+n}}{(2m+n)!} +\\ +&= +i^n +\sum_{m=0}^\infty +\frac{(-1)^m}{(2m+n)!} +\frac{(2m+n)!}{m!\,(2m+n-m)!} +\biggl(\frac{\xi}{2}\biggr)^{2m+n} +\\ +&= +i^n +\sum_{m=0}^\infty +\frac{(-1)^m} +{m!\,\Gamma(m+n+1)} +\biggl(\frac{\xi}{2}\biggr)^{2m+n} += +i^n J_n(\xi). +\end{align*} +Die Funktionen $F_n(\xi)$ sind daher bis auf einen Phasenfaktor der +Wert $J_n(\xi)$ einer Bessel-Funktion. + +\subsubsection{Berechnung der Fourier-Transformation mit Bessel-Funktionen} +Mit allen oben zusammengestellten Notationen kann die Fourier-Transformation +jetzt in Polarkoordinaten als +\[ +\tilde{F}(R,\vartheta) += +\sum_{n\in\mathbb{Z}} +e^{in\vartheta} +\int_0^\infty +\hat{f}_n(r) +i^n +J_n(rR) +r\,dr +\] +geschrieben werden. +Dies hat tatsächlich die Form eines Skalarproduktes der Funktion +$\tilde{f}(r,\varphi)$ mit einer Funktion der Form +\[ +\tilde{e}_{n,R}(r,\varphi) += +e^{in\varphi} +J_n(rR). +\] +Letzeres sind die in Abschnitt~\ref{buch:fourier:section:2d} +versprochenen Basisfunktionen. + +\subsubsection{Fourier-Reihe von $e^{i\xi\cos\varphi}$} +Die Funktionen $F_n(\xi)$ sind wegen +\[ +F_n(\xi) += +\frac{1}{2\pi} +\int_0^{2\pi} +e^{in\varphi} +e^{i\xi\cos\varphi} +\,d\varphi, +\] +daraus kann man die Fourier-Reihe von $e^{i\xi\cos\varphi}$ +berechnen, dies wird im folgenden Satz durchgeführt. + + +\begin{satz} +\label{buch:fourier:satz:expinphi} +Die komplexe Fourier-Reihe der Funktion +$\varphi\mapsto \exp(i\xi\cos\varphi)$ +ist +\begin{align} +e^{i\xi\cos\varphi} +&= +J_0(\xi) ++ +2\sum_{n=1}^\infty i^n J_n(\xi) \cos n\varphi. +\label{buch:fourier:eqn:expinphicomplex}. +\intertext{Real- und Imaginärteil davon sind die Fourier-Reihen} +\cos(\xi\cos\varphi) +&= +J_0(\xi) + 2\sum_{m=1}^\infty (-1)^m J_{2m}(\xi) \cos2m\varphi +\label{buch:fourier:eqn:expinphireal} +\\ +\sin(\xi\cos\varphi) +&= +2\sum_{m=0}^\infty (-1)^m J_{2m+1}(\xi) \cos(2m+1)\varphi. +\label{buch:fourier:eqn:expinphiimaginary} +\end{align} +\end{satz} + +\begin{proof}[Beweis] +Die Fourier-Koeffizienten $F_n(\xi)$ der Funktion $e^{i\xi\cos\varphi}$ +führen auf die Fourier-Reihe +\begin{align*} +e^{i\xi\cos\varphi} +&= +\sum_{n\in\mathbb{Z}} F_n(\xi) e^{in\varphi} += +\sum_{n\in\mathbb{Z}} i^n J_n(\xi) e^{in\varphi}. +\end{align*} +Terme mit $\pm n$ können wegen +\[ +\left. +\begin{aligned} +J_{-n}(\xi) &= (-1)^n J_n(\xi) +\\ +i^{-n}&=(-1)^n i^n +\end{aligned} +\quad +\right\} +\qquad\Rightarrow\qquad +i^{-n}J_{-n}(\xi) = i^n J_n(\xi) +\] +zusammengefasst werden, auf diese Weise erhält man +\begin{align*} +e^{i\xi\cos\varphi} +&= +J_0(\xi) ++ +\sum_{n=1}^\infty i^n J_n(\xi) (e^{in\varphi}+e^{-in\varphi}) += +2\sum_{n=1}^\infty i^n J_n(\xi) \cos n\varphi. +\end{align*} +Dies beweist +\eqref{buch:fourier:eqn:expinphicomplex}. + +Indem man Real- und Imaginärteil trennt, kann man daraus auch +die Fourier-Reihen von $\cos(\xi\cos\varphi)$ und +$\sin(\xi\cos\varphi)$ gewinnen, sie sind +\begin{align*} +\exp(\xi\cos\varphi) +&= +J_0(\xi) + 2\sum_{n=1}^\infty i^{n} J_{n}(\xi) \cos n\varphi +\\ +&= +J_0(\xi) ++ +2\sum_{m=1}^\infty i^{2m}J_{2m}(\xi)\cos 2m\varphi ++ +2\sum_{m=0}^\infty i^{2m+1}J_{2m+1}(\xi)\cos(2m+1)\varphi +\\ +&= +J_0(\xi) ++ +2\sum_{m=1}^\infty (-1)^{m}J_{2m}(\xi)\cos 2m\varphi ++ +2i\sum_{m=0}^\infty (-1)^{m}J_{2m+1}(\xi)\cos(2m+1)\varphi +\\ +\cos(\xi\cos\varphi) +&= +J_0(\xi) ++ +2\sum_{m=1}^\infty (-1)^{m}J_{2m}(\xi)\cos 2m\varphi +\\ +\sin(\xi\cos\varphi) +&= +2\sum_{m=0}^\infty (-1)^m J_{2m+1}(\xi) \cos(2m+1)\varphi. +\end{align*} +Damit sind auch die Formeln +\eqref{buch:fourier:eqn:expinphireal} +und +\eqref{buch:fourier:eqn:expinphiimaginary} +für die reellen Fourier-Reihen bewiesen. +\end{proof} + +% +% Integraldarstellung der Bessel-Funktion +% +\subsection{Integraldarstellung der Bessel-Funktion} +Aus \eqref{buch:fourier:eqn:Fncosphi} kann jetzt die Integraldarstelltung +der Bessel-Funktionen gewonnen werden. +Dazu substituiert man $\varphi$ durch $\tau$ mit +$\varphi = \frac{\pi}2-\tau$ +oder +$\tau=\frac{\pi}2-\varphi$ +und $d\tau = -d\varphi$ +im Integral und berechnet +\begin{align*} +J_n(\xi) +&= +(-i)^n +\frac{1}{2\pi} +\int_0^{2\pi} +e^{in\varphi+i\xi \cos\varphi} +\,d\varphi +\\ +&= +- +(-i)^n +\frac{1}{2\pi} +\int_{\frac{\pi}2}^{-\frac{3\pi}2} +e^{in(\frac{\pi}2-\tau) + i\xi\cos(\frac{\pi}2-\tau)} +\,d\tau +\\ +&= +(-i)^n +\frac{1}{2\pi} +\int^{\frac{\pi}2}_{-\frac{3\pi}2} +i^n +e^{-in\tau + i\xi\sin\tau)} +\,d\tau. +\intertext{Da der Integrand $2\pi$-periodisch ist, kann das +Integrationsintervall auf $[-\pi,\pi]$ verschoben werden, was} +&= +\frac{1}{2\pi} +\int_{-\pi}^{\pi} +e^{-in\tau + i\xi\sin\tau)} +\,d\tau. +\intertext{ergibt. +Das Integral kann in zwei Integrale} +&= +\frac{1}{2\pi} +\int_0^\pi +e^{-in\tau + i\xi\sin\tau} +\,d\tau ++ +\frac{1}{2\pi} +\int_0^\pi +e^{in\tau - i\xi\sin\tau} +\,d\tau +\intertext{aufgeteilt werden, +} +&= +\frac{1}{\pi} +\int_0^\pi +\frac{ +e^{-in\tau + i\xi\sin\tau} ++ +e^{in\tau - i\xi\sin\tau} +}{2} +\,d\tau +\\ +&= +\frac{1}{\pi} +\int_0^\pi +\frac{ +e^{i(-n\tau + \xi\sin\tau)} ++ +e^{-i(-n\tau + \xi\sin\tau)} +}{2} +\,d\tau +\\ +&= +\frac{1}{\pi} +\int_0^\pi +\cos(n\tau - \xi\sin\tau) +\,d\tau. +\end{align*} +Damit haben wir den folgenden Satz bewiesen: + +\begin{satz}[Integraldarstelltung der Bessel-Funktionen] +\label{buch:fourier:satz:bessel-integraldarstellung} +Die Bessel-Funktionen $J_n$ mit ganzzahliger Ordnung $n$ haben +die Integraldarstellung +\begin{equation} +J_n(\xi) += +\frac{1}{\pi} +\int_0^\pi +\cos(n\tau - \xi\sin\tau) +\,d\tau. +\label{buch:fourier:eqn:bessel-integraldarstellung} +\end{equation} +\end{satz} + + + + diff --git a/buch/chapters/075-fourier/chapter.tex b/buch/chapters/075-fourier/chapter.tex index 341d8df..681a1c0 100644 --- a/buch/chapters/075-fourier/chapter.tex +++ b/buch/chapters/075-fourier/chapter.tex @@ -13,7 +13,8 @@ führen zu neuen speziellen Funktionen. In diesem Kapitel soll als Beispiel die Fourier-Transformation der Bessel-Funktionen untersucht werden. -%\input{chapters/075-fourier/bessel.tex} +\input{chapters/075-fourier/2d.tex} +\input{chapters/075-fourier/bessel.tex} %\section{TODO} %\begin{itemize} -- cgit v1.2.1 From 45241da09f3dc8b8d700d505edea2d38a26a517c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 1 Apr 2022 17:45:58 +0200 Subject: add Vorlesung Komplexe Funktionen --- buch/chapters/110-elliptisch/Makefile.inc | 1 + buch/chapters/110-elliptisch/chapter.tex | 12 ++++++------ 2 files changed, 7 insertions(+), 6 deletions(-) (limited to 'buch') diff --git a/buch/chapters/110-elliptisch/Makefile.inc b/buch/chapters/110-elliptisch/Makefile.inc index 0ca1392..538db68 100644 --- a/buch/chapters/110-elliptisch/Makefile.inc +++ b/buch/chapters/110-elliptisch/Makefile.inc @@ -8,4 +8,5 @@ CHAPTERFILES = $(CHAPTERFILES) \ chapters/110-elliptisch/ellintegral.tex \ chapters/110-elliptisch/jacobi.tex \ chapters/110-elliptisch/lemniskate.tex \ + chapters/110-elliptisch/uebungsaufgaben/001.tex \ chapters/110-geometrie/chapter.tex diff --git a/buch/chapters/110-elliptisch/chapter.tex b/buch/chapters/110-elliptisch/chapter.tex index a03ce24..e09fa53 100644 --- a/buch/chapters/110-elliptisch/chapter.tex +++ b/buch/chapters/110-elliptisch/chapter.tex @@ -20,11 +20,11 @@ aufgebaute Integrale in dieser Familie zu finden. \input{chapters/110-elliptisch/jacobi.tex} \input{chapters/110-elliptisch/lemniskate.tex} -%\section*{Übungsaufgaben} -%\rhead{Übungsaufgaben} -%\aufgabetoplevel{chapters/020-exponential/uebungsaufgaben} -%\begin{uebungsaufgaben} +\section*{Übungsaufgaben} +\rhead{Übungsaufgaben} +\aufgabetoplevel{chapters/110-elliptisch/uebungsaufgaben} +\begin{uebungsaufgaben} %\uebungsaufgabe{0} -%\uebungsaufgabe{1} -%\end{uebungsaufgaben} +\uebungsaufgabe{1} +\end{uebungsaufgaben} -- cgit v1.2.1 From f4ce26a24fbb50621ca52316209bbffd25a60794 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 2 Apr 2022 08:38:00 +0200 Subject: add missing exercise --- buch/chapters/110-elliptisch/uebungsaufgaben/1.tex | 312 +++++++++++++++++++++ .../110-elliptisch/uebungsaufgaben/Makefile | 8 + .../uebungsaufgaben/anharmonisch.pdf | Bin 0 -> 19279 bytes .../uebungsaufgaben/anharmonisch.tex | 62 ++++ 4 files changed, 382 insertions(+) create mode 100644 buch/chapters/110-elliptisch/uebungsaufgaben/1.tex create mode 100644 buch/chapters/110-elliptisch/uebungsaufgaben/Makefile create mode 100644 buch/chapters/110-elliptisch/uebungsaufgaben/anharmonisch.pdf create mode 100644 buch/chapters/110-elliptisch/uebungsaufgaben/anharmonisch.tex (limited to 'buch') diff --git a/buch/chapters/110-elliptisch/uebungsaufgaben/1.tex b/buch/chapters/110-elliptisch/uebungsaufgaben/1.tex new file mode 100644 index 0000000..8e4b39f --- /dev/null +++ b/buch/chapters/110-elliptisch/uebungsaufgaben/1.tex @@ -0,0 +1,312 @@ +In einem anharmonische Oszillator oszilliert eine Masse $m$ unter dem +Einfluss einer Kraft, die nach dem Gesetz +\[ +F(x) = -\kappa x + \delta x^3 +\] +von der Auslenkung aus der Ruhelage abhängt. +Nehmen Sie im Folgenden an, dass $\delta >0$ ist, +dass also die rücktreibende Kraft $F(x)$ kleiner ist als bei einem +harmonischen Oszillator. +Ziel der folgenden Teilaufgaben ist, die Lösung $x(t)$ schrittweise +dadurch zu bestimmen, dass die Bewegungsgleichung in die Differentialgleichung +der Jacobischen elliptischen Funktion $\operatorname{sn}(u,k)$ umgeformt +wird. +\begin{teilaufgaben} +\item +Berechnen Sie die Auslenkung $x_0$, bei der die rücktreibende Kraft +verschwindet. +Eine beschränkte Schwingung kann diese Amplitude nicht überschreiten. +\item +Berechnen Sie die potentielle Energie in Abhängigkeit von der +Auslenkung. +\item +\label{buch:1101:basic-dgl} +Formulieren Sie den Energieerhaltungssatz für die Gesamtenergie $E$ +dieses Oszillators. +Leiten Sie daraus eine nichtlineare Differentialgleichung erster Ordnung +for den anharmonischen Oszillator ab, die sie in der Form +$\frac12m\dot{x}^2 = f(x)$ schreiben. +\item +Die Amplitude der Schwingung ist derjenige $x$-Wert, für den die +Geschwindigkeit verschwindet. +Leiten Sie die Amplitude aus der Differentialgleichung von +\ref{buch:1101:basic-dgl} ab. +Sie erhalten zwei Werte $x_{\pm}$, wobei der kleinere $x_-$ +die Amplitude einer beschränkten Schwingung beschreibt, +während die $x_+$ die minimale Ausgangsamplitude einer gegen +$\infty$ divergenten Lösung ist. +\item +Rechnen Sie nach, dass +\[ +\frac{x_+^2+x_-^2}{2} += +x_0^2 +\qquad\text{und}\qquad +x_-^2x_+^2 += +\frac{4E}{\delta}. +\] +\item +Faktorisieren Sie die Funktion $f(x)$ in der Differentialgleichung +von Teilaufgabe c) mit Hilfe der in Teilaufgabe d) bestimmten +Nullstellen $x_{\pm}^2$. +\item +Dividieren Sie die Differentialgleichung durch $x_-^2$, schreiben +Sie $X=x/x_-$ und bringen Sie die Differentialgleichung in die +Form +\begin{equation} +A \dot{X}^2 += +(1-X^2) +(1-k^2X^2), +\label{buch:1101:eqn:dgl3} +\end{equation} +wobei $k^2=x_-^2/x_+^2$ und $A$ geeignet gewählt werden müssen. +\item +\label{buch:1101:teilaufgabe:dgl3} +Verwenden Sie $t(\tau) = \alpha\tau$ +und +$Y(\tau)=X(t(\tau))$ um eine Differentialgleichung für die Funktion +$Y(\tau)$ zu gewinnen, die die Form der Differentialgleichung +von $\operatorname{sn}(u,k)$ hat, für die also $A=0$ in +\eqref{buch:1101:eqn:dgl3} ist. +\item +Verwenden Sie die Lösung $\operatorname{sn}(u,k)$ der in +\ref{buch:1101:teilaufgabe:dgl3} erhaltenen Differentialgleichung, +um die Lösung $x(t)$ der ursprünglichen Gleichung aufzuschreiben. +\end{teilaufgaben} + +\begin{loesung} +\begin{figure} +\centering +\includegraphics{chapters/110-elliptisch/uebungsaufgaben/anharmonisch.pdf} +\caption{Rechte Seite der Differentialgleichung +\eqref{buch:1101:eqn:dglf}. +Eine beschränkte Lösung bewegt sich im Bereich $xx_+$ die Kraft abstossend ist und zu einer +divergenten Lösung führt. +\label{buch:1101:fig:potential} +} +\end{figure} +\begin{teilaufgaben} +\item +Wegen +\[ +F(x) += +-\kappa x\biggl(1-\frac{\delta}{\kappa}x^2\biggr) += +-Ix +\biggl(1-\sqrt{\frac{\delta}{\kappa}}x\biggr) +\biggl(1+\sqrt{\frac{\delta}{\kappa}}x\biggr) +\] +folgt, dass die rücktreibende Kraft bei der Auslenkung $\pm x_0$ mit +\[ +x_0^2 += +\frac{\kappa}{\delta} +\qquad\text{oder}\qquad +x_0 = \sqrt{\frac{\kappa}{\delta}} +\] +verschwindet. +\item +Die potentielle Energie ist die Arbeit, die gegen die rücktreibende Kraft +geleistet wird, um die Auslenkung $x$ zu erreichen. +Sie entsteht durch Integrieren der Kraft über +das Auslenkungsinterval, also +\[ +E_{\text{pot}} += +- +\int_0^x F(\xi) \,d\xi += +\int_0^x \kappa\xi-\delta\xi^3\,d\xi += +\biggl[ +\kappa\frac{\xi^2}{2} +- +\delta +\frac{\xi^4}{4} +\biggr]_0^x += +\kappa\frac{x^2}{2} +- +\delta\frac{x^4}{4}. +\] +\item +Die kinetische Energie ist gegeben durch +\[ +E_{\text{kin}} += +\frac12m\dot{x}^2. +\] +Die Gesamtenergie ist damit +\[ +E += +\frac12m\dot{x}^2 ++ +\kappa +\frac{x^2}{2} +- +\delta +\frac{x^4}{4}. +\] +Die verlangte Umformung ergibt +\begin{align} +\frac12m\dot{x}^2 +&= +E +- +\kappa\frac{x^2}{2} ++ +\delta\frac{x^4}{4} +\label{buch:1101:eqn:dglf} +\end{align} +als Differentialgleichung für $x$. +Die Ableitung $\dot{x}$ hat positives Vorzeichen wenn die Kraft +abstossend ist und negatives Vorzeichen dort, wo die Kraft anziehend ist. +% +\item +Die Amplitude der Schwingung ist derjenige $x$-Wert, für den +die Geschwindigkeit verschwindet, also eine Lösung der Gleichung +\[ +0 += +\frac{2E}{m} -\frac{\kappa}{m}x^2 + \frac{\delta}{2m}x^4. +\] +Der gemeinsame Nenner $m$ spielt offenbar keine Rolle. +Die Gleichung hat die zwei Lösungen +\[ +x_{\pm}^2 += +\frac{\kappa \pm \sqrt{\kappa^2-4E\delta}}{\delta} += +\frac{\kappa}{\delta} +\pm +\sqrt{ +\biggl(\frac{\kappa}{\delta}\biggr)^2 +- +\frac{4E}{\delta} +}. +\] +Die Situation ist in Abbildung~\ref{buch:1101:fig:potential} +Für $x>x_+$ ist die Kraft abstossend, die Lösung divergiert. +Die Lösung mit dem negativen Zeichen $x_-$ bleibt dagegen beschränkt, +dies ist die Lösung, die wir suchen. + +\item +Die beiden Formeln ergeben sich aus den Regeln von Vieta für die +Lösungen einer quadratischen Gleichungg der Form $x^4+px^2+q$. +Die Nullstellen haben den Mittelwert $-p/2$ und das Produkt $q$. + +\item +Die rechte Seite der Differentialgleichung lässt sich mit Hilfe +der beiden Nullstellen $x_{\pm}^2$ faktorisieren und bekommt die Form +\[ +\frac12m\dot{x}^2 += +\frac{\delta}{4}(x_+^2-x^2)(x_-^2-x^2). +\] + +\item +Indem die ganze Gleichung durch $x_-^2$ dividiert wird, entsteht +\[ +\frac12m +\biggl(\frac{\dot{x}}{x_-}\biggr)^2 += +\frac{\delta}{4} +(x_+^2-x^2) +\biggl(1-\frac{x^2}{x_-^2}\biggr). +\] +Schreiben wir $X=x/x_-$ wird daraus +\[ +\frac1{2}m\dot{X}^2 += +\frac{\delta}{4} +\biggl(x_+^2-x_-^2 X^2\biggr) +(1-X^2). +\] +Durch Ausklammern von $x_+^2$ im ersten Faktor wir daraus +\[ +\frac1{2}m\dot{X}^2 += +\frac{\delta}{4} +x_+^2 +\biggl(1-\frac{x_-^2}{x_+^2} X^2\biggr) +(1-X^2). +\] +Mit der Schreibweise $k^2 = x_-^2/x_+^2$ wird die Differentialgleichung +zu +\begin{equation} +\frac{2m}{\delta x_+^2} \dot{X}^2 += +(1-X^2)(1-k^2X^2), +\label{buch:1101:eqn:dgl2} +\end{equation} +was der Differentialgleichung für die Jacobische elliptische Funktion +$\operatorname{sn}(u,k)$ bereits sehr ähnlich sieht. +\item +Bis auf den Faktor vor $\dot{X}^2$ ist +\eqref{buch:1101:eqn:dgl2} +die Differentialgleichung +von +$\operatorname{sn}(u,k)$. +Um den Faktor zum Verschwinden zu bringen, schreiben wir +$t(\tau) = \alpha\tau$. +Die Ableitung von $Y(\tau)=X(t(\tau))$ nach $\tau$ ist +\[ +\frac{dY}{d\tau} += +\dot{X}(t(\tau))\frac{dt}{d\tau} += +\alpha +\dot{X}(t(\tau)) +\qquad\Rightarrow\qquad +\frac{1}{\alpha^2}\frac{dY}{d\tau} += +\dot{X}(t(\tau)). +\] +Die Differentialgleichung für $Y(\tau)$ ist +\[ +\frac{2mk^2}{\delta x_+^2\alpha^2} +\frac{dY}{d\tau} += +(1-Y^2)(1-k^2Y^2). +\] +Der Koeffizient vor der Ableitung wird $1$, wenn man +\[ +\alpha^2 += +\frac{2mk^2}{\delta x_+^2} +\] +wählt. +Diese Differentialgleichug hat die Lösung +\[ +Y(\tau) = \operatorname{sn}(\tau,k). +\] +\item +Indem man die gefunden Grössen einsetzt kann man jetzt die Lösung +der Differentialgleichung in geschlossener Form als +\begin{align*} +x(t) +&= +x_- X(t) += +x_- \operatorname{sn}\biggl( +t\sqrt{\frac{\delta x_+^2}{2mk^2} } +,k +\biggr) +\end{align*} +Das Produkt $\delta x_+^2$ kann auch als +\[ +\delta x_+^2 += +\kappa+\sqrt{\kappa -4\delta E} +\] +geschrieben werden. +\qedhere +\end{teilaufgaben} +\end{loesung} + + diff --git a/buch/chapters/110-elliptisch/uebungsaufgaben/Makefile b/buch/chapters/110-elliptisch/uebungsaufgaben/Makefile new file mode 100644 index 0000000..0ca5234 --- /dev/null +++ b/buch/chapters/110-elliptisch/uebungsaufgaben/Makefile @@ -0,0 +1,8 @@ +# +# Makefile +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +anharmonisch.pdf: anharmonisch.tex + pdflatex anharmonisch.tex diff --git a/buch/chapters/110-elliptisch/uebungsaufgaben/anharmonisch.pdf b/buch/chapters/110-elliptisch/uebungsaufgaben/anharmonisch.pdf new file mode 100644 index 0000000..4b00f4d Binary files /dev/null and b/buch/chapters/110-elliptisch/uebungsaufgaben/anharmonisch.pdf differ diff --git a/buch/chapters/110-elliptisch/uebungsaufgaben/anharmonisch.tex b/buch/chapters/110-elliptisch/uebungsaufgaben/anharmonisch.tex new file mode 100644 index 0000000..a00c393 --- /dev/null +++ b/buch/chapters/110-elliptisch/uebungsaufgaben/anharmonisch.tex @@ -0,0 +1,62 @@ +% +% anharmonisch.tex -- Potential einer anharmonischen Schwingung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\def\E{3} +\def\K{0.2} +\def\D{0.0025} + +\pgfmathparse{sqrt(\K/\D)} +\xdef\xnull{\pgfmathresult} + +\pgfmathparse{sqrt((\K+sqrt(\K*\K-4*\E*\D))/\D)} +\xdef\xplus{\pgfmathresult} +\pgfmathparse{sqrt((\K-sqrt(\K*\K-4*\E*\D))/\D)} +\xdef\xminus{\pgfmathresult} + +\def\xmax{13} + +\fill[color=darkgreen!20] (0,-1.5) rectangle (\xminus,4.7); +\node[color=darkgreen] at ({0.5*\xminus},4.7) [below] {anziehende Kraft\strut}; + +\fill[color=orange!20] (\xplus,-1.5) rectangle (\xmax,4.7); +\node[color=orange] at ({0.5*(\xplus+\xmax)},4.7) [below] {abstossende\strut}; +\node[color=orange] at ({0.5*(\xplus+\xmax)},4.3) [below] {Kraft\strut}; + +\node[color=gray] at (\xnull,4.7) [below] {verbotener Bereich\strut}; + +\draw (-0.1,\E) -- (0.1,\E); +\node at (-0.1,\E) [left] {$E$}; + +\draw[color=red,line width=1pt] + plot[domain=0:13,samples=100] + ({\x},{\E-(0.5*\K-0.25*\D*\x*\x)*\x*\x}); + +\draw[->] (-0.1,0) -- ({\xmax+0.3},0) coordinate[label={$x$}]; +\draw[->] (0,-1.5) -- (0,5) coordinate[label={right:$f(x)$}]; + +\fill[color=blue] (\xminus,0) circle[radius=0.08]; +\node[color=blue] at (\xminus,0) [below left] {$x_-\mathstrut$}; + +\fill[color=blue] (\xplus,0) circle[radius=0.08]; +\node[color=blue] at (\xplus,0) [below right] {$x_+\mathstrut$}; + +\fill[color=blue] (\xnull,0) circle[radius=0.08]; +\node[color=blue] at (\xnull,0) [below] {$x_0\mathstrut$}; + +\end{tikzpicture} +\end{document} + -- cgit v1.2.1 From 670555039265d83945b0d3e205aefb020425585b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Patrik=20M=C3=BCller?= Date: Wed, 6 Apr 2022 08:00:09 +0200 Subject: Start definition.tex --- buch/.gitignore | 37 ++ buch/papers/laguerre/definition.tex | 150 +++++-- buch/papers/laguerre/images/wasserstoff_model.tex | 58 +++ buch/papers/laguerre/main.tex | 4 +- buch/papers/laguerre/packages.tex | 1 - buch/papers/laguerre/scripts/gamma_approx.ipynb | 431 +++++++++++++++++++++ buch/papers/laguerre/scripts/laguerre_plot.py | 39 ++ .../laguerre/scripts/lanczos_approximation.py | 47 +++ buch/papers/laguerre/scripts/quadrature_gama.py | 178 +++++++++ buch/papers/laguerre/wasserstoff.tex | 147 ++++++- buch/standalone.tex | 35 ++ 11 files changed, 1077 insertions(+), 50 deletions(-) create mode 100644 buch/.gitignore create mode 100644 buch/papers/laguerre/images/wasserstoff_model.tex create mode 100644 buch/papers/laguerre/scripts/gamma_approx.ipynb create mode 100644 buch/papers/laguerre/scripts/laguerre_plot.py create mode 100644 buch/papers/laguerre/scripts/lanczos_approximation.py create mode 100644 buch/papers/laguerre/scripts/quadrature_gama.py create mode 100644 buch/standalone.tex (limited to 'buch') diff --git a/buch/.gitignore b/buch/.gitignore new file mode 100644 index 0000000..4d056e5 --- /dev/null +++ b/buch/.gitignore @@ -0,0 +1,37 @@ +*.acn +*.acr +*.alg +*.aux +*.bbl +*.blg +*.dvi +*.fdb_latexmk +*.glg +*.glo +*.gls +*.idx +*.ilg +*.ind +*.ist +*.lof +*.log +*.lot +*.maf +*.mtc +*.mtc0 +*.nav +*.nlo +*.out +*.pdfsync +.vscode/* +*.fls +*.xdv +*.ps +*.snm +*.synctex.gz +*.toc +*.vrb +*.xdy +*.tdo +*-blx.bib +*.synctex \ No newline at end of file diff --git a/buch/papers/laguerre/definition.tex b/buch/papers/laguerre/definition.tex index 5f6d8bd..84a26cf 100644 --- a/buch/papers/laguerre/definition.tex +++ b/buch/papers/laguerre/definition.tex @@ -6,43 +6,133 @@ \section{Definition \label{laguerre:section:definition}} \rhead{Definition} - +Die Laguerre-Differentialgleichung ist gegeben durch \begin{align} - x y''(x) + (1 - x) y'(x) + n y(x) - = - 0 - \label{laguerre:dgl} +x y''(x) + (1 - x) y'(x) + n y(x) += +0 +, \quad +n \in \mathbb{N}_0 +, \quad +x \in \mathbb{R} +. +\label{laguerre:dgl} \end{align} - +Zur Lösung der Gleichung \eqref{laguerre:dgl} +verwenden wir einen Potenzreihenansatz. +Setzt man nun den Ansatz +\begin{align*} +y(x) +&= +\sum_{k=0}^\infty a_k x^k +\\ +y'(x) +& = +\sum_{k=1}^\infty k a_k x^{k-1} += +\sum_{k=0}^\infty (k+1) a_{k+1} x^k +\\ +y''(x) +&= +\sum_{k=2}^\infty k (k-1) a_k x^{k-2} += +\sum_{k=1}^\infty (k+1) k a_{k+1} x^{k-1} +\end{align*} +in die Differentialgleichung ein, erhält man: +\begin{align*} +\sum_{k=1}^\infty (k+1) k a_{k+1} x^k ++ \sum_{k=0}^\infty (k+1) a_{k+1} x^k +- \sum_{k=0}^\infty k a_k x^k ++ n \sum_{k=0}^\infty a_k x^k +&= +0\\ +\sum_{k=0}^\infty +\left[ (k+1) k a_{k+1} + (k+1) a_{k+1} - k a_k + n a_k \right] x^k +&= +0. +\end{align*} +Daraus lässt sich die Rekursionsbeziehung +\begin{align*} +a_{k+1} +&= +\frac{k-n}{(k+1) ^ 2} a_k +\end{align*} +ableiten. +Für ein konstantes $n$ erhalten wir als Potenzreihenlösung ein Polynom vom Grad $n$, +denn für $k=n$ wird $a_{n+1} = 0$ und damit auch $a_{n+2}=a_{n+3}=\ldots=0$. +Aus der Rekursionsbeziehung ist zudem ersichtlich, +dass $a_0 \neq 0$ beliebig gewählt werden kann. +Wählen wir nun $c_0 = 1$, dann folgt für die Koeffizienten $a_1, a_2, a_3$ +\begin{align*} +a_1 += +-\frac{n}{1^2} +,&& +a_2 += +\frac{(n-1)n}{1^2 2^2} +,&& +a_3 += +-\frac{(n-2)(n-1)n}{1^2 2^2 3^2} +\end{align*} +und allgemein +\begin{align*} +k&\leq n: +& +a_k +&= +(-1)^k \frac{n!}{(n-k)!} \frac{1}{(k!)^2} += +\frac{(-1)^k}{k!} +\begin{pmatrix} +n +\\ +k +\end{pmatrix} +\\ +k&>n: +& +a_k +&= +0. +\end{align*} +Somit haben wir die Laguerre-Polynome $L_n(x)$ erhalten: \begin{align} - L_n(x) - = - \sum_{k=0}^{n} - \frac{(-1)^k}{k!} - \begin{pmatrix} - n \\ - k - \end{pmatrix} - x^k - \label{laguerre:polynom} +L_n(x) += +\sum_{k=0}^{n} +\frac{(-1)^k}{k!} +\begin{pmatrix} +n \\ +k +\end{pmatrix} +x^k +\label{laguerre:polynom} \end{align} +\subsection{Assoziierte Laguerre-Polynome +\label{laguerre:subsection:assoz_laguerre} +} \begin{align} - x y''(x) + (\alpha + 1 - x) y'(x) + n y(x) - = - 0 - \label{laguerre:generell_dgl} +x y''(x) + (\alpha + 1 - x) y'(x) + n y(x) += +0 +\label{laguerre:generell_dgl} \end{align} \begin{align} - L_n^\alpha (x) - = - \sum_{k=0}^{n} - \frac{(-1)^k}{k!} - \begin{pmatrix} - n + \alpha \\ - n - k - \end{pmatrix} - x^k - \label{laguerre:polynom} +L_n^\alpha (x) += +\sum_{k=0}^{n} +\frac{(-1)^k}{k!} +\begin{pmatrix} +n + \alpha \\ +n - k +\end{pmatrix} +x^k +\label{laguerre:polynom} \end{align} + +% https://www.math.kit.edu/iana1/lehre/hm3phys2012w/media/laguerre.pdf +% http://www.physics.okayama-u.ac.jp/jeschke_homepage/E4/kapitel4.pdf diff --git a/buch/papers/laguerre/images/wasserstoff_model.tex b/buch/papers/laguerre/images/wasserstoff_model.tex new file mode 100644 index 0000000..fe838c3 --- /dev/null +++ b/buch/papers/laguerre/images/wasserstoff_model.tex @@ -0,0 +1,58 @@ +\documentclass{standalone} + +\usepackage{pgfplots} +\usepackage{tikz-3dplot} + +\tdplotsetmaincoords{60}{115} +\pgfplotsset{compat=newest} + +\begin{document} + +\newcommand{\drawcircle}[4]{ +\shade[ball color=#3, opacity=#4] (#1) circle (#2 cm); +\tdplotsetrotatedcoords{0}{0}{0}; +\draw[dashed, tdplot_rotated_coords, #3!40!black] (#1) circle (#2); +} + +\begin{tikzpicture}[tdplot_main_coords, scale = 2] +\def\r{1.0} +\def\rp{0.2} +\def\rn{0.05} +\def\rvec{1.0} +\def\thetavec{45} +\def\phivec{60} + +\coordinate (O) at (0, 0, 0); +\tdplotsetcoord{P}{\rvec}{\thetavec}{\phivec} + +% Labels +\node[inner sep=1pt] at (0, -4.0*\rp, 1.0*\r) (plabel){Proton}; +\draw (plabel) -- (O); +\node[inner sep=1pt] at (-0.*\r, 1.0*\r, 1.3*\r) (elabel){Elektron}; +\draw (elabel) -- (P); +% Draw proton +\drawcircle{O}{\rp}{red}{1.0} + +% Draw spherical coordinates of electron +\draw (O) -- node[anchor=north west, yshift=4pt]{$r$} (P); +\draw[dashed] (O) -- (Pxy); +\draw[dashed] (P) -- (Pxy); +\tdplotdrawarc{(O)}{0.6}{0}{\phivec}{anchor=north}{$\varphi$} +\tdplotsetthetaplanecoords{\phivec} +\tdplotdrawarc[tdplot_rotated_coords]{(0,0,0)}{0.5}{0}% +{\thetavec}{anchor=south west, xshift=-2pt, yshift=-2pt}{$\vartheta$} + +% Draw electron +\drawcircle{P}{\rn}{blue}{1.0} + +% Draw surrounding sphere +\drawcircle{O}{\r}{gray}{0.3} + +% Draw cartesian coordinate system +\draw[-stealth, thick] (O) -- (1.8*\r,0,0) node[below left] {$x$}; +\draw[-stealth, thick] (O) -- (0,1.3*\r,0) node[below right] {$y$}; +\draw[-stealth, thick] (O) -- (0,0,1.3*\r) node[above] {$z$}; + +\end{tikzpicture} + +\end{document} \ No newline at end of file diff --git a/buch/papers/laguerre/main.tex b/buch/papers/laguerre/main.tex index 1fe0f8b..3db67d5 100644 --- a/buch/papers/laguerre/main.tex +++ b/buch/papers/laguerre/main.tex @@ -13,8 +13,8 @@ Hier kommt eine Einleitung. \input{papers/laguerre/definition} \input{papers/laguerre/eigenschaften} \input{papers/laguerre/quadratur} -\input{papers/laguerre/transformation} -\input{papers/laguerre/wasserstoff} +% \input{papers/laguerre/transformation} +% \input{papers/laguerre/wasserstoff} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/laguerre/packages.tex b/buch/papers/laguerre/packages.tex index ab55228..4ebc172 100644 --- a/buch/papers/laguerre/packages.tex +++ b/buch/papers/laguerre/packages.tex @@ -7,4 +7,3 @@ % if your paper needs special packages, add package commands as in the % following example \usepackage{derivative} - diff --git a/buch/papers/laguerre/scripts/gamma_approx.ipynb b/buch/papers/laguerre/scripts/gamma_approx.ipynb new file mode 100644 index 0000000..9a1fee6 --- /dev/null +++ b/buch/papers/laguerre/scripts/gamma_approx.ipynb @@ -0,0 +1,431 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Gauss-Laguerre Quadratur für die Gamma-Funktion\n", + "\n", + "$$\n", + " \\Gamma(z)\n", + " = \n", + " \\int_0^\\infty t^{z-1}e^{-t}dt\n", + "$$\n", + "\n", + "$$\n", + " \\int_0^\\infty f(x) e^{-x} dx \n", + " \\approx \n", + " \\sum_{i=1}^{N} f(x_i) w_i\n", + " \\qquad\\text{ wobei }\n", + " w_i = \\frac{x_i}{(n+1)^2 [L_{n+1}(x_i)]^2}\n", + "$$\n", + "und $x_i$ sind Nullstellen des Laguerre Polynoms $L_n(x)$\n", + "\n", + "Der Fehler ist gegeben als\n", + "\n", + "$$\n", + " E \n", + " =\n", + " \\frac{(n!)^2}{(2n)!} f^{(2n)}(\\xi) \n", + " = \n", + " (-2n + z)_{2n} \\frac{(n!)^2}{(2n)!} \\xi^{z - 2n - 1}\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from cmath import exp, pi, sin, sqrt\n", + "import scipy.special\n", + "\n", + "EPSILON = 1e-07\n" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "lanczos_p = [\n", + " 676.5203681218851,\n", + " -1259.1392167224028,\n", + " 771.32342877765313,\n", + " -176.61502916214059,\n", + " 12.507343278686905,\n", + " -0.13857109526572012,\n", + " 9.9843695780195716e-6,\n", + " 1.5056327351493116e-7,\n", + "]\n", + "\n", + "\n", + "def drop_imag(z):\n", + " if abs(z.imag) <= EPSILON:\n", + " z = z.real\n", + " return z\n", + "\n", + "\n", + "def lanczos_gamma(z):\n", + " z = complex(z)\n", + " if z.real < 0.5:\n", + " y = pi / (sin(pi * z) * lanczos_gamma(1 - z)) # Reflection formula\n", + " else:\n", + " z -= 1\n", + " x = 0.99999999999980993\n", + " for (i, pval) in enumerate(lanczos_p):\n", + " x += pval / (z + i + 1)\n", + " t = z + len(lanczos_p) - 0.5\n", + " y = sqrt(2 * pi) * t ** (z + 0.5) * exp(-t) * x\n", + " return drop_imag(y)\n" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "zeros = np.array(\n", + " [\n", + " 1.70279632305101000e-1,\n", + " 9.03701776799379912e-1,\n", + " 2.25108662986613069e0,\n", + " 4.26670017028765879e0,\n", + " 7.04590540239346570e0,\n", + " 1.07585160101809952e1,\n", + " 1.57406786412780046e1,\n", + " 2.28631317368892641e1,\n", + " ]\n", + ")\n", + "\n", + "weights = np.array(\n", + " [\n", + " 3.69188589341637530e-1,\n", + " 4.18786780814342956e-1,\n", + " 1.75794986637171806e-1,\n", + " 3.33434922612156515e-2,\n", + " 2.79453623522567252e-3,\n", + " 9.07650877335821310e-5,\n", + " 8.48574671627253154e-7,\n", + " 1.04800117487151038e-9,\n", + " ]\n", + ")\n", + "\n", + "\n", + "def pochhammer(z, n):\n", + " return np.prod(z + np.arange(n))\n", + "\n", + "\n", + "def find_shift(z, target):\n", + " factor = 1.0\n", + " steps = int(np.floor(target - np.real(z)))\n", + " zs = z + steps\n", + " if steps > 0:\n", + " factor = 1 / pochhammer(z, steps)\n", + " elif steps < 0:\n", + " factor = pochhammer(zs, -steps)\n", + " return zs, factor\n", + "\n", + "\n", + "def laguerre_gamma(z, x, w, target=11):\n", + " # res = 0.0\n", + " z = complex(z)\n", + " if z.real < 1e-3:\n", + " res = pi / (\n", + " sin(pi * z) * laguerre_gamma(1 - z, x, w, target)\n", + " ) # Reflection formula\n", + " else:\n", + " z_shifted, correction_factor = find_shift(z, target)\n", + " res = np.sum(x ** (z_shifted - 1) * w)\n", + " res *= correction_factor\n", + " res = drop_imag(res)\n", + " return res\n" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "def eval_laguerre(x, target=12):\n", + " return np.array([laguerre_gamma(xi, zeros, weights, target) for xi in x])\n", + "\n", + "\n", + "def eval_lanczos(x):\n", + " return np.array([lanczos_gamma(xi) for xi in x])\n", + "\n", + "\n", + "def eval_mean_laguerre(x, targets):\n", + " return np.mean([eval_laguerre(x, target) for target in targets], 0)\n", + "\n", + "\n", + "def calc_rel_error(x, y):\n", + " return (y - x) / x\n", + "\n", + "\n", + "def evaluate(x, target=12):\n", + " lanczos_gammas = eval_lanczos(x)\n", + " laguerre_gammas = eval_laguerre(x, target)\n", + " rel_error = calc_rel_error(lanczos_gammas, laguerre_gammas)\n", + " return rel_error\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Test with real values" + ] + }, + { + "cell_type": "code", + "execution_count": 60, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "targets = np.arange(8, 15)\n", + "mean_targets = ((11, 12),)\n", + "x = np.linspace(EPSILON, 1 - EPSILON, 101)\n", + "_, axs = plt.subplots(\n", + " 2, sharex=True, clear=True, constrained_layout=True, figsize=(12, 12)\n", + ")\n", + "\n", + "lanczos = eval_lanczos(x)\n", + "for mean_target in mean_targets:\n", + " vals = eval_mean_laguerre(x, mean_target)\n", + " rel_error_mean = calc_rel_error(lanczos, vals)\n", + " axs[0].plot(x, rel_error_mean, label=mean_target)\n", + " axs[1].semilogy(x, np.abs(rel_error_mean), label=mean_target)\n", + "\n", + "for target in targets:\n", + " rel_error = evaluate(x, target)\n", + " axs[0].plot(x, rel_error, label=target)\n", + " axs[1].semilogy(x, np.abs(rel_error), label=target)\n", + "# axs[0].set_ylim(*(np.array([-1, 1]) * 3.5e-8))\n", + "\n", + "axs[0].set_xlim(x[0], x[-1])\n", + "axs[1].set_ylim(1e-10, 2e-7)\n", + "for ax in axs:\n", + " ax.legend()\n", + " ax.grid(which=\"both\")\n" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(-7.5, 25.0)" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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", 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", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "targets = (11, 12)\n", + "xmax = 15\n", + "x = np.linspace(-xmax + EPSILON, xmax - EPSILON, 1000)\n", + "\n", + "mean_lag = eval_mean_laguerre(x, targets)\n", + "lanczos = eval_lanczos(x)\n", + "rel_error = calc_rel_error(lanczos, mean_lag)\n", + "rel_error_simple = evaluate(x, targets[-1])\n", + "# rel_error = evaluate(x, target)\n", + "\n", + "_, axs = plt.subplots(\n", + " 2, sharex=True, clear=True, constrained_layout=True, figsize=(12, 12)\n", + ")\n", + "axs[0].plot(x, rel_error, label=targets)\n", + "axs[1].semilogy(x, np.abs(rel_error), label=targets)\n", + "axs[0].plot(x, rel_error_simple, label=targets[-1])\n", + "axs[1].semilogy(x, np.abs(rel_error_simple), label=targets[-1])\n", + "axs[0].set_xlim(x[0], x[-1])\n", + "# axs[0].set_ylim(*(np.array([-1, 1]) * 4.2e-8))\n", + "axs[1].set_ylim(1e-10, 5e-8)\n", + "for ax in axs:\n", + " ax.legend()\n", + "\n", + "x2 = np.linspace(-5 + EPSILON, 5, 4001)\n", + "_, ax = plt.subplots(constrained_layout=True, figsize=(8, 6))\n", + "ax.plot(x2, eval_mean_laguerre(x2, targets))\n", + "ax.set_xlim(x2[0], x2[-1])\n", + "ax.set_ylim(-7.5, 25)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Test with complex values" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "targets = (11, 12)\n", + "vals = np.linspace(-5 + EPSILON, 5, 100)\n", + "x, y = np.meshgrid(vals, vals)\n", + "mesh = x + 1j * y\n", + "input = mesh.flatten()\n", + "\n", + "mean_lag = eval_mean_laguerre(input, targets).reshape(mesh.shape)\n", + "lanczos = eval_lanczos(input).reshape(mesh.shape)\n", + "rel_error = np.abs(calc_rel_error(lanczos, mean_lag))\n", + "\n", + "lag = eval_laguerre(input, targets[-1]).reshape(mesh.shape)\n", + "rel_error_simple = np.abs(calc_rel_error(lanczos, lag))\n", + "# rel_error = evaluate(x, target)\n", + "\n", + "fig, axs = plt.subplots(\n", + " 2,\n", + " 2,\n", + " sharex=True,\n", + " sharey=True,\n", + " clear=True,\n", + " constrained_layout=True,\n", + " figsize=(12, 10),\n", + ")\n", + "_c = axs[0, 1].pcolormesh(x, y, np.log10(np.abs(lanczos - mean_lag)), shading=\"gouraud\")\n", + "_c = axs[0, 0].pcolormesh(x, y, np.log10(np.abs(lanczos - lag)), shading=\"gouraud\")\n", + "fig.colorbar(_c, ax=axs[0, :])\n", + "_c = axs[1, 1].pcolormesh(x, y, np.log10(rel_error), shading=\"gouraud\")\n", + "_c = axs[1, 0].pcolormesh(x, y, np.log10(rel_error_simple), shading=\"gouraud\")\n", + "fig.colorbar(_c, ax=axs[1, :])\n", + "_ = axs[0, 0].set_title(\"Absolute Error\")\n", + "_ = axs[1, 0].set_title(\"Relative Error\")\n" + ] + }, + { + "cell_type": "code", + "execution_count": 67, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "# TODO: Macht kei Sinn!\n", + "n = 8\n", + "ms = np.arange(4, 5)\n", + "xi = np.linspace(EPSILON, 20, 201)[:, None]\n", + "z = np.arange(6, 16)[None]+0.1\n", + "c = scipy.special.factorial(n) ** 2 / scipy.special.factorial(2 * n)\n", + "\n", + "\n", + "_, ax = plt.subplots(clear=True, constrained_layout=True, figsize=(12, 8))\n", + "ax.grid(1)\n", + "for m, color in zip(ms, ['r', 'b', 'g', 'c', 'm', 'y']):\n", + " e = np.abs(\n", + " scipy.special.poch(z - 2 * n, 2 * n)\n", + " / scipy.special.poch(z - m, m)\n", + " * c\n", + " * xi ** (z - 2 * n + m - 1)\n", + " )\n", + " # ax.semilogy(xi, e, color=color)\n", + " ax.semilogy(xi, e)\n", + " ax.set_xticks(np.arange(xi[-1] +1))\n", + " ax.set_ylim(1e-8, 1e5)\n", + " _ = ax.legend([f'z={zi}' for zi in z[0]])\n" + ] + } + ], + "metadata": { + "interpreter": { + "hash": "767d51c1340bd893661ea55ea3124f6de3c7a262a8b4abca0554b478b1e2ff90" + }, + "kernelspec": { + "display_name": "Python 3.8.10 64-bit", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.10" + }, + "orig_nbformat": 4 + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/buch/papers/laguerre/scripts/laguerre_plot.py b/buch/papers/laguerre/scripts/laguerre_plot.py new file mode 100644 index 0000000..cd90df1 --- /dev/null +++ b/buch/papers/laguerre/scripts/laguerre_plot.py @@ -0,0 +1,39 @@ +#!/usr/bin/env python3 +# -*- coding:utf-8 -*- +"""Some plots for Laguerre Polynomials.""" + +from pathlib import Path + +import matplotlib.pyplot as plt +import numpy as np +import scipy.special as ss + +N = 1000 +t = np.linspace(0, 12.5, N)[:, None] +root = str(Path(__file__).parent) + +fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(6, 4)) +for n in np.arange(0, 10): + k = np.arange(0, n + 1)[None] + L = np.sum((-1) ** k * ss.binom(n, k) / ss.factorial(k) * t ** k, -1) + ax.plot(t, L, label=f"n={n}") +ax.set_xticks(np.arange(1, t[-1])) +ax.set_xlim(t[0], t[-1] + 0.1*(t[1] - t[0])) +ax.set_ylim(-20, 20) +ax.legend(ncol=2) +# set the x-spine +ax.spines['left'].set_position('zero') +ax.spines['right'].set_visible(False) +ax.spines['bottom'].set_position('zero') +ax.spines['top'].set_visible(False) +ax.xaxis.set_ticks_position('bottom') +ax.yaxis.set_ticks_position('left') + +# make arrows +# ax.plot((1), (0), ls="", marker=">", ms=10, color="k", +# transform=ax.get_yaxis_transform(), clip_on=False) +# ax.plot((0), (1), ls="", marker="^", ms=10, color="k", +# transform=ax.get_xaxis_transform(), clip_on=False) +# ax.grid(1) +fig.savefig(f'{root}/laguerre_polynomes.pdf') +# plt.show() diff --git a/buch/papers/laguerre/scripts/lanczos_approximation.py b/buch/papers/laguerre/scripts/lanczos_approximation.py new file mode 100644 index 0000000..3c48266 --- /dev/null +++ b/buch/papers/laguerre/scripts/lanczos_approximation.py @@ -0,0 +1,47 @@ +from cmath import exp, pi, sin, sqrt + +p = [ + 676.5203681218851, + -1259.1392167224028, + 771.32342877765313, + -176.61502916214059, + 12.507343278686905, + -0.13857109526572012, + 9.9843695780195716e-6, + 1.5056327351493116e-7, +] + +EPSILON = 1e-07 + + +def drop_imag(z): + if abs(z.imag) <= EPSILON: + z = z.real + return z + + +def gamma(z): + z = complex(z) + if z.real < 0.5: + y = pi / (sin(pi * z) * gamma(1 - z)) # Reflection formula + else: + z -= 1 + x = 0.99999999999980993 + for (i, pval) in enumerate(p): + x += pval / (z + i + 1) + t = z + len(p) - 0.5 + y = sqrt(2 * pi) * t ** (z + 0.5) * exp(-t) * x + return drop_imag(y) + + +""" +The above use of the reflection (thus the if-else structure) is necessary, even though +it may look strange, as it allows to extend the approximation to values of z where +Re(z) < 0.5, where the Lanczos method is not valid. +""" + +print(gamma(1)) +print(gamma(5)) +print(gamma(0.5)) +print(gamma(0.5* (1 + 1j))) +print(gamma(-0.5)) diff --git a/buch/papers/laguerre/scripts/quadrature_gama.py b/buch/papers/laguerre/scripts/quadrature_gama.py new file mode 100644 index 0000000..37a9cd8 --- /dev/null +++ b/buch/papers/laguerre/scripts/quadrature_gama.py @@ -0,0 +1,178 @@ +#!/usr/bin/env python3 +# -*- coding:utf-8 -*- +"""Use Gauss-Laguerre quadrature to calculate gamma function.""" +# import sympy +from cmath import exp, pi, sin, sqrt + +import matplotlib.pyplot as plt +import numpy as np +import scipy.special as ss + +p = [ + 676.5203681218851, + -1259.1392167224028, + 771.32342877765313, + -176.61502916214059, + 12.507343278686905, + -0.13857109526572012, + 9.9843695780195716e-6, + 1.5056327351493116e-7, +] + +EPSILON = 1e-07 + + +def drop_imag(z): + if abs(z.imag) <= EPSILON: + z = z.real + return z + + +def gamma(z): + z = complex(z) + if z.real < 0.5: + y = pi / (sin(pi * z) * gamma(1 - z)) # Reflection formula + else: + z -= 1 + x = 0.99999999999980993 + for (i, pval) in enumerate(p): + x += pval / (z + i + 1) + t = z + len(p) - 0.5 + y = sqrt(2 * pi) * t ** (z + 0.5) * exp(-t) * x + return drop_imag(y) + + +zeros = np.array( + [ + 3.22547689619392312e-1, + 1.74576110115834658e0, + 4.53662029692112798e0, + 9.39507091230113313e0, + ], + np.longdouble, +) +weights = np.array( + [ + 6.03154104341633602e-1, + 3.57418692437799687e-1, + 3.88879085150053843e-2, + 5.39294705561327450e-4, + ], + np.longdouble, +) + +zeros = np.array( + [ + 1.70279632305101000e-1, + 9.03701776799379912e-1, + 2.25108662986613069e0, + 4.26670017028765879e0, + 7.04590540239346570e0, + 1.07585160101809952e1, + 1.57406786412780046e1, + 2.28631317368892641e1, + ], + np.longdouble, +) + +weights = np.array( + [ + 3.69188589341637530e-1, + 4.18786780814342956e-1, + 1.75794986637171806e-1, + 3.33434922612156515e-2, + 2.79453623522567252e-3, + 9.07650877335821310e-5, + 8.48574671627253154e-7, + 1.04800117487151038e-9, + ], + np.longdouble, +) + + +def calc_gamma(z, n, x, w): + res = 0.0 + z = complex(z) + for xi, wi in zip(x, w): + res += xi ** (z + n - 1) * wi + for i in range(int(n)): + res /= z + i + res = drop_imag(res) + return res + +small = 1e-3 +Z = np.linspace(small, 1-small, 101) + +# Z = [-3/2, -1/2, 1/2, 3/2] +# target = +# targets = np.array([gamma(z) for z in Z]) +targets1 = ss.gamma(Z) +targets2 = np.array([gamma(z) for z in Z]) +approxs = np.array([calc_gamma(z, 11, zeros, weights) for z in Z]) +rel_error1 = np.abs(targets1 - approxs) / targets1 +rel_error2 = np.abs(targets2 - approxs) / targets2 + +_, axs = plt.subplots(2, num=1, clear=True, constrained_layout=True) +axs[0].plot(Z, rel_error1) +axs[1].semilogy(Z, rel_error1) +axs[0].plot(Z, rel_error2) +axs[1].semilogy(Z, rel_error2) +axs[1].semilogy(Z, np.abs(targets1-targets2)/targets1) +plt.show() +# values = np.array([calc_gamma]) +# _ = [ +# print( +# n, +# [ +# float( +# f"{np.abs((calc_gamma(z, n, zeros, weights) - gamma(z)) / gamma(z)):.3g}" +# ) +# for z in Z +# ], +# ) +# for n in range(21) +# ] + + +# target = ss.gamma(z) +# target = np.sqrt(np.pi) + +# _, ax = plt.subplots(num=1, clear=True, constrained_layout=True) +# for i, degree in enumerate(degrees): +# samples_points, weights = np.polynomial.laguerre.laggauss(degree) +# values = np.sum( +# samples_points[:, None] ** (z + shifts[None] - 1) * weights[:, None], 0 +# ) / ss.poch(z, shifts) +# # print(np.abs(target - values)) +# print(values) +# ax.plot(shifts, values, label=f"N={degree}") +# ax.legend() +# plt.show() + + +# def count_equal_digits(x, y): +# for i in range(1, 13): +# try: +# np.testing.assert_almost_equal(x, y, i) +# except AssertionError: +# break +# return i + + +# Z = np.linspace(1.0, 11.0, 11) +# # degrees = [2, 4, 8, 16, 32, 64, 100] +# d = 100 +# X = np.zeros(len(Z)) +# for i, z in enumerate(Z): +# samples_points, weights = np.polynomial.laguerre.laggauss(d) +# X[i] = np.sum(samples_points ** (z - 1) * weights) +# # X[i] = np.sum(np.sin(z * samples_points) * weights) +# Y = ss.gamma(Z) +# # Y = Z / (Z ** 2 + 1) +# ed = [count_equal_digits(x, y) for x, y in zip(X, Y)] +# for x,y in zip(X,Y): +# print(x,y) + +# _, ax = plt.subplots(num=1, clear=True, constrained_layout=True) +# ax.plot(Z, ed) +# plt.show() diff --git a/buch/papers/laguerre/wasserstoff.tex b/buch/papers/laguerre/wasserstoff.tex index caaa6af..0da8be3 100644 --- a/buch/papers/laguerre/wasserstoff.tex +++ b/buch/papers/laguerre/wasserstoff.tex @@ -6,24 +6,137 @@ \section{Radialer Schwingungsanteil eines Wasserstoffatoms \label{laguerre:section:radial_h_atom}} +Das Wasserstoffatom besteht aus einem Proton im Kern +mit Masse $M$ und Ladung $+e$. +Ein Elektron mit Masse $m$ und Ladung $-e$ umkreist das Proton +(vgl. Abbildung~\ref{laguerre:fig:wasserstoff_model}). +Für das folgende Model werden folgende Annahmen getroffen: + +\begin{figure} +\centering +\includegraphics{papers/laguerre/images/wasserstoff_model.pdf} +\caption{Skizze eines Wasserstoffatoms. +Kartesische, wie auch Kugelkoordinaten sind eingezeichnet. +} +\label{laguerre:fig:wasserstoff_model} +\end{figure} + +\begin{enumerate} +\item +Das Elektron wird als nicht-relativistisches Teilchen betrachtet, +das heisst, +relativistische Effekte sind vernachlässigbar. +\item +Der Spin des Elektrons und des Protons +und das damit verbundene magnetische Moment +wird vernachlässigt. +\item +Fluktuationen des Vakuums werden nicht berücksichtigt. +\item +Wechselwirkung zwischen Elektron und Proton +ist durch die Coulombwechselwirkung gegeben. +Somit entspricht die potentielle Energie der Coulombenergie $V_C(r)$ +und nimmt damit die folgende Form an +\begin{align} + V_C(r) + = + -\frac{e^2}{4 \pi \epsilon_0 r} + \text{ mit } + r + = + \lvert\vec{r}\rvert + = + \sqrt{x^2 + y^2 + z^2} + . + \label{laguerre:coulombenergie} +\end{align} +Im Falle das der Kern einen endlichen Radius $r_0$ besitzt, +ist die $1/r$-Abhängigkeit in Gleichung \eqref{laguerre:coulombenergie} +als Näherung zu betrachten. +Diese Näherung darf nur angewendet werden, wenn die +Aufenthaltswahrscheinlicheit des Elektrons +innerhalb $r_0$ vernachlässigbar ist. +Für das Wasserstoffatom ist diese Näherung für alle Zustände gerechtfertigt. +\item +Da $M \gg m$, kann das Proton als in Ruhe angenommen werden. +\end{enumerate} + +\subsection{Herleitung zeitunabhängige Schrödinger-Gleichung} +\label{laguerre:subsection:herleitung_schroedinger} +Das Problem ist kugelsymmetrisch, +darum transformieren wir das Problem in Kugelkoordinaten. +Somit gilt: + +\begin{align*} + r + & = + \sqrt{x^2 + y^2 + z^2}\\ + \vartheta + & = + \arccos\left(\frac{z}{r}\right)\\ + \varphi + & = + \arctan\left(\frac{y}{x}\right) +\end{align*} + +Die potentielle Energie $V_C(r)$ hat keine direkte Zeitabhängigkeit. +Daraus folgt, dass die konstant ist Gesamtenergie $E$ +und es existieren stationäre Zustände + \begin{align} - \nonumber - - \frac{\hbar^2}{2m} - & - \left( - \frac{1}{r^2} \pdv{}{r} - \left( r^2 \pdv{}{r} \right) - + - \frac{1}{r^2 \sin \vartheta} \pdv{}{\vartheta} - \left( \sin \vartheta \pdv{}{\vartheta} \right) - + - \frac{1}{r^2 \sin^2 \vartheta} \pdv[2]{}{\varphi} - \right) - u(r, \vartheta, \varphi) - \\ - & - - \frac{e^2}{4 \pi \epsilon_0 r} u(r, \vartheta, \varphi) + \psi(r, \vartheta, \varphi, t) + = + u(r, \vartheta, \varphi) e^{-i E t / h}, +\end{align} +wobei $u(r, \vartheta, \varphi)$ +die zeitunabhängige Schrödinger-Gleichung erfüllt. + +\begin{align} + -\frac{\hbar^2}{2m} \Delta u(r, \vartheta, \varphi) + + V_C(r) u(r, \vartheta, \varphi) = E u(r, \vartheta, \varphi) - \label{laguerre:pdg_h_atom} + \label{laguerre:schroedinger} +\end{align} + +Für Kugelkoordinaten hat der Laplace-Operator $\Delta$ die Form + +\begin{align} + \Delta + = + \frac{1}{r^2} \pdv{}{r} \left( r^2 \pdv{}{r} \right) + + \frac{1}{r^2 \sin\vartheta} \pdv{}{\vartheta} + \left(\sin\vartheta \pdv{}{\vartheta}\right) + + \frac{1}{r^2 \sin^2\vartheta} \pdv[2]{}{\varphi} + \label{laguerre:laplace_kugel} \end{align} + +Setzt man nun +\eqref{laguerre:coulombenergie} und \eqref{laguerre:laplace_kugel} +in \eqref{laguerre:schroedinger} ein, +erhält man die zeitunabhängige Schrödinger-Gleichung für Kugelkoordinaten + +\begin{align} +\nonumber +- \frac{\hbar^2}{2m} +& +\left( +\frac{1}{r^2} \pdv{}{r} +\left( r^2 \pdv{}{r} \right) ++ +\frac{1}{r^2 \sin \vartheta} \pdv{}{\vartheta} +\left( \sin \vartheta \pdv{}{\vartheta} \right) ++ +\frac{1}{r^2 \sin^2 \vartheta} \pdv[2]{}{\varphi} +\right) +u(r, \vartheta, \varphi) +\\ +& - +\frac{e^2}{4 \pi \epsilon_0 r} u(r, \vartheta, \varphi) += +E u(r, \vartheta, \varphi). +\label{laguerre:pdg_h_atom} +\end{align} + +\subsection{Separation der Schrödinger-Gleichung} +\label{laguerre:subsection:seperation_schroedinger} diff --git a/buch/standalone.tex b/buch/standalone.tex new file mode 100644 index 0000000..e681460 --- /dev/null +++ b/buch/standalone.tex @@ -0,0 +1,35 @@ +% +% packages.tex -- common packages +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass{book} +\def\IncludeBookCover{0} +\input{common/packages.tex} +% additional packages used by the individual papers, add a line for +% each paper +\input{papers/common/addpackages.tex} + +% workaround for biblatex bug +\makeatletter +\def\blx@maxline{77} +\makeatother +\addbibresource{chapters/references.bib} + +% Bibresources for each article +\input{papers/common/addbibresources.tex} + +% make sure the last index starts on an odd page +\AtEndDocument{\clearpage\ifodd\value{page}\else\null\clearpage\fi} +\makeindex + +%\pgfplotsset{compat=1.12} +\setlength{\headheight}{15pt} % fix headheight warning +\DeclareGraphicsRule{*}{mps}{*}{} + +\begin{document} + \input{common/macros.tex} + \def\chapterauthor#1{{\large #1}\bigskip\bigskip} + + \input{papers/laguerre/main} +\end{document} \ No newline at end of file -- cgit v1.2.1 From 77e1d7652711e18ab381f0eaf2059675689d7304 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 9 Apr 2022 12:48:35 +0200 Subject: Singularitaeten --- buch/chapters/050-differential/besselhyper.maxima | 37 ++ buch/chapters/080-funktionentheorie/Makefile.inc | 1 + .../chapters/080-funktionentheorie/anwendungen.tex | 1 + .../080-funktionentheorie/images/operator-1.pdf | Bin 0 -> 6228 bytes .../080-funktionentheorie/images/operator.mp | 46 +++ .../080-funktionentheorie/singularitaeten.tex | 427 +++++++++++++++++++++ 6 files changed, 512 insertions(+) create mode 100644 buch/chapters/050-differential/besselhyper.maxima create mode 100644 buch/chapters/080-funktionentheorie/images/operator-1.pdf create mode 100644 buch/chapters/080-funktionentheorie/images/operator.mp create mode 100644 buch/chapters/080-funktionentheorie/singularitaeten.tex (limited to 'buch') diff --git a/buch/chapters/050-differential/besselhyper.maxima b/buch/chapters/050-differential/besselhyper.maxima new file mode 100644 index 0000000..0a67819 --- /dev/null +++ b/buch/chapters/050-differential/besselhyper.maxima @@ -0,0 +1,37 @@ +/* + * besselhyper.maxima + */ +gradef(y(x), yp(x)); +gradef(yp(x), ypp(x)); + +w(x) := x^alpha * y(-x^2/4); + +/* Zusammenhang zwischen Y und W */ +Y: x^(-alpha) * W; + +/* erste Ableitung Yp ausgedrückt durch W und W' */ +e: Wp=diff(w(x),x) $ +e: ratsimp(e); +e: subst(W * x^(-alpha), y(-x^2/4), e) $ +e: subst(Yp, yp(-x^2/4), e) $ +s: solve(e, Yp) $ +Yp: rhs(s[1]) $ +Yp: ratsimp(Yp); +ratsimp(subst(0,W,Yp)); +ratsimp(subst(0,Wp,Yp)); + +/* zweite Ableitung Yp ausgedrückt durch W, W' und W'' */ +e: Wpp = ratsimp(diff(diff(w(x),x),x)); +e: subst(W * x^(-alpha), y(-x^2/4), e) $ +e: subst(Yp, yp(-x^2/4), e) $ +e: subst(Ypp, ypp(-x^2/4), e) $ +e: ratsimp(e) $ +Ypp: rhs(solve(e, Ypp)[1]) $ +Ypp: ratsimp(Ypp); +ratsimp(subst(0, W, subst(0, Wp, Ypp))); +ratsimp(subst(0, W, subst(0, Wpp, Ypp))); +ratsimp(subst(0, Wp, subst(0, Wpp, Ypp))); + + +B: (-x^2/4) * Ypp + (alpha+1)*Yp - Y; +expand(-x^(alpha+2) * B); diff --git a/buch/chapters/080-funktionentheorie/Makefile.inc b/buch/chapters/080-funktionentheorie/Makefile.inc index 813865f..affaa94 100644 --- a/buch/chapters/080-funktionentheorie/Makefile.inc +++ b/buch/chapters/080-funktionentheorie/Makefile.inc @@ -12,6 +12,7 @@ CHAPTERFILES = $(CHAPTERFILES) \ chapters/080-funktionentheorie/anwendungen.tex \ chapters/080-funktionentheorie/gammareflektion.tex \ chapters/080-funktionentheorie/carlson.tex \ + chapters/080-funktionentheorie/singularitaeten.tex \ chapters/080-funktionentheorie/uebungsaufgaben/1.tex \ chapters/080-funktionentheorie/uebungsaufgaben/2.tex \ chapters/080-funktionentheorie/chapter.tex diff --git a/buch/chapters/080-funktionentheorie/anwendungen.tex b/buch/chapters/080-funktionentheorie/anwendungen.tex index e02fb3e..4cdf9be 100644 --- a/buch/chapters/080-funktionentheorie/anwendungen.tex +++ b/buch/chapters/080-funktionentheorie/anwendungen.tex @@ -8,3 +8,4 @@ \input{chapters/080-funktionentheorie/gammareflektion.tex} \input{chapters/080-funktionentheorie/carlson.tex} +\input{chapters/080-funktionentheorie/singularitaeten.tex} diff --git a/buch/chapters/080-funktionentheorie/images/operator-1.pdf b/buch/chapters/080-funktionentheorie/images/operator-1.pdf new file mode 100644 index 0000000..4ba1346 Binary files /dev/null and b/buch/chapters/080-funktionentheorie/images/operator-1.pdf differ diff --git a/buch/chapters/080-funktionentheorie/images/operator.mp b/buch/chapters/080-funktionentheorie/images/operator.mp new file mode 100644 index 0000000..35f4303 --- /dev/null +++ b/buch/chapters/080-funktionentheorie/images/operator.mp @@ -0,0 +1,46 @@ +% +% operatormp -- Seitz-Kull-Operator in Metapost +% +% (c) 2016 Prof Dr Andreas Mueller, Hochschule Rapperswil +% +verbatimtex +\documentclass{book} +\usepackage{times} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsfonts} +\usepackage{txfonts} +\begin{document} +etex; + +beginfig(1) + +label(btex $A$ etex, (0,0)); + +path circle; + +numeric r; +r := 4.7; +numeric b; +b := 0.45; + +circle := r * (cosd(40), sind(40)); + +for alpha = 41 step 1 until 370: + circle := circle--(r * (cosd(alpha), sind(alpha))); +endfor; + +path head; +head := (0,0)--(5,-3)--(0,6)--(-5,-3)--cycle; + +z1 = (-0.3,-0.4); + +pickup pencircle scaled b; +draw circle shifted z1; +fill head scaled 0.2 rotated 10 shifted (r,0) rotated 10 shifted z1; + +endfig; + +end + + diff --git a/buch/chapters/080-funktionentheorie/singularitaeten.tex b/buch/chapters/080-funktionentheorie/singularitaeten.tex new file mode 100644 index 0000000..71d1844 --- /dev/null +++ b/buch/chapters/080-funktionentheorie/singularitaeten.tex @@ -0,0 +1,427 @@ +% +% singularitaeten.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\newcommand*\sk{\vcenter{\hbox{\includegraphics[scale=0.8]{chapters/080-funktionentheorie/images/operator-1.pdf}}}} + +\subsection{Lösungen von linearen Differentialgleichungen mit Singularitäten +\label{buch:funktionentheorie:subsection:dglsing}} +Die Potenzreihenmethode hat ermöglicht, mindestens eine Lösung gewisser +linearer Differentialgleichungen zu finden. +Bei Differentialgleichungen wie der Besselschen Differentialgleichung, +deren Koeffizienten Singularitäten aufweisen, konnte aber nur eine +Lösung gefunden werden, während die Theorie verlangt, dass eine +Differentialgleichung zweiter Ordnung zwei linear unabhängige Lösungen +haben muss. + +Ziel dieses Abschnitts ist zu zeigen, warum dies nicht möglich war und +wie diese Schwierigkeit mit Hilfe der analytischen Fortsetzung überwunden +werden kann. + +\subsubsection{Differentialgleichungen mit Singularitäten} +Mit der Besselschen +Differentialgleichung~\eqref{buch:differentialgleichungen:eqn:bessel} +ist es nicht möglich, die zweite Ableitung $y''(0)$ an der Stelle $x=0$ +zu bestimmen. +Die Differentialgleichung kann an der Stelle $x=0$ nicht nach $y''$ +aufgelöst werden. +Wenn man die Differentialgleichung in ein Differntialgleichungssystem +\[ +\frac{d}{dx} +\begin{pmatrix} +y_1\\y_2 +\end{pmatrix} += +\begin{pmatrix} +0&1\\ +1-\frac{\alpha^2}{x^2} +& +-\frac{1}{x} +\end{pmatrix} +\begin{pmatrix} +y_1\\y_2 +\end{pmatrix} +\] +erster Ordnung umwandelt, zeigt sich an der Stelle $x=0$ eine +Singularität in der Matrix, die Ableitung kann also für $x=0$ +nicht bestimmt werden. +In einer Umgebung von $x=0$ erfüllt die Differentialgleichung +die Voraussetzungen bekannter Existenz- und Eindeutigkeitssätze +für gewöhnliche Differentialgleichungen nicht. + +Ein ähnliches Problem tritt bei jeder hypergeometrischen +Differentialgleichung auf. +Diese werden gemäss Abschnitt +\ref{buch:differentialgleichungen:section:hypergeometrisch} +aus den Differentialoperatoren +\[ +D_a=z\frac{d}{dz} + a +\] +zusammengesetzt. +Die Ableitung höchster Ordnung eines Produktes solcher Operationen ist +\[ +D_{a_1} +\cdots +D_{a_p} += +z^p\frac{d^p}{dz^p} + \text{Ableitungen niedrigerer Ordnung}. +\] +Dies zeigt, dass für $p>0$ oder $q>0$ ein Faktor $x$ bei der +Ableitung höchster Ordnung unvermeidlich ist, die Differentialgleichung +kann also wieder nicht nach dieser Ableitung aufgelöst werden und +erfüllt die Voraussetzungen der Existenz- und Eindeutigkeitssätze +in einer Umgebung von $x=0$ wieder nicht. + +Die Besselsche Differentialgleichung +hat auch nicht die Form $y''+p(x)xy'+q(x)=0$, die der Theorie der +Indexgleichung zugrunde lag. +Daher kann es auch keine Garantie geben, dass die Methode der +verallgemeinerten Potenzreihen zwei linear unabhängige Lösungen +liefern kann. +Tatsächlich wurde für ganzzahlige $n$ wegen $J_n(x) = (-1)^n J_{-n}(x)$ +nur eine Lösung statt der erwarteten zwei linear unabhängigen +Lösungen gefunden. + +Sind die Koeffizienten einer linearen Differentialgleichungen wie +in den genannten Beispielen singulär bei $x=0$, kann man auch nicht +erwarten, dass die Lösungen singulär sind. +Dies war schliesslich die Motivation, einen Lösungsansatz mit einer +verallgemeinerten Potenzreihe zu versuchen. +Mit den Funktion $x^\varrho$ lässt sich bereits eine recht grosse +Klasse von Singularitäten beschreiben, aber es ist nicht klar, +welche weiteren Arten von Singularitäten berücksichtigt werden sollten. +Dies soll im Folgenden geklärt werden. + +\subsubsection{Der Lösungsraum einer Differentialgleichung zweiter Ordnung} +Eine Differentialgleichung $n$-ter Ordnung hat lokal einen $n$-dimensionalen +Vektorraum als Lösungsraum. + +\begin{definition} +Sei +\begin{equation} +\sum_{k=0}^n a_k(x) y^{(n)}(x) = 0 +\label{buch:funktionentheorie:singularitaeten:eqn:defdgl} +\end{equation} +eine Differentialgleichung $n$-ter Ordnung mit analytischen Koeffizienten +und $x_0\in \mathbb{C}$. +Dann ist +\[ +\mathbb{L}_{x_0} += +\left\{ +y(x) +\;\left|\; +\begin{minipage}{6cm} +$y$ ist Lösung der Differentialgleichung +\eqref{buch:funktionentheorie:singularitaeten:eqn:defdgl} +in einer Umgebung von $x_0$ +\end{minipage} +\right. +\right\} +\] +der Lösungsraum der Differentialgleichung +\eqref{buch:funktionentheorie:singularitaeten:eqn:defdgl}. +Wenn der Punkt $x_0$ aus dem Kontext klar ist, kann er auch weggelassen +werden: $\mathbb{L}_{x_0}=\mathbb{L}$. +\end{definition} + +\subsubsection{Analytische Fortsetzung auf einem Weg um $0$} +Die betrachteten Differentialgleichungen haben holomorphe +Koeffizienten, Lösungen der Differentialgleichung lassen sich +daher immer in die komplexe Ebene fortsetzen, solange man die +Singularitäten der Koeffizienten vermeidet. +Hat eine Funktion $y(z)$ eine Laurent-Reihe +\[ +y(z) = \sum_{k=-\infty}^\infty a_kz^k, +\] +dann ist sie automatisch in einer Umgebung von $0$ definiert +ausser in $0$. +Die analytische Fortsetzung entlang eines Pfades, der $0$ +umschliesst, ist die Funktion $y(z)$ selbst. + +Für die Wurzelfunktion $y(z)=z^{\frac1n}$ ist dies nicht möglich. +Die analytische Fortsetzung von $\sqrt[n]{x}$ auf der positiven reellen +Achse entlang einer Kurve, die $0$ umschliesst, +produziert die Funktion +\[ +\sqrt[n]{z} += +\sqrt[n]{re^{i\varphi}} += +\sqrt[n]{r}e^{i\frac{\varphi}n}, +\] +die für $\varphi=2\pi$ zu $e^{i\frac{2\pi}n}\sqrt{x}$ wird. +Verallgemeinerte Potenzreihen als Lösungen zeigen daher, dass +die analytische Fortsetzung der Lösung entlang eines Pfades um +eine Singularität nicht mit der Lösung übereinstimmen muss. +Das Studium dieser analytischen Fortsetzung dürfte daher zusätzliche +Informationen über die Lösung hervorbringen. + +\begin{definition} +Der {\em Fortsetzungsoperator} $\sk$ ist der lineare Operator, der eine +in einem Punkt $x\in\mathbb{R}^+$ analytische Funktion $f(x)$ entlang eines +geschlossenen Weges fortsetzt, der $0$ im Gegenuhrzeigersinn umläuft. +Die Einschränkung der analytischen Fortsetzung auf $\mathbb{R}^+$ wird +mit $\sk f(x)$ bezeichnet. +\end{definition} + +Die obengenannten Beispiele lassen sich mit dem Operator $\sk$ als +\[ +\begin{aligned} +\sk z^n +&= +z^n +&\qquad& n \in \mathbb{Z} +\\ +\sk +\sum_{k=-\infty}^\infty a_kz^k +&= +\sum_{k=-\infty}^\infty a_kz^k +\\ +\sk z^\varrho +&= +e^{2\pi i\varrho} z^\varrho +\end{aligned} +\] +schreiben. + +\subsubsection{Rechenregeln für die analytische Fortsetzung} +Der Operator $\sk$ ist ein Algebrahomomorphismus, d.~h.~für zwei analytische +Funktionen $f$ und $g$ gilt +\[ +\begin{aligned} +\sk(\lambda f + \mu g) +&= +\lambda \sk f + \mu \sk g +\\ +\sk(fg) +&= +(\sk f)(\sk g) +\end{aligned} +\] +für beliebige $\lambda,\mu\in\mathbb{C}$. +Ist $f$ eine in ganz $\mathbb{C}$ holomorphe Funktion, dann lässt sie +sich mit Hilfe einer Potenzreihe berechnen. +Der Wert $f(g(z))$ entsteht durch Einsetzen von $g(z)$ in die Potenzreihe. +Analytische Fortsetzung mit $\sk$ reproduziert jeden einzelnen Term +der Potenzreihe, es folgt +$\sk f(g(z)) = f(\sk g(z))$. +Ebenso folgt auch, dass der Operator $\sk$ mit der Ableitung +vertauscht, dass also +\[ +\frac{d^n}{dz^n}(\sk f) += +\sk(f^{(n)}). +\] + + +\subsubsection{Analytische Fortsetzung von Lösungen einer Differentialgleichung} +Wir untersuchen jetzt die Wirkung des Operators $\sk$ auf +den Lösungsraum $\mathbb{L}$ einer Differentialgleichung mit +analytischen Koeffizienten, die in einer Umgebung von $0$ +definiert sind. +Auf den Koeffizienten wirkt $\sk$ als die Identität. +Ist $y(x)$ eine Lösung der Differentialgleichung, dann gilt +\[ +0 += +\sk\biggl( +\sum_{k=0}^n a_k(x) y^{(n)}(x) +\biggr) += +\sum_{k=0}^n (\sk a_k)(x) \cdot (\sk y)^{(n)}(x) += +\sum_{k=0}^n a_k(x) \cdot (\sk y)^{(n)}(x), +\] +somit ist $\sk y$ ebenfalls eine Lösung. +Wir schliessen daraus, dass $\sk$ eine lineare Abbildung +$\mathbb{L}\to\mathbb{L}$ ist. + +Der Lösungsraum einer Differentialgleichung $n$-ter Ordnung +ist $n$-dimensional. +Nach Wahl einer Basis des Lösungsraums kann der Operator $\sk$ +mit Hilfe einer Matrix $A\in M_{n\times n}(\mathbb{C})$ beschrieben werden. +Sei $\mathscr{W}=\{w_1,\dots,w_n\}$ eine Basis des Lösungsraums, dann +kann $\sk w_j$ wieder eine Lösung der Differentialgleichung +und kann daher geschrieben werden als Linearkombination +\begin{equation} +\sk w_j += +\sum_{k=1}^n +a_{jk} w_k +\end{equation} +der Funktionen in $\mathscr{W}$. + +Die Matrix $A$ mit den Einträgen $a_{jk}$ kann durch Wahl einer +geeigneten Basis in besonders einfache Form gebracht. +Wir führen diese Diskussion im folgenden nur für eine Differentialgleichung +zweiter Ordnung $n=2$. + + +\subsubsection{Fall $A$ diagonalisierbar: verallgemeinerte Potenzreihen} +In diesem Fall kann man die Lösungsfunktionen $w_1$ und $w_2$ so +wählen, dass die Matrix +\[ +A=\begin{pmatrix}\lambda_1&0\\0&\lambda_2\end{pmatrix} +\] +diagonal wird mit Eigenwerten $\lambda_j$, $j=1,2$. +Dies bedeutet, dass $\sk w_j = \lambda_j w_j$. +Wir schreiben +\[ +\varrho_j = \frac{1}{2\pi i} \log\lambda_j. +\] +Der Logarithmus ist nicht eindeutig, er ist nur bis auf ein Vielfaches +von $2\pi i$ bestimmt. +Folglich aus auch $\varrho_j$ nicht eindeutig bestimmt, eine +andere Wahl des Logarithmus ändert $\varrho_j$ aber um eine ganze Zahl. + +Die Funktion $z^{\varrho_j}$ wird unter der Wirkung von $\sk$ zu +\[ +\sk z^{\varrho_j} += +e^{2\pi i\varrho_j} z^{\varrho_j} += +e^{\log \lambda_j} z^{\varrho_j} += +\lambda_j z^{\varrho_j}. +\] +Auf den Funktionen $z^{\varrho_j}$ und $w_j$ wirkt der Operator $\sk$ +also die gleich durch Multiplikation mit $\lambda_j$. +Deren Quotient +\[ +f(z) = \frac{w_j(z)}{z^{\varrho_j}} +\qquad\text{erfüllt}\qquad +\sk f += +\frac{\sk w_j}{\sk z^{\varrho_j}} += +\frac{\lambda_j w_j}{\lambda_j z^{\varrho_j}} += +\frac{w_j}{z^{\varrho_j}} += +f. +\] +Die Funktion $f$ kann daher als Laurent-Reihe +\[ +f(z) += +\sum_{k=-\infty}^\infty a_kz^k +\] +geschrieben werden. +Die Lösung $w_2(z)$ muss daher die Form +\begin{equation} +w_j(z) += +z^{\varrho_j} f(z) += +z^{\varrho_j} \sum_{k=-\infty}^\infty a_kz^k +\end{equation} +haben, also die einer verallgemeinerten Potenzreihe. +Auch hier zeigt sich, dass die Wahl des Logarithmus in der Definition +von $\varrho_j$ unbedeutend ist, sie äussert sich nur in einer +Verschiebung der Koeffizienten $a_k$. + +Falls der Operator $\sk$ also diagonalisierbar ist, dann gibt es +zwei linear unabhängige Lösungen der Differentialgleichung in der +Form einer verallgemeinerten Potenzreihe. + +\subsubsection{Fall $A$ nicht diagonalisierbar: logarithmische Lösungen} +Falls die Matrix $A$ nicht diagonalisierbar ist, hat sie nur einen +Eigenwert $\lambda$ und kann durch geeignete Wahl einer Basis in +Jordansche Normalform +\[ +A += +\begin{pmatrix} +\lambda & 1 \\ + 0 & \lambda +\end{pmatrix} +\] +gebracht werden. +Dies bedeutet, dass +\begin{align*} +\sk w_1 &= \lambda w_1 + w_2 +\\ +\sk w_2 &= \lambda w_2. +\end{align*} +Die Funktion $w_2$ hat unter $\sk$ die gleichen Eigenschaften +wie im diagonalisierbaren Fall, man kann also wieder schliessen, +dass $w_2$ durch eine verallgemeinerte Potenzreihe mit +\[ +\varrho=\frac{1}{2\pi i} \log \lambda +\] +dargestellt werden kann. + +Für den Quotienten $w_1/w_2$ findet man jetzt das Bild +\begin{equation} +\sk \frac{w_1}{w_2} += +\frac{\sk w_1}{\sk w_2} += +\frac{\lambda w_1+w_2}{\lambda w_2} += +\frac{w_1}{w_2} + \frac{1}{\lambda} +\label{buch:funktionentheorie:singularitaeten:sklog} +\end{equation} +Das Verhalten von $w_1$ unter $\sk$ in +\eqref{buch:funktionentheorie:singularitaeten:sklog} +ist dasselbe wie bei $\log(z)/\lambda$, denn +\[ +\sk \frac{\log(z)}{\lambda} += +\frac{\log(z)}{\lambda} + 1. +\] +Die Differenz $w_1-\log(z)/\lambda$ wird bei der analytischen +Fortsetzung zu +\[ +\sk\biggl( +\frac{w_1}{w_2}-\frac{\log(z)}{\lambda} +\biggr) += +\sk \frac{w_1}{w_2} - \sk\frac{\log(z)}{\lambda} += +\frac{w_1}{w_2} + \frac{1}{\lambda} +- +\frac{\log(z)}{\lambda} +-\frac{1}{\lambda} += +\frac{w_1}{w_2}-\frac{\log(z)}{\lambda}. +\] +Die Differenz ist daher wieder als Laurent-Reihe +\[ +\frac{w_1}{w_2}-\frac{\log(z)}{\lambda} += +\sum_{k=-\infty}^\infty b_kz^k +\] +darstellbar, was nach $w_1$ aufgelöst +\[ +w_1(z) += +\frac{1}{\lambda} \log(z) w_2(z) ++ +w_2(z) \sum_{k=-\infty}^\infty b_kz^k +\] +ergibt. +Da $w_2$ eine verallgemeinerte Potenzreihe ist, kann man dies auch +als +\begin{equation} +w_1(z) += +c \log(z) w_2(z) ++ +z^{\varrho} +\sum_{k=-\infty}^{\infty} c_kz^k +\label{buch:funktionentheorie:singularitäten:eqn:w1} +\end{equation} +schreiben, wobei Konstanten $c$ und $c_k$ noch bestimmt werden müssen. +Setzt man +\eqref{buch:funktionentheorie:singularitäten:eqn:w1} +in die ursprüngliche Differentialgleichung ein, verschwindet der +$\log(z)$-Term und für die verbleibenden Koeffizienten kann die +bekannte Methode des Koeffizientenvergleichs verwendet werden. + +\subsubsection{Bessel-Funktionen zweiter Art} + + + -- cgit v1.2.1 From 0344a846c083c11e9ed93ddc5898dd55c6dd1022 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 20 Apr 2022 10:30:56 +0200 Subject: lemniscate sine stuff --- buch/chapters/110-elliptisch/Makefile.inc | 3 + buch/chapters/110-elliptisch/chapter.tex | 21 +- buch/chapters/110-elliptisch/ellintegral.tex | 208 +- buch/chapters/110-elliptisch/images/Makefile | 9 +- .../chapters/110-elliptisch/images/jacobiplots.pdf | Bin 56975 -> 56975 bytes buch/chapters/110-elliptisch/images/lemniskate.pdf | Bin 9914 -> 14339 bytes buch/chapters/110-elliptisch/images/lemniskate.tex | 15 +- buch/chapters/110-elliptisch/images/slcl.cpp | 128 + buch/chapters/110-elliptisch/images/slcl.pdf | Bin 0 -> 28269 bytes buch/chapters/110-elliptisch/images/slcl.tex | 88 + buch/chapters/110-elliptisch/jacobi.tex | 3264 ++++++++++---------- buch/chapters/110-elliptisch/lemniskate.tex | 299 +- 12 files changed, 2431 insertions(+), 1604 deletions(-) create mode 100644 buch/chapters/110-elliptisch/images/slcl.cpp create mode 100644 buch/chapters/110-elliptisch/images/slcl.pdf create mode 100644 buch/chapters/110-elliptisch/images/slcl.tex (limited to 'buch') diff --git a/buch/chapters/110-elliptisch/Makefile.inc b/buch/chapters/110-elliptisch/Makefile.inc index 538db68..b23df52 100644 --- a/buch/chapters/110-elliptisch/Makefile.inc +++ b/buch/chapters/110-elliptisch/Makefile.inc @@ -7,6 +7,9 @@ CHAPTERFILES = $(CHAPTERFILES) \ chapters/110-elliptisch/ellintegral.tex \ chapters/110-elliptisch/jacobi.tex \ + chapters/110-elliptisch/elltrigo.tex \ + chapters/110-elliptisch/dglsol.tex \ + chapters/110-elliptisch/mathpendel.tex \ chapters/110-elliptisch/lemniskate.tex \ chapters/110-elliptisch/uebungsaufgaben/001.tex \ chapters/110-geometrie/chapter.tex diff --git a/buch/chapters/110-elliptisch/chapter.tex b/buch/chapters/110-elliptisch/chapter.tex index e09fa53..e05f3bd 100644 --- a/buch/chapters/110-elliptisch/chapter.tex +++ b/buch/chapters/110-elliptisch/chapter.tex @@ -10,18 +10,33 @@ \rhead{} Der Versuch, die Länge eines Ellipsenbogens zu berechnen, hat -in Abschnitt~\ref{buch:geometrie:subsection:hyperbeln-und-ellipsen} +in Abschnitt~\ref{buch:geometrie:subsection:kegelschnitte} zu Integralen geführt, die nicht in geschlossener Form ausgewertet werden können. Neben den dort gefundenen Integralen sind noch weitere, ähnlich aufgebaute Integrale in dieser Familie zu finden. +Auf die trigonometrischen Funktionen stösst man, indem man Funktion +der Bogenlänge umkehrt. +Ein analoges Vorgehen bei den elliptischen Integralen führt auf +die Jacobischen elliptischen Funktionen, die in +Abschnitt~\ref{buch:elliptisch:section:jacobi} allerdings auf +eine eher geometrische Art eingeführt werden. +Die Verbindung zu den elliptischen Integralen wird dann in +Abschnitt~\ref{buch:elliptisch:subsection:differentialgleichungen} +wieder hergestellt. + \input{chapters/110-elliptisch/ellintegral.tex} + \input{chapters/110-elliptisch/jacobi.tex} +\input{chapters/110-elliptisch/elltrigo.tex} +\input{chapters/110-elliptisch/dglsol.tex} +\input{chapters/110-elliptisch/mathpendel.tex} + \input{chapters/110-elliptisch/lemniskate.tex} -\section*{Übungsaufgaben} -\rhead{Übungsaufgaben} +\section*{Übungsaufgabe} +\rhead{Übungsaufgabe} \aufgabetoplevel{chapters/110-elliptisch/uebungsaufgaben} \begin{uebungsaufgaben} %\uebungsaufgabe{0} diff --git a/buch/chapters/110-elliptisch/ellintegral.tex b/buch/chapters/110-elliptisch/ellintegral.tex index 46659cd..4cb2ba3 100644 --- a/buch/chapters/110-elliptisch/ellintegral.tex +++ b/buch/chapters/110-elliptisch/ellintegral.tex @@ -7,7 +7,7 @@ \label{buch:elliptisch:section:integral}} \rhead{Elliptisches Integral} Bei der Berechnung des Ellipsenbogens in -Abschnitt~\ref{buch:geometrie:subsection:hyperbeln-und-ellipsen} +Abschnitt~\ref{buch:geometrie:subsection:kegelschnitte} sind wir auf ein Integral gestossen, welches sich nicht in geschlossener Form ausdrücken liess. Um solche Integrale in den Griff zu bekommen, ist es nötig, sie als @@ -172,7 +172,188 @@ die {\em Jacobi-Normalform} heisst. \subsubsection{Vollständige elliptische Integrale als hypergeometrische Funktionen} -XXX Als hypergeometrische Funktionen \url{https://www.youtube.com/watch?v=j0t1yWrvKmE} \\ +%XXX Als hypergeometrische Funktionen \url{https://www.youtube.com/watch?v=j0t1yWrvKmE} \\ +Das vollständige elliptische Integral $K(k)$ kann mit Hilfe der +Binomialreihe umgeformt werden in eine hypergeometrische Reihe. +Da im Integral nur $k^2$ auftaucht, wird sich $K(k)$ als +hypergeometrische Funktion von $k^2$ ausdrücken lassen. + +\begin{satz} +\label{buch:elliptisch:satz:hyperK} +Das vollständige elliptische Integral $K(k)$ lässt sich durch die +hypergeometrische Funktion $\mathstrut_2F_1$ als +\[ +K(k) += +\frac{\pi}2 +\cdot +\mathstrut_2F_1\biggl( +\begin{matrix}\frac12,\frac12\\1\end{matrix};1;k^2 +\biggr) +\] +ausdrücken. +\end{satz} + +\begin{proof}[Beweis] +Zunächst ist das vollständige elliptische Integral in der Legendre-Form +\begin{align} +K(k) +&= +\int_0^{\frac{\pi}2} +\frac{d\vartheta}{\sqrt{1-k^2\sin^2\vartheta}} +%\notag +%\\ +%& += +\int_0^{\frac{\pi}2} +\bigl( +1-(k\sin\vartheta)^2 +\bigr)^{-\frac12}\,d\vartheta. +\notag +\intertext{Die Wurzel im letzten Integral kann mit Hilfe der binomischen +Reihe vereinfacht werden zu} +&= +\sum_{n=0}^\infty +(-1)^n k^2\binom{-\frac12}{n} +\int_0^{\frac{\pi}2} +\sin^{2n}\vartheta +\,d\vartheta. +\label{buch:elliptisch:beweis:ellharm2} +\end{align} +Der verallgemeinerte Binomialkoeffizient lässt sich nach +\begin{align*} +\binom{-\frac12}{n} +&= +\frac{(-\frac12)(-\frac32)(-\frac52)\cdot\ldots\cdot(-\frac12-n+1)}{n!} += +(-1)^n +\cdot +\frac{1}{n!} +\cdot +\frac12\cdot\frac32\cdot\frac52\cdot\ldots\cdot\biggl(\frac12+n-1\biggr) += +(-1)^n\frac{(\frac12)_n}{n!} +\end{align*} +vereinfachen. +Setzt man dies in \eqref{buch:elliptisch:beweis:ellharm2} ein, erhält +man +\begin{align*} +K(k) +&= +\sum_{n=0}^\infty +(-1)^n k^{2n} +\cdot +(-1)^n +\frac{(\frac12)_n}{n!} +\cdot +\int_0^{\frac{\pi}2} \sin^{2n}\vartheta\,d\vartheta += +\sum_{n=0}^\infty +\frac{(\frac12)_n}{n!} +\int_0^{\frac{\pi}2} \sin^{2n}\vartheta\,d\vartheta +\cdot (k^2)^n. +\end{align*} +Es muss jetzt also nur noch das Integral von $\sin^{2n}\vartheta$ +berechnet werden. +Mit partieller Integration kann man +\begin{align*} +\int \sin^m\vartheta\,d\vartheta +&= +\int +\underbrace{\sin \vartheta}_{\uparrow} +\underbrace{\sin^{m-1}\vartheta}_{\downarrow} +\,d\vartheta +\\ +&= +-\cos\vartheta\sin^{m-1}\vartheta ++ +\int \cos^2\vartheta (m-1)\sin^{m-2}\vartheta\,d\vartheta +\\ +&= +-\cos\vartheta \sin^{m-1}\vartheta ++ +(m-1) +\int +(1-\sin^2\vartheta) +\sin^{m-2}\vartheta\,d\vartheta. +\end{align*} +Wegen $\sin 0=0$ und +$\cos\frac{\pi}2=0$ verschwindet der erste Term im bestimmten Integral +und der zweite wird +\begin{align*} +\int_0^{\frac{\pi}2} +\sin^{m} \vartheta +\,d\vartheta +&= +(m-1) +\int_0^{\frac{\pi}2} +\sin^{m-2}\vartheta\,d\vartheta +- +(m-1) +\int_0^{\frac{\pi}2} +\sin^m \vartheta\,d\vartheta +\\ +m +\int_0^{\frac{\pi}2} +\sin^{m} \vartheta\,d\vartheta +&= +(m-1) +\int_0^{\frac{\pi}2} +\sin^{m-2} \vartheta\,d\vartheta +\\ +\int_0^{\frac{\pi}2} +\sin^{m} \vartheta\,d\vartheta +&= +\frac{m-1}{m} +\int_0^{\frac{\pi}2} +\sin^{m-2} \vartheta\,d\vartheta. +\end{align*} +Mit dieser Rekursionsformel kann jetzt das Integral berechnet werden. +Es folgt +\begin{align*} +\int_0^{\frac{\pi}2} +\sin^{2n}\vartheta\,d\vartheta +&= +\frac{2n-1}{2n} +\int_0^{\frac{\pi}2} +\sin^{2n-2}\vartheta\,d\vartheta +\\ +&= +\frac{2n-1}{2n} +\frac{2n-3}{2n-2} +\frac{2n-5}{2n-4} +\cdots +\frac{2n-(2n-1)}{2(n-1)} +\int_0^{\frac{\pi}2} +\sin^{2n-4}\vartheta\,d\vartheta +\\ +&= +\frac{ +(n-\frac12)(n-\frac32)(n-\frac52)\cdot\ldots\cdot\frac32\cdot\frac12 +}{ +n! +} +\int_0^{\frac{\pi}2} 1\,d\vartheta +\\ +&= +\frac{(\frac12)_n}{n!} +\cdot +\frac{\pi}2. +\end{align*} +Damit wird die Reihenentwicklung für $K(k)$ jetzt zu +\[ +K(k) += +\frac{\pi}2 +\sum_{n=0}^\infty +\frac{(\frac12)_n(\frac12)_n}{n!} \cdot \frac{(k^2)^n}{n!} += +\frac{\pi}2 +\cdot +\mathstrut_2F_1\biggl(\begin{matrix}\frac12,\frac12\\1\end{matrix};k^2\biggr), +\] +dies beweist die Behauptung. +\end{proof} @@ -247,6 +428,29 @@ Für den extremen Wert $\varepsilon=0$ entsteht der Umfang einer Ellipse, also $E(0)=\frac{\pi}2$. Für $\varepsilon=1$ ist $a=0$, es entsteht eine Strecke mit Länge $E(1)=1$. +\begin{satz} +\label{buch:elliptisch:satz:hyperE} +Das volständige elliptische Integral $E(k)$ ist +\[ +E(k) += +\int_0^{\frac{\pi}2} \sqrt{1-k^2\sin^2\vartheta}\,d\vartheta += +\frac{\pi}2 +\cdot +\mathstrut_2F_1\biggl( +\begin{matrix}-\frac12,\frac12\\1\end{matrix}; +k^2 +\biggr). +\] +\end{satz} + +\begin{proof}[Beweis] +Die Identität kann wie im Satz~\ref{buch:elliptisch:satz:hyperK} mit +Hilfe einer Entwicklung der Wurzel mit der Binomialreihe gefunden +werden. +\end{proof} + \subsubsection{Komplementäre Integrale} \subsubsection{Ableitung} diff --git a/buch/chapters/110-elliptisch/images/Makefile b/buch/chapters/110-elliptisch/images/Makefile index 68322b6..a7c9e74 100644 --- a/buch/chapters/110-elliptisch/images/Makefile +++ b/buch/chapters/110-elliptisch/images/Makefile @@ -5,7 +5,7 @@ # all: lemniskate.pdf ellipsenumfang.pdf unvollstaendig.pdf rechteck.pdf \ ellipse.pdf pendel.pdf jacobiplots.pdf jacobidef.pdf jacobi12.pdf \ - sncnlimit.pdf + sncnlimit.pdf slcl.pdf lemniskate.pdf: lemniskate.tex pdflatex lemniskate.tex @@ -71,3 +71,10 @@ jacobi12.pdf: jacobi12.tex sncnlimit.pdf: sncnlimit.tex pdflatex sncnlimit.tex +slcl: slcl.cpp + g++ -O -Wall -std=c++11 slcl.cpp -o slcl `pkg-config --cflags gsl` `pkg-config --libs gsl` + +slcldata.tex: slcl + ./slcl --outfile=slcldata.tex --a=0 --b=13.4 --steps=200 +slcl.pdf: slcl.tex slcldata.tex + pdflatex slcl.tex diff --git a/buch/chapters/110-elliptisch/images/jacobiplots.pdf b/buch/chapters/110-elliptisch/images/jacobiplots.pdf index 88cf119..f0e6e78 100644 Binary files a/buch/chapters/110-elliptisch/images/jacobiplots.pdf and b/buch/chapters/110-elliptisch/images/jacobiplots.pdf differ diff --git a/buch/chapters/110-elliptisch/images/lemniskate.pdf b/buch/chapters/110-elliptisch/images/lemniskate.pdf index 063a3e1..9e02c3c 100644 Binary files a/buch/chapters/110-elliptisch/images/lemniskate.pdf and b/buch/chapters/110-elliptisch/images/lemniskate.pdf differ diff --git a/buch/chapters/110-elliptisch/images/lemniskate.tex b/buch/chapters/110-elliptisch/images/lemniskate.tex index f74a81f..fe90631 100644 --- a/buch/chapters/110-elliptisch/images/lemniskate.tex +++ b/buch/chapters/110-elliptisch/images/lemniskate.tex @@ -27,13 +27,16 @@ \draw[color=red,line width=2.0pt] plot[domain=45:\a,samples=100] ({\x}:{sqrt(2*cos(2*\x))}); -\draw[->] (-1.5,0) -- (1.5,0) coordinate[label={$x$}]; -\draw[->] (0,-0.7) -- (0,0.7) coordinate[label={right:$y$}]; +\draw[->] (-1.5,0) -- (1.7,0) coordinate[label={$X$}]; +\draw[->] (0,-0.7) -- (0,0.7) coordinate[label={right:$Y$}]; \fill[color=white] (1,0) circle[radius=0.02]; \draw (1,0) circle[radius=0.02]; +\node at ({1},0) [below] {$\displaystyle a\mathstrut$}; + \fill[color=white] (-1,0) circle[radius=0.02]; \draw (-1,0) circle[radius=0.02]; +\node at ({-1},0) [below] {$\displaystyle\llap{$-$}a\mathstrut$}; \node[color=blue] at (\a:{0.6*sqrt(2*cos(2*\a))}) [below] {$r$}; \node[color=red] at ({\b}:{sqrt(2*cos(2*\b))}) [above] {$s$}; @@ -41,6 +44,14 @@ \fill[color=white] (\a:{sqrt(2*cos(2*\a))}) circle[radius=0.02]; \draw[color=red] (\a:{sqrt(2*cos(2*\a))}) circle[radius=0.02]; +\draw ({sqrt(2)},{-0.1/\skala}) -- ({sqrt(2)},{0.1/\skala}); +\node at ({sqrt(2)},0) [below right] + {$\displaystyle a\mathstrut\sqrt{2}$}; +\draw ({-sqrt(2)},{-0.1/\skala}) -- ({-sqrt(2)},{0.1/\skala}); +\node at ({-sqrt(2)},0) [below left] + {$\displaystyle -a\mathstrut\sqrt{2}$}; + + \end{tikzpicture} \end{document} diff --git a/buch/chapters/110-elliptisch/images/slcl.cpp b/buch/chapters/110-elliptisch/images/slcl.cpp new file mode 100644 index 0000000..8584e94 --- /dev/null +++ b/buch/chapters/110-elliptisch/images/slcl.cpp @@ -0,0 +1,128 @@ +/* + * slcl.cpp + * + * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule + */ +#include +#include +#include +#include +#include +#include +#include +#include +#include + +namespace slcl { + +static struct option longopts[] { +{ "outfile", required_argument, NULL, 'o' }, +{ "a", required_argument, NULL, 'a' }, +{ "b", required_argument, NULL, 'b' }, +{ "steps", required_argument, NULL, 'n' }, +{ NULL, 0, NULL, 0 } +}; + +class plot { + typedef std::pair point_t; + typedef std::vector curve_t; + curve_t _sl; + curve_t _cl; + double _a; + double _b; + int _steps; +public: + double a() const { return _a; } + double b() const { return _b; } + int steps() const { return _steps; } +public: + plot(double a, double b, int steps) : _a(a), _b(b), _steps(steps) { + double l = sqrt(2); + double k = 1 / l; + double m = k * k; + double h = (b - a) / steps; + for (int i = 0; i <= steps; i++) { + double x = a + h * i; + double sn, cn, dn; + gsl_sf_elljac_e(x, m, &sn, &cn, &dn); + _sl.push_back(std::make_pair(l * x, k * sn / dn)); + _cl.push_back(std::make_pair(l * x, cn)); + } + } +private: + std::string point(const point_t p) const { + char buffer[128]; + snprintf(buffer, sizeof(buffer), "({%.4f*\\dx},{%.4f*\\dy})", + p.first, p.second); + return std::string(buffer); + } + std::string path(const curve_t& curve) const { + std::ostringstream out; + auto i = curve.begin(); + out << point(*(i++)); + do { + out << std::endl << " -- " << point(*(i++)); + } while (i != curve.end()); + out.flush(); + return out.str(); + } +public: + std::string slpath() const { + return path(_sl); + } + std::string clpath() const { + return path(_cl); + } +}; + +/** + * \brief Main function for the slcl program + */ +int main(int argc, char *argv[]) { + int longindex; + int c; + double a = 0; + double b = 10; + int steps = 100; + std::ostream *out = &std::cout; + while (EOF != (c = getopt_long(argc, argv, "a:b:o:n:", + longopts, &longindex))) + switch (c) { + case 'a': + a = std::stod(optarg); + break; + case 'b': + b = std::stod(optarg) / sqrt(2); + break; + case 'n': + steps = std::stol(optarg); + break; + case 'o': + out = new std::ofstream(optarg); + break; + } + + plot p(a, b, steps); + (*out) << "\\def\\slpath{ " << p.slpath(); + (*out) << std::endl << " }" << std::endl; + (*out) << "\\def\\clpath{ " << p.clpath(); + (*out) << std::endl << " }" << std::endl; + + out->flush(); + //out->close(); + return EXIT_SUCCESS; +} + +} // namespace slcl + +int main(int argc, char *argv[]) { + try { + return slcl::main(argc, argv); + } catch (const std::exception& e) { + std::cerr << "terminated by exception: " << e.what(); + std::cerr << std::endl; + } catch (...) { + std::cerr << "terminated by unknown exception" << std::endl; + } + return EXIT_FAILURE; +} diff --git a/buch/chapters/110-elliptisch/images/slcl.pdf b/buch/chapters/110-elliptisch/images/slcl.pdf new file mode 100644 index 0000000..493b5fa Binary files /dev/null and b/buch/chapters/110-elliptisch/images/slcl.pdf differ diff --git a/buch/chapters/110-elliptisch/images/slcl.tex b/buch/chapters/110-elliptisch/images/slcl.tex new file mode 100644 index 0000000..08241ac --- /dev/null +++ b/buch/chapters/110-elliptisch/images/slcl.tex @@ -0,0 +1,88 @@ +% +% tikztemplate.tex -- template for standalon tikz images +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\input{slcldata.tex} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +% add image content here +\def\lemniscateconstant{2.6220575542} +\pgfmathparse{(3.1415926535/2)/\lemniscateconstant} +\xdef\scalechange{\pgfmathresult} + +\pgfmathparse{\scalechange*(180/3.1415926535)} +\xdef\ts{\pgfmathresult} + +\def\dx{1} +\def\dy{3} + +\draw[line width=0.3pt] + ({\lemniscateconstant*\dx},0) + -- + ({\lemniscateconstant*\dx},{1*\dy}); +\draw[line width=0.3pt] + ({2*\lemniscateconstant*\dx},0) + -- + ({2*\lemniscateconstant*\dx},{-1*\dy}); +\draw[line width=0.3pt] + ({3*\lemniscateconstant*\dx},0) + -- + ({3*\lemniscateconstant*\dx},{-1*\dy}); +\draw[line width=0.3pt] + ({4*\lemniscateconstant*\dx},0) + -- + ({4*\lemniscateconstant*\dx},{1*\dy}); +\draw[line width=0.3pt] + ({5*\lemniscateconstant*\dx},0) + -- + ({5*\lemniscateconstant*\dx},{1*\dy}); + +\draw[color=red!20,line width=1.4pt] + plot[domain=0:13,samples=200] ({\x},{\dy*sin(\ts*\x)}); +\draw[color=blue!20,line width=1.4pt] + plot[domain=0:13,samples=200] ({\x},{\dy*cos(\ts*\x)}); + +\draw[color=red,line width=1.4pt] \slpath; +\draw[color=blue,line width=1.4pt] \clpath; + +\draw[->] (0,{-1*\dy-0.1}) -- (0,{1*\dy+0.4}) coordinate[label={right:$r$}]; +\draw[->] (-0.1,0) -- (13.7,0) coordinate[label={$s$}]; + +\foreach \i in {1,2,3,4,5}{ + \draw ({\lemniscateconstant*\i},-0.1) -- ({\lemniscateconstant*\i},0.1); +} +\node at ({\lemniscateconstant*\dx},0) [below left] {$ \varpi\mathstrut$}; +\node at ({2*\lemniscateconstant*\dx},0) [below left] {$2\varpi\mathstrut$}; +\node at ({3*\lemniscateconstant*\dx},0) [below right] {$3\varpi\mathstrut$}; +\node at ({4*\lemniscateconstant*\dx},0) [below right] {$4\varpi\mathstrut$}; +\node at ({5*\lemniscateconstant*\dx},0) [below left] {$5\varpi\mathstrut$}; + +\node[color=red] at ({1.6*\lemniscateconstant*\dx},{0.6*\dy}) + [below left] {$\operatorname{sl}(s)$}; +\node[color=red!50] at ({1.5*\lemniscateconstant*\dx},{sin(1.5*90)*\dy*0.90}) + [above right] {$\sin \bigl(\frac{\pi}{2\varpi}s\bigr)$}; + +\node[color=blue] at ({1.4*\lemniscateconstant*\dx},{-0.6*\dy}) + [above right] {$\operatorname{cl}(s)$}; +\node[color=blue!50] at ({1.5*\lemniscateconstant*\dx},{cos(1.5*90)*\dy*0.90}) + [below left] {$\cos\bigl(\frac{\pi}{2\varpi}s\bigr)$}; + +\draw (-0.1,{1*\dy}) -- (0.1,{1*\dy}); +\draw (-0.1,{-1*\dy}) -- (0.1,{-1*\dy}); +\node at (0,{1*\dy}) [left] {$1\mathstrut$}; +\node at (0,0) [left] {$0\mathstrut$}; +\node at (0,{-1*\dy}) [left] {$-1\mathstrut$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/110-elliptisch/jacobi.tex b/buch/chapters/110-elliptisch/jacobi.tex index f1e0987..e1fbc00 100644 --- a/buch/chapters/110-elliptisch/jacobi.tex +++ b/buch/chapters/110-elliptisch/jacobi.tex @@ -22,1597 +22,1743 @@ dann muss man die Umkehrfunktionen der elliptischen Integrale dafür ins Auge fassen. +%% +%% elliptische Funktionen als Trigonometrie +%% +%\subsection{Elliptische Funktionen als Trigonometrie} +%\begin{figure} +%\centering +%\includegraphics{chapters/110-elliptisch/images/ellipse.pdf} +%\caption{Kreis und Ellipse zum Vergleich und zur Herleitung der +%elliptischen Funktionen von Jacobi als ``trigonometrische'' Funktionen +%auf einer Ellipse. +%\label{buch:elliptisch:fig:ellipse}} +%\end{figure} +%% based on Willliam Schwalm, Elliptic functions and elliptic integrals +%% https://youtu.be/DCXItCajCyo % -% ellpitische Funktionen als Trigonometrie +%% +%% Geometrie einer Ellipse +%% +%\subsubsection{Geometrie einer Ellipse} +%Eine {\em Ellipse} ist die Menge der Punkte der Ebene, für die die Summe +%\index{Ellipse}% +%der Entfernungen von zwei festen Punkten $F_1$ und $F_2$, +%den {\em Brennpunkten}, konstant ist. +%\index{Brennpunkt}% +%In Abbildung~\ref{buch:elliptisch:fig:ellipse} eine Ellipse +%mit Brennpunkten in $F_1=(-e,0)$ und $F_2=(e,0)$ dargestellt, +%die durch die Punkte $(\pm a,0)$ und $(0,\pm b)$ auf den Achsen geht. +%Der Punkt $(a,0)$ hat die Entfernungen $a+e$ und $a-e$ von den beiden +%Brennpunkten, also die Entfernungssumme $a+e+a-e=2a$. +%Jeder andere Punkt auf der Ellipse muss ebenfalls diese Entfernungssumme +%haben, insbesondere auch der Punkt $(0,b)$. +%Seine Entfernung zu jedem Brennpunkt muss aus Symmetriegründen gleich gross, +%also $a$ sein. +%Aus dem Satz von Pythagoras liest man daher ab, dass +%\[ +%b^2+e^2=a^2 +%\qquad\Rightarrow\qquad +%e^2 = a^2-b^2 +%\] +%sein muss. +%Die Strecke $e$ heisst auch {\em (lineare) Exzentrizität} der Ellipse. +%Das Verhältnis $\varepsilon= e/a$ heisst die {\em numerische Exzentrizität} +%der Ellipse. % -\subsection{Elliptische Funktionen als Trigonometrie} -\begin{figure} -\centering -\includegraphics{chapters/110-elliptisch/images/ellipse.pdf} -\caption{Kreis und Ellipse zum Vergleich und zur Herleitung der -elliptischen Funktionen von Jacobi als ``trigonometrische'' Funktionen -auf einer Ellipse. -\label{buch:elliptisch:fig:ellipse}} -\end{figure} -% based on Willliam Schwalm, Elliptic functions and elliptic integrals -% https://youtu.be/DCXItCajCyo - +%% +%% Die Ellipsengleichung +%% +%\subsubsection{Ellipsengleichung} +%Der Punkt $P=(x,y)$ auf der Ellipse hat die Entfernungen +%\begin{equation} +%\begin{aligned} +%\overline{PF_1}^2 +%&= +%y^2 + (x+e)^2 +%\\ +%\overline{PF_2}^2 +%&= +%y^2 + (x-e)^2 +%\end{aligned} +%\label{buch:elliptisch:eqn:wurzelausdruecke} +%\end{equation} +%von den Brennpunkten, für die +%\begin{equation} +%\overline{PF_1}+\overline{PF_2} +%= +%2a +%\label{buch:elliptisch:eqn:pf1pf2a} +%\end{equation} +%gelten muss. +%Man kann nachrechnen, dass ein Punkt $P$, der die Gleichung +%\[ +%\frac{x^2}{a^2} + \frac{y^2}{b^2}=1 +%\] +%erfüllt, auch die Eigenschaft~\eqref{buch:elliptisch:eqn:pf1pf2a} +%erfüllt. +%Zur Vereinfachung setzen wir $l_1=\overline{PF_1}$ und $l_2=\overline{PF_2}$. +%$l_1$ und $l_2$ sind Wurzeln aus der rechten Seite von +%\eqref{buch:elliptisch:eqn:wurzelausdruecke}. +%Das Quadrat von $l_1+l_2$ ist +%\[ +%l_1^2 + 2l_1l_2 + l_2^2 = 4a^2. +%\] +%Um die Wurzeln ganz zu eliminieren, bringt man das Produkt $l_1l_2$ alleine +%auf die rechte Seite und quadriert. +%Man muss also verifizieren, dass +%\[ +%(l_1^2 + l_2^2 -4a^2)^2 = 4l_1^2l_2^2. +%\] +%In den entstehenden Ausdrücken muss man ausserdem $e=\sqrt{a^2-b^2}$ und +%\[ +%y=b\sqrt{1-\frac{x^2}{a^2}} +%\] +%substituieren. +%Diese Rechnung führt man am einfachsten mit Hilfe eines +%Computeralgebraprogramms durch, welches obige Behauptung bestätigt. % -% Geometrie einer Ellipse +%% +%% Normierung +%% +%\subsubsection{Normierung} +%Die trigonometrischen Funktionen sind definiert als Verhältnisse +%von Seiten rechtwinkliger Dreiecke. +%Dadurch, dass man den die Hypothenuse auf Länge $1$ normiert, +%kann man die Sinus- und Kosinus-Funktion als Koordinaten eines +%Punktes auf dem Einheitskreis interpretieren. % -\subsubsection{Geometrie einer Ellipse} -Eine {\em Ellipse} ist die Menge der Punkte der Ebene, für die die Summe -\index{Ellipse}% -der Entfernungen von zwei festen Punkten $F_1$ und $F_2$, -den {\em Brennpunkten}, konstant ist. -\index{Brennpunkt}% -In Abbildung~\ref{buch:elliptisch:fig:ellipse} eine Ellipse -mit Brennpunkten in $F_1=(-e,0)$ und $F_2=(e,0)$ dargestellt, -die durch die Punkte $(\pm a,0)$ und $(0,\pm b)$ auf den Achsen geht. -Der Punkt $(a,0)$ hat die Entfernungen $a+e$ und $a-e$ von den beiden -Brennpunkten, also die Entfernungssumme $a+e+a-e=2a$. -Jeder andere Punkt auf der Ellipse muss ebenfalls diese Entfernungssumme -haben, insbesondere auch der Punkt $(0,b)$. -Seine Entfernung zu jedem Brennpunkt muss aus Symmetriegründen gleich gross, -also $a$ sein. -Aus dem Satz von Pythagoras liest man daher ab, dass -\[ -b^2+e^2=a^2 -\qquad\Rightarrow\qquad -e^2 = a^2-b^2 -\] -sein muss. -Die Strecke $e$ heisst auch {\em (lineare) Exzentrizität} der Ellipse. -Das Verhältnis $\varepsilon= e/a$ heisst die {\em numerische Exzentrizität} -der Ellipse. - +%Für die Koordinaten eines Punktes auf der Ellipse ist dies nicht so einfach, +%weil es nicht nur eine Ellipse gibt, sondern für jede numerische Exzentrizität +%mindestens eine mit Halbeachse $1$. +%Wir wählen die Ellipsen so, dass $a$ die grosse Halbachse ist, also $a>b$. +%Als Normierungsbedingung verwenden wir, dass $b=1$ sein soll, wie in +%Abbildung~\ref{buch:elliptisch:fig:jacobidef}. +%Dann ist $a=1/\varepsilon>1$. +%In dieser Normierung haben Punkte $(x,y)$ auf der Ellipse $y$-Koordinaten +%zwischen $-1$ und $1$ und $x$-Koordinaten zwischen $-a$ und $a$. % -% Die Ellipsengleichung +%Im Zusammenhang mit elliptischen Funktionen wird die numerische Exzentrizität +%$\varepsilon$ auch mit +%\[ +%k +%= +%\varepsilon +%= +%\frac{e}{a} +%= +%\frac{\sqrt{a^2-b^2}}{a} +%= +%\frac{\sqrt{a^2-1}}{a}, +%\] +%die Zahl $k$ heisst auch der {\em Modulus}. +%Man kann $a$ auch durch $k$ ausdrücken, durch Quadrieren und Umstellen +%findet man +%\[ +%k^2a^2 = a^2-1 +%\quad\Rightarrow\quad +%1=a^2(k^2-1) +%\quad\Rightarrow\quad +%a=\frac{1}{\sqrt{k^2-1}}. +%\] % -\subsubsection{Ellipsengleichung} -Der Punkt $P=(x,y)$ auf der Ellipse hat die Entfernungen -\begin{equation} -\begin{aligned} -\overline{PF_1}^2 -&= -y^2 + (x+e)^2 -\\ -\overline{PF_2}^2 -&= -y^2 + (x-e)^2 -\end{aligned} -\label{buch:elliptisch:eqn:wurzelausdruecke} -\end{equation} -von den Brennpunkten, für die -\begin{equation} -\overline{PF_1}+\overline{PF_2} -= -2a -\label{buch:elliptisch:eqn:pf1pf2a} -\end{equation} -gelten muss. -Man kann nachrechnen, dass ein Punkt $P$, der die Gleichung -\[ -\frac{x^2}{a^2} + \frac{y^2}{b^2}=1 -\] -erfüllt, auch die Eigenschaft~\eqref{buch:elliptisch:eqn:pf1pf2a} -erfüllt. -Zur Vereinfachung setzen wir $l_1=\overline{PF_1}$ und $l_2=\overline{PF_2}$. -$l_1$ und $l_2$ sind Wurzeln aus der rechten Seite von -\eqref{buch:elliptisch:eqn:wurzelausdruecke}. -Das Quadrat von $l_1+l_2$ ist -\[ -l_1^2 + 2l_1l_2 + l_2^2 = 4a^2. -\] -Um die Wurzeln ganz zu eliminieren, bringt man das Produkt $l_1l_2$ alleine -auf die rechte Seite und quadriert. -Man muss also verifizieren, dass -\[ -(l_1^2 + l_2^2 -4a^2)^2 = 4l_1^2l_2^2. -\] -In den entstehenden Ausdrücken muss man ausserdem $e=\sqrt{a^2-b^2}$ und -\[ -y=b\sqrt{1-\frac{x^2}{a^2}} -\] -substituieren. -Diese Rechnung führt man am einfachsten mit Hilfe eines -Computeralgebraprogramms durch, welches obige Behauptung bestätigt. - +%Die Gleichung der ``Einheitsellipse'' zu diesem Modulus ist +%\[ +%\frac{x^2}{a^2}+y^2=1 +%\qquad\text{oder}\qquad +%x^2(k^2-1) + y^2 = 1. +%\] % -% Normierung +%% +%% Definition der elliptischen Funktionen +%% +%\begin{figure} +%\centering +%\includegraphics{chapters/110-elliptisch/images/jacobidef.pdf} +%\caption{Definition der elliptischen Funktionen als Trigonometrie +%an einer Ellipse mit Halbachsen $a$ und $1$. +%\label{buch:elliptisch:fig:jacobidef}} +%\end{figure} +%\subsubsection{Definition der elliptischen Funktionen} +%Die elliptischen Funktionen für einen Punkt $P$ auf der Ellipse mit Modulus $k$ +%können jetzt als Verhältnisse der Koordinaten des Punktes definieren. +%Es stellt sich aber die Frage, was man als Argument verwenden soll. +%Es soll so etwas wie den Winkel $\varphi$ zwischen der $x$-Achse und dem +%Radiusvektor zum Punkt $P$ +%darstellen, aber wir haben hier noch eine Wahlfreiheit, die wir später +%ausnützen möchten. +%Im Moment müssen wir die Frage noch nicht beantworten und nennen das +%noch unbestimmte Argument $u$. +%Wir kümmern uns später um die Frage, wie $u$ von $\varphi$ abhängt. % -\subsubsection{Normierung} -Die trigonometrischen Funktionen sind definiert als Verhältnisse -von Seiten rechtwinkliger Dreiecke. -Dadurch, dass man den die Hypothenuse auf Länge $1$ normiert, -kann man die Sinus- und Kosinus-Funktion als Koordinaten eines -Punktes auf dem Einheitskreis interpretieren. - -Für die Koordinaten eines Punktes auf der Ellipse ist dies nicht so einfach, -weil es nicht nur eine Ellipse gibt, sondern für jede numerische Exzentrizität -mindestens eine mit Halbeachse $1$. -Wir wählen die Ellipsen so, dass $a$ die grosse Halbachse ist, also $a>b$. -Als Normierungsbedingung verwenden wir, dass $b=1$ sein soll, wie in -Abbildung~\ref{buch:elliptisch:fig:jacobidef}. -Dann ist $a=1/\varepsilon>1$. -In dieser Normierung haben Punkte $(x,y)$ auf der Ellipse $y$-Koordinaten -zwischen $-1$ und $1$ und $x$-Koordinaten zwischen $-a$ und $a$. - -Im Zusammenhang mit elliptischen Funktionen wird die numerische Exzentrizität -$\varepsilon$ auch mit -\[ -k -= -\varepsilon -= -\frac{e}{a} -= -\frac{\sqrt{a^2-b^2}}{a} -= -\frac{\sqrt{a^2-1}}{a}, -\] -die Zahl $k$ heisst auch der {\em Modulus}. -Man kann $a$ auch durch $k$ ausdrücken, durch Quadrieren und Umstellen -findet man -\[ -k^2a^2 = a^2-1 -\quad\Rightarrow\quad -1=a^2(k^2-1) -\quad\Rightarrow\quad -a=\frac{1}{\sqrt{k^2-1}}. -\] - -Die Gleichung der ``Einheitsellipse'' zu diesem Modulus ist -\[ -\frac{x^2}{a^2}+y^2=1 -\qquad\text{oder}\qquad -x^2(k^2-1) + y^2 = 1. -\] - +%Die Funktionen, die wir definieren wollen, hängen ausserdem auch +%vom Modulus ab. +%Falls der verwendete Modulus aus dem Zusammenhang klar ist, lassen +%wir das $k$-Argument weg. % -% Definition der elliptischen Funktionen +%Die Punkte auf dem Einheitskreis haben alle den gleichen Abstand vom +%Nullpunkt, dies ist gleichzeitig die definierende Gleichung $r^2=x^2+y^2=1$ +%des Kreises. +%Die Punkte auf der Ellipse erfüllen die Gleichung $x^2/a^2+y^2=1$, +%die Entfernung der Punkte $r=\sqrt{x^2+y^2}$ vom Nullpunkt variert aber. % -\begin{figure} -\centering -\includegraphics{chapters/110-elliptisch/images/jacobidef.pdf} -\caption{Definition der elliptischen Funktionen als Trigonometrie -an einer Ellipse mit Halbachsen $a$ und $1$. -\label{buch:elliptisch:fig:jacobidef}} -\end{figure} -\subsubsection{Definition der elliptischen Funktionen} -Die elliptischen Funktionen für einen Punkt $P$ auf der Ellipse mit Modulus $k$ -können jetzt als Verhältnisse der Koordinaten des Punktes definieren. -Es stellt sich aber die Frage, was man als Argument verwenden soll. -Es soll so etwas wie den Winkel $\varphi$ zwischen der $x$-Achse und dem -Radiusvektor zum Punkt $P$ -darstellen, aber wir haben hier noch eine Wahlfreiheit, die wir später -ausnützen möchten. -Im Moment müssen wir die Frage noch nicht beantworten und nennen das -noch unbestimmte Argument $u$. -Wir kümmern uns später um die Frage, wie $u$ von $\varphi$ abhängt. - -Die Funktionen, die wir definieren wollen, hängen ausserdem auch -vom Modulus ab. -Falls der verwendete Modulus aus dem Zusammenhang klar ist, lassen -wir das $k$-Argument weg. - -Die Punkte auf dem Einheitskreis haben alle den gleichen Abstand vom -Nullpunkt, dies ist gleichzeitig die definierende Gleichung $r^2=x^2+y^2=1$ -des Kreises. -Die Punkte auf der Ellipse erfüllen die Gleichung $x^2/a^2+y^2=1$, -die Entfernung der Punkte $r=\sqrt{x^2+y^2}$ vom Nullpunkt variert aber. - -In Analogie zu den trigonometrischen Funktionen setzen wir jetzt für -die Funktionen -\[ -\begin{aligned} -&\text{sinus amplitudinis:}& -{\color{red}\operatorname{sn}(u,k)}&= y \\ -&\text{cosinus amplitudinis:}& -{\color{blue}\operatorname{cn}(u,k)}&= \frac{x}{a} \\ -&\text{delta amplitudinis:}& -{\color{darkgreen}\operatorname{dn}(u,k)}&=\frac{r}{a}, -\end{aligned} -\] -die auch in Abbildung~\ref{buch:elliptisch:fig:jacobidef} -dargestellt sind. -Aus der Gleichung der Ellipse folgt sofort, dass -\[ -\operatorname{sn}(u,k)^2 + \operatorname{cn}(u,k)^2 = 1 -\] -ist. -Der Satz von Pythagoras kann verwendet werden, um die Entfernung zu -berechnen, also gilt -\begin{equation} -r^2 -= -a^2 \operatorname{dn}(u,k)^2 -= -x^2 + y^2 -= -a^2\operatorname{cn}(u,k)^2 + \operatorname{sn}(u,k)^2 -\quad -\Rightarrow -\quad -a^2 \operatorname{dn}(u,k)^2 -= -a^2\operatorname{cn}(u,k)^2 + \operatorname{sn}(u,k)^2. -\label{buch:elliptisch:eqn:sncndnrelation} -\end{equation} -Ersetzt man -$ -a^2\operatorname{cn}(u,k)^2 -= -a^2-a^2\operatorname{sn}(u,k)^2 -$, ergibt sich -\[ -a^2 \operatorname{dn}(u,k)^2 -= -a^2-a^2\operatorname{sn}(u,k)^2 -+ -\operatorname{sn}(u,k)^2 -\quad -\Rightarrow -\quad -\operatorname{dn}(u,k)^2 -+ -\frac{a^2-1}{a^2}\operatorname{sn}(u,k)^2 -= -1, -\] -woraus sich die Identität -\[ -\operatorname{dn}(u,k)^2 + k^2 \operatorname{sn}(u,k)^2 = 1 -\] -ergibt. -Ebenso kann man aus~\eqref{buch:elliptisch:eqn:sncndnrelation} -die Funktion $\operatorname{cn}(u,k)$ eliminieren, was auf -\[ -a^2\operatorname{dn}(u,k)^2 -= -a^2\operatorname{cn}(u,k)^2 -+1-\operatorname{cn}(u,k)^2 -= -(a^2-1)\operatorname{cn}(u,k)^2 -+1. -\] -Nach Division durch $a^2$ ergibt sich -\begin{align*} -\operatorname{dn}(u,k)^2 -- -k^2\operatorname{cn}(u,k)^2 -&= -\frac{1}{a^2} -= -\frac{a^2-a^2+1}{a^2} -= -1-k^2 =: k^{\prime 2}. -\end{align*} -Wir stellen die hiermit gefundenen Relationen zwischen den grundlegenden -Jacobischen elliptischen Funktionen für später zusammen in den Formeln -\begin{equation} -\begin{aligned} -\operatorname{sn}^2(u,k) -+ -\operatorname{cn}^2(u,k) -&= -1 -\\ -\operatorname{dn}^2(u,k) + k^2\operatorname{sn}^2(u,k) -&= -1 -\\ -\operatorname{dn}^2(u,k) -k^2\operatorname{cn}^2(u,k) -&= -k^{\prime 2}. -\end{aligned} -\label{buch:elliptisch:eqn:jacobi-relationen} -\end{equation} -zusammen. -So wie es möglich ist, $\sin\alpha$ durch $\cos\alpha$ auszudrücken, -ist es mit -\eqref{buch:elliptisch:eqn:jacobi-relationen} -jetzt auch möglich jede grundlegende elliptische Funktion durch -jede anderen auszudrücken. -Die Resultate sind in der Tabelle~\ref{buch:elliptisch:fig:jacobi-relationen} -zusammengestellt. - -\begin{table} -\centering -\renewcommand{\arraystretch}{2.1} -\begin{tabular}{|>{$\displaystyle}c<{$}|>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}|} -\hline -&\operatorname{sn}(u,k) -&\operatorname{cn}(u,k) -&\operatorname{dn}(u,k)\\ -\hline -\operatorname{sn}(u,k) -&\operatorname{sn}(u,k) -&\sqrt{1-\operatorname{cn}^2(u,k)} -&\frac1k\sqrt{1-\operatorname{dn}^2(u,k)} -\\ -\operatorname{cn}(u,k) -&\sqrt{1-\operatorname{sn}^2(u,k)} -&\operatorname{cn}(u,k) -&\frac{1}{k}\sqrt{\operatorname{dn}^2(u,k)-k^{\prime2}} -\\ -\operatorname{dn}(u,k) -&\sqrt{1-k^2\operatorname{sn}^2(u,k)} -&\sqrt{k^{\prime2}+k^2\operatorname{cn}^2(u,k)} -&\operatorname{dn}(u,k) -\\ -\hline -\end{tabular} -\caption{Jede der Jacobischen elliptischen Funktionen lässt sich -unter Verwendung der Relationen~\eqref{buch:elliptisch:eqn:jacobi-relationen} -durch jede andere ausdrücken. -\label{buch:elliptisch:fig:jacobi-relationen}} -\end{table} - +%In Analogie zu den trigonometrischen Funktionen setzen wir jetzt für +%die Funktionen +%\[ +%\begin{aligned} +%&\text{sinus amplitudinis:}& +%{\color{red}\operatorname{sn}(u,k)}&= y \\ +%&\text{cosinus amplitudinis:}& +%{\color{blue}\operatorname{cn}(u,k)}&= \frac{x}{a} \\ +%&\text{delta amplitudinis:}& +%{\color{darkgreen}\operatorname{dn}(u,k)}&=\frac{r}{a}, +%\end{aligned} +%\] +%die auch in Abbildung~\ref{buch:elliptisch:fig:jacobidef} +%dargestellt sind. +%Aus der Gleichung der Ellipse folgt sofort, dass +%\[ +%\operatorname{sn}(u,k)^2 + \operatorname{cn}(u,k)^2 = 1 +%\] +%ist. +%Der Satz von Pythagoras kann verwendet werden, um die Entfernung zu +%berechnen, also gilt +%\begin{equation} +%r^2 +%= +%a^2 \operatorname{dn}(u,k)^2 +%= +%x^2 + y^2 +%= +%a^2\operatorname{cn}(u,k)^2 + \operatorname{sn}(u,k)^2 +%\quad +%\Rightarrow +%\quad +%a^2 \operatorname{dn}(u,k)^2 +%= +%a^2\operatorname{cn}(u,k)^2 + \operatorname{sn}(u,k)^2. +%\label{buch:elliptisch:eqn:sncndnrelation} +%\end{equation} +%Ersetzt man +%$ +%a^2\operatorname{cn}(u,k)^2 +%= +%a^2-a^2\operatorname{sn}(u,k)^2 +%$, ergibt sich +%\[ +%a^2 \operatorname{dn}(u,k)^2 +%= +%a^2-a^2\operatorname{sn}(u,k)^2 +%+ +%\operatorname{sn}(u,k)^2 +%\quad +%\Rightarrow +%\quad +%\operatorname{dn}(u,k)^2 +%+ +%\frac{a^2-1}{a^2}\operatorname{sn}(u,k)^2 +%= +%1, +%\] +%woraus sich die Identität +%\[ +%\operatorname{dn}(u,k)^2 + k^2 \operatorname{sn}(u,k)^2 = 1 +%\] +%ergibt. +%Ebenso kann man aus~\eqref{buch:elliptisch:eqn:sncndnrelation} +%die Funktion $\operatorname{cn}(u,k)$ eliminieren, was auf +%\[ +%a^2\operatorname{dn}(u,k)^2 +%= +%a^2\operatorname{cn}(u,k)^2 +%+1-\operatorname{cn}(u,k)^2 +%= +%(a^2-1)\operatorname{cn}(u,k)^2 +%+1. +%\] +%Nach Division durch $a^2$ ergibt sich +%\begin{align*} +%\operatorname{dn}(u,k)^2 +%- +%k^2\operatorname{cn}(u,k)^2 +%&= +%\frac{1}{a^2} +%= +%\frac{a^2-a^2+1}{a^2} +%= +%1-k^2 =: k^{\prime 2}. +%\end{align*} +%Wir stellen die hiermit gefundenen Relationen zwischen den grundlegenden +%Jacobischen elliptischen Funktionen für später zusammen in den Formeln +%\begin{equation} +%\begin{aligned} +%\operatorname{sn}^2(u,k) +%+ +%\operatorname{cn}^2(u,k) +%&= +%1 +%\\ +%\operatorname{dn}^2(u,k) + k^2\operatorname{sn}^2(u,k) +%&= +%1 +%\\ +%\operatorname{dn}^2(u,k) -k^2\operatorname{cn}^2(u,k) +%&= +%k^{\prime 2}. +%\end{aligned} +%\label{buch:elliptisch:eqn:jacobi-relationen} +%\end{equation} +%zusammen. +%So wie es möglich ist, $\sin\alpha$ durch $\cos\alpha$ auszudrücken, +%ist es mit +%\eqref{buch:elliptisch:eqn:jacobi-relationen} +%jetzt auch möglich jede grundlegende elliptische Funktion durch +%jede anderen auszudrücken. +%Die Resultate sind in der Tabelle~\ref{buch:elliptisch:fig:jacobi-relationen} +%zusammengestellt. % -% Ableitungen der Jacobi-ellpitischen Funktionen -% -\subsubsection{Ableitung} -Die trigonometrischen Funktionen sind deshalb so besonders nützlich -für die Lösung von Schwingungsdifferentialgleichungen, weil sie die -Beziehungen -\[ -\frac{d}{d\varphi} \cos\varphi = -\sin\varphi -\qquad\text{und}\qquad -\frac{d}{d\varphi} \sin\varphi = \cos\varphi -\] -erfüllen. -So einfach können die Beziehungen natürlich nicht sein, sonst würde sich -durch Integration ja wieder nur die trigonometrischen Funktionen ergeben. -Durch geschickte Wahl des Arguments $u$ kann man aber erreichen, dass -sie ähnlich nützliche Beziehungen zwischen den Ableitungen ergeben. - -Gesucht ist jetzt also eine Wahl für das Argument $u$ zum Beispiel in -Abhängigkeit von $\varphi$, dass sich einfache und nützliche -Ableitungsformeln ergeben. -Wir setzen daher $u(\varphi)$ voraus und beachten, dass $x$ und $y$ -ebenfalls von $\varphi$ abhängen, es ist -$y=\sin\varphi$ und $x=a\cos\varphi$. -Die Ableitungen von $x$ und $y$ nach $\varphi$ sind -\begin{align*} -\frac{dy}{d\varphi} -&= -\cos\varphi -= -\frac{1}{a} x -= -\operatorname{cn}(u,k) -\\ -\frac{dx}{d\varphi} -&= --a\sin\varphi -= --a y -= --a\operatorname{sn}(u,k). -\end{align*} -Daraus kann man jetzt die folgenden Ausdrücke für die Ableitungen der -elliptischen Funktionen nach $\varphi$ ableiten: -\begin{align*} -\frac{d}{d\varphi} \operatorname{sn}(u,z) -&= -\frac{d}{d\varphi} y(\varphi) -= -\cos\varphi -= -\frac{x}{a} -= -\operatorname{cn}(u,k) -&&\Rightarrow& -\frac{d}{du} -\operatorname{sn}(u,k) -&= -\operatorname{cn}(u,k) \frac{d\varphi}{du} -\\ -\frac{d}{d\varphi} \operatorname{cn}(u,z) -&= -\frac{d}{d\varphi} \frac{x(\varphi)}{a} -= --\sin\varphi -= --\operatorname{sn}(u,k) -&&\Rightarrow& -\frac{d}{du}\operatorname{cn}(u,k) -&= --\operatorname{sn}(u,k) \frac{d\varphi}{du} -\\ -\frac{d}{d\varphi} \operatorname{dn}(u,z) -&= -\frac{1}{a}\frac{dr}{d\varphi} -= -\frac{1}{a}\frac{d\sqrt{x^2+y^2}}{d\varphi} -\\ -&= -\frac{x}{ar} \frac{dx}{d\varphi} -+ -\frac{y}{ar} \frac{dy}{d\varphi} -\\ -&= -\frac{x}{ar} (-a\operatorname{sn}(u,k)) -+ -\frac{y}{ar} \operatorname{cn}(u,k) -\\ -&= -\frac{x}{ar}(-ay) -+ -\frac{y}{ar} \frac{x}{a} -= -\frac{xy(-1+\frac{1}{a^2})}{r} -\\ -&= --\frac{xy(a^2-1)}{a^2r} -\\ -&= --\frac{a^2-1}{ar} -\operatorname{cn}(u,k) \operatorname{sn}(u,k) -\\ -&=-k^2 -\frac{a}{r} -\operatorname{cn}(u,k) \operatorname{sn}(u,k) -\\ -&= --k^2\frac{\operatorname{cn}(u,k)\operatorname{sn}(u,k)}{\operatorname{dn}(u,k)} -&&\Rightarrow& -\frac{d}{du} \operatorname{dn}(u,k) -&= --k^2\frac{\operatorname{cn}(u,k) -\operatorname{sn}(u,k)}{\operatorname{dn}(u,k)} -\frac{d\varphi}{du} -\end{align*} -Die einfachsten Beziehungen ergeben sich offenbar, wenn man $u$ so -wählt, dass -\[ -\frac{d\varphi}{du} -= -\operatorname{dn}(u,k) -= -\frac{r}{a} -\] -Damit haben wir die grundlegenden Ableitungsregeln -\begin{align*} -\frac{d}{du}\operatorname{sn}(u,k) -&= -\phantom{-}\operatorname{cn}(u,k)\operatorname{dn}(u,k) -\\ -\frac{d}{du}\operatorname{cn}(u,k) -&= --\operatorname{sn}(u,k)\operatorname{dn}(u,k) -\\ -\frac{d}{du}\operatorname{dn}(u,k) -&= --k^2\operatorname{sn}(u,k)\operatorname{cn}(u,k) -\end{align*} -der elliptischen Funktionen nach Jacobi. - +%\begin{table} +%\centering +%\renewcommand{\arraystretch}{2.1} +%\begin{tabular}{|>{$\displaystyle}c<{$}|>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}|} +%\hline +%&\operatorname{sn}(u,k) +%&\operatorname{cn}(u,k) +%&\operatorname{dn}(u,k)\\ +%\hline +%\operatorname{sn}(u,k) +%&\operatorname{sn}(u,k) +%&\sqrt{1-\operatorname{cn}^2(u,k)} +%&\frac1k\sqrt{1-\operatorname{dn}^2(u,k)} +%\\ +%\operatorname{cn}(u,k) +%&\sqrt{1-\operatorname{sn}^2(u,k)} +%&\operatorname{cn}(u,k) +%&\frac{1}{k}\sqrt{\operatorname{dn}^2(u,k)-k^{\prime2}} +%\\ +%\operatorname{dn}(u,k) +%&\sqrt{1-k^2\operatorname{sn}^2(u,k)} +%&\sqrt{k^{\prime2}+k^2\operatorname{cn}^2(u,k)} +%&\operatorname{dn}(u,k) +%\\ +%\hline +%\end{tabular} +%\caption{Jede der Jacobischen elliptischen Funktionen lässt sich +%unter Verwendung der Relationen~\eqref{buch:elliptisch:eqn:jacobi-relationen} +%durch jede andere ausdrücken. +%\label{buch:elliptisch:fig:jacobi-relationen}} +%\end{table} % -% Der Grenzfall $k=1$ +%% +%% Ableitungen der Jacobi-ellpitischen Funktionen +%% +%\subsubsection{Ableitung} +%Die trigonometrischen Funktionen sind deshalb so besonders nützlich +%für die Lösung von Schwingungsdifferentialgleichungen, weil sie die +%Beziehungen +%\[ +%\frac{d}{d\varphi} \cos\varphi = -\sin\varphi +%\qquad\text{und}\qquad +%\frac{d}{d\varphi} \sin\varphi = \cos\varphi +%\] +%erfüllen. +%So einfach können die Beziehungen natürlich nicht sein, sonst würde sich +%durch Integration ja wieder nur die trigonometrischen Funktionen ergeben. +%Durch geschickte Wahl des Arguments $u$ kann man aber erreichen, dass +%sie ähnlich nützliche Beziehungen zwischen den Ableitungen ergeben. % -\subsubsection{Der Grenzwert $k\to1$} -\begin{figure} -\centering -\includegraphics{chapters/110-elliptisch/images/sncnlimit.pdf} -\caption{Grenzfälle der Jacobischen elliptischen Funktionen -für die Werte $0$ und $1$ des Parameters $k$. -\label{buch:elliptisch:fig:sncnlimit}} -\end{figure} -Für $k=1$ ist $k^{\prime2}=1-k^2=$ und es folgt aus den -Relationen~\eqref{buch:elliptisch:eqn:jacobi-relationen} -\[ -\operatorname{cn}^2(u,k) -- -k^2 -\operatorname{dn}^2(u,k) -= -k^{\prime2} -= -0 -\qquad\Rightarrow\qquad -\operatorname{cn}^2(u,1) -= -\operatorname{dn}^2(u,1), -\] -die beiden Funktionen -$\operatorname{cn}(u,k)$ -und -$\operatorname{dn}(u,k)$ -fallen also zusammen. -Die Ableitungsregeln werden dadurch vereinfacht: -\begin{align*} -\operatorname{sn}'(u,1) -&= -\operatorname{cn}(u,1) -\operatorname{dn}(u,1) -= -\operatorname{cn}^2(u,1) -= -1-\operatorname{sn}^2(u,1) -&&\Rightarrow& y'&=1-y^2 -\\ -\operatorname{cn}'(u,1) -&= -- -\operatorname{sn}(u,1) -\operatorname{dn}(u,1) -= -- -\operatorname{sn}(u,1)\operatorname{cn}(u,1) -&&\Rightarrow& -\frac{z'}{z}&=(\log z)' = -y -\end{align*} -Die erste Differentialgleichung für $y$ lässt sich separieren, man findet -die Lösung -\[ -\frac{y'}{1-y^2} -= -1 -\quad\Rightarrow\quad -\int \frac{dy}{1-y^2} = \int \,du -\quad\Rightarrow\quad -\operatorname{artanh}(y) = u -\quad\Rightarrow\quad -\operatorname{sn}(u,1)=\tanh u. -\] -Damit kann man jetzt auch $z$ berechnen: -\begin{align*} -(\log \operatorname{cn}(u,1))' -&= -\tanh u -&&\Rightarrow& -\log\operatorname{cn}(u,1) -&= --\int\tanh u\,du -= --\log\cosh u -\\ -& -&&\Rightarrow& -\operatorname{cn}(u,1) -&= -\frac{1}{\cosh u} -= -\operatorname{sech}u. -\end{align*} -Die Grenzfunktionen sind in Abbildung~\ref{buch:elliptisch:fig:sncnlimit} -dargestellt. - +%Gesucht ist jetzt also eine Wahl für das Argument $u$ zum Beispiel in +%Abhängigkeit von $\varphi$, dass sich einfache und nützliche +%Ableitungsformeln ergeben. +%Wir setzen daher $u(\varphi)$ voraus und beachten, dass $x$ und $y$ +%ebenfalls von $\varphi$ abhängen, es ist +%$y=\sin\varphi$ und $x=a\cos\varphi$. +%Die Ableitungen von $x$ und $y$ nach $\varphi$ sind +%\begin{align*} +%\frac{dy}{d\varphi} +%&= +%\cos\varphi +%= +%\frac{1}{a} x +%= +%\operatorname{cn}(u,k) +%\\ +%\frac{dx}{d\varphi} +%&= +%-a\sin\varphi +%= +%-a y +%= +%-a\operatorname{sn}(u,k). +%\end{align*} +%Daraus kann man jetzt die folgenden Ausdrücke für die Ableitungen der +%elliptischen Funktionen nach $\varphi$ ableiten: +%\begin{align*} +%\frac{d}{d\varphi} \operatorname{sn}(u,z) +%&= +%\frac{d}{d\varphi} y(\varphi) +%= +%\cos\varphi +%= +%\frac{x}{a} +%= +%\operatorname{cn}(u,k) +%&&\Rightarrow& +%\frac{d}{du} +%\operatorname{sn}(u,k) +%&= +%\operatorname{cn}(u,k) \frac{d\varphi}{du} +%\\ +%\frac{d}{d\varphi} \operatorname{cn}(u,z) +%&= +%\frac{d}{d\varphi} \frac{x(\varphi)}{a} +%= +%-\sin\varphi +%= +%-\operatorname{sn}(u,k) +%&&\Rightarrow& +%\frac{d}{du}\operatorname{cn}(u,k) +%&= +%-\operatorname{sn}(u,k) \frac{d\varphi}{du} +%\\ +%\frac{d}{d\varphi} \operatorname{dn}(u,z) +%&= +%\frac{1}{a}\frac{dr}{d\varphi} +%= +%\frac{1}{a}\frac{d\sqrt{x^2+y^2}}{d\varphi} +%%\\ +%%& +%\rlap{$\displaystyle\mathstrut +%= +%\frac{x}{ar} \frac{dx}{d\varphi} +%+ +%\frac{y}{ar} \frac{dy}{d\varphi} +%%\\ +%%& +%= +%\frac{x}{ar} (-a\operatorname{sn}(u,k)) +%+ +%\frac{y}{ar} \operatorname{cn}(u,k) +%$} +%\\ +%& +%\rlap{$\displaystyle\mathstrut +%= +%\frac{x}{ar}(-ay) +%+ +%\frac{y}{ar} \frac{x}{a} +%%\rlap{$\displaystyle +%= +%\frac{xy(-1+\frac{1}{a^2})}{r} +%%$} +%%\\ +%%& +%= +%-\frac{xy(a^2-1)}{a^2r} +%$} +%\\ +%&= +%-\frac{a^2-1}{ar} +%\operatorname{cn}(u,k) \operatorname{sn}(u,k) +%%\\ +%%& +%\rlap{$\displaystyle\mathstrut +%= +%-k^2 +%\frac{a}{r} +%\operatorname{cn}(u,k) \operatorname{sn}(u,k) +%$} +%\\ +%&= +%-k^2\frac{\operatorname{cn}(u,k)\operatorname{sn}(u,k)}{\operatorname{dn}(u,k)} +%&&\Rightarrow& +%\frac{d}{du} \operatorname{dn}(u,k) +%&= +%-k^2\frac{\operatorname{cn}(u,k) +%\operatorname{sn}(u,k)}{\operatorname{dn}(u,k)} +%\frac{d\varphi}{du}. +%\end{align*} +%Die einfachsten Beziehungen ergeben sich offenbar, wenn man $u$ so +%wählt, dass +%\[ +%\frac{d\varphi}{du} +%= +%\operatorname{dn}(u,k) +%= +%\frac{r}{a}. +%\] +%Damit haben wir die grundlegenden Ableitungsregeln % -% Das Argument u +%\begin{satz} +%\label{buch:elliptisch:satz:ableitungen} +%Die Jacobischen elliptischen Funktionen haben die Ableitungen +%\begin{equation} +%\begin{aligned} +%\frac{d}{du}\operatorname{sn}(u,k) +%&= +%\phantom{-}\operatorname{cn}(u,k)\operatorname{dn}(u,k) +%\\ +%\frac{d}{du}\operatorname{cn}(u,k) +%&= +%-\operatorname{sn}(u,k)\operatorname{dn}(u,k) +%\\ +%\frac{d}{du}\operatorname{dn}(u,k) +%&= +%-k^2\operatorname{sn}(u,k)\operatorname{cn}(u,k). +%\end{aligned} +%\label{buch:elliptisch:eqn:ableitungsregeln} +%\end{equation} +%\end{satz} % -\subsubsection{Das Argument $u$} -Die Gleichung -\begin{equation} -\frac{d\varphi}{du} -= -\operatorname{dn}(u,k) -\label{buch:elliptisch:eqn:uableitung} -\end{equation} -ermöglicht, $\varphi$ in Abhängigkeit von $u$ zu berechnen, ohne jedoch -die geometrische Bedeutung zu klären. -Das beginnt bereits damit, dass der Winkel $\varphi$ nicht nicht der -Polarwinkel des Punktes $P$ in Abbildung~\ref{buch:elliptisch:fig:jacobidef} -ist, diesen nennen wir $\vartheta$. -Der Zusammenhang zwischen $\varphi$ und $\vartheta$ ist -\begin{equation} -\frac1{a}\tan\varphi = \tan\vartheta -\label{buch:elliptisch:eqn:phitheta} -\end{equation} - -Um die geometrische Bedeutung besser zu verstehen, nehmen wir jetzt an, -dass die Ellipse mit einem Parameter $t$ parametrisiert ist, dass also -$\varphi(t)$, $\vartheta(t)$ und $u(t)$ Funktionen von $t$ sind. -Die Ableitung von~\eqref{buch:elliptisch:eqn:phitheta} ist -\[ -\frac1{a}\cdot \frac{1}{\cos^2\varphi}\cdot \dot{\varphi} -= -\frac{1}{\cos^2\vartheta}\cdot \dot{\vartheta}. -\] -Daraus kann die Ableitung von $\vartheta$ nach $\varphi$ bestimmt -werden, sie ist -\[ -\frac{d\vartheta}{d\varphi} -= -\frac{\dot{\vartheta}}{\dot{\varphi}} -= -\frac{1}{a} -\cdot -\frac{\cos^2\vartheta}{\cos^2\varphi} -= -\frac{1}{a} -\cdot -\frac{(x/r)^2}{(x/a)^2} -= -\frac{1}{a}\cdot -\frac{a^2}{r^2} -= -\frac{1}{a}\cdot\frac{1}{\operatorname{dn}^2(u,k)}. -\] -Damit kann man jetzt mit Hilfe von~\eqref{buch:elliptisch:eqn:uableitung} -Die Ableitung von $\vartheta$ nach $u$ ermitteln, sie ist -\[ -\frac{d\vartheta}{du} -= -\frac{d\vartheta}{d\varphi} -\cdot -\frac{d\varphi}{du} -= -\frac{1}{a}\cdot\frac{1}{\operatorname{dn}^2(u,k)} -\cdot -\operatorname{dn}(u,k) -= -\frac{1}{a} -\cdot -\frac{1}{\operatorname{dn}(u,k)} -= -\frac{1}{a} -\cdot\frac{a}{r} -= -\frac{1}{r}, -\] -wobei wir auch die Definition der Funktion $\operatorname{dn}(u,k)$ -verwendet haben. - -In der Parametrisierung mit dem Parameter $t$ kann man jetzt die Ableitung -von $u$ nach $t$ berechnen als -\[ -\frac{du}{dt} -= -\frac{du}{d\vartheta} -\frac{d\vartheta}{dt} -= -r -\dot{\vartheta}. -\] -Darin ist $\dot{\vartheta}$ die Winkelgeschwindigkeit des Punktes um -das Zentrum $O$ und $r$ ist die aktuelle Entfernung des Punktes $P$ -von $O$. -$r\dot{\vartheta}$ ist also die Geschwindigkeitskomponenten des Punktes -$P$ senkrecht auf den aktuellen Radiusvektor. -Der Parameter $u$, der zum Punkt $P$ gehört, ist also das Integral -\[ -u(P) = \int_0^P r\,d\vartheta. -\] -Für einen Kreis ist die Geschwindigkeit von $P$ immer senkrecht -auf dem Radiusvektor und der Radius ist konstant, so dass -$u(P)=\vartheta(P)$ ist. - +%% +%% Der Grenzfall $k=1$ +%% +%\subsubsection{Der Grenzwert $k\to1$} +%\begin{figure} +%\centering +%\includegraphics{chapters/110-elliptisch/images/sncnlimit.pdf} +%\caption{Grenzfälle der Jacobischen elliptischen Funktionen +%für die Werte $0$ und $1$ des Parameters $k$. +%\label{buch:elliptisch:fig:sncnlimit}} +%\end{figure} +%Für $k=1$ ist $k^{\prime2}=1-k^2=$ und es folgt aus den +%Relationen~\eqref{buch:elliptisch:eqn:jacobi-relationen} +%\[ +%\operatorname{cn}^2(u,k) +%- +%k^2 +%\operatorname{dn}^2(u,k) +%= +%k^{\prime2} +%= +%0 +%\qquad\Rightarrow\qquad +%\operatorname{cn}^2(u,1) +%= +%\operatorname{dn}^2(u,1), +%\] +%die beiden Funktionen +%$\operatorname{cn}(u,k)$ +%und +%$\operatorname{dn}(u,k)$ +%fallen also zusammen. +%Die Ableitungsregeln werden dadurch vereinfacht: +%\begin{align*} +%\operatorname{sn}'(u,1) +%&= +%\operatorname{cn}(u,1) +%\operatorname{dn}(u,1) +%= +%\operatorname{cn}^2(u,1) +%= +%1-\operatorname{sn}^2(u,1) +%&&\Rightarrow& y'&=1-y^2 +%\\ +%\operatorname{cn}'(u,1) +%&= +%- +%\operatorname{sn}(u,1) +%\operatorname{dn}(u,1) +%= +%- +%\operatorname{sn}(u,1)\operatorname{cn}(u,1) +%&&\Rightarrow& +%\frac{z'}{z}&=(\log z)' = -y +%\end{align*} +%Die erste Differentialgleichung für $y$ lässt sich separieren, man findet +%die Lösung +%\[ +%\frac{y'}{1-y^2} +%= +%1 +%\quad\Rightarrow\quad +%\int \frac{dy}{1-y^2} = \int \,du +%\quad\Rightarrow\quad +%\operatorname{artanh}(y) = u +%\quad\Rightarrow\quad +%\operatorname{sn}(u,1)=\tanh u. +%\] +%Damit kann man jetzt auch $z$ berechnen: +%\begin{align*} +%(\log \operatorname{cn}(u,1))' +%&= +%\tanh u +%&&\Rightarrow& +%\log\operatorname{cn}(u,1) +%&= +%-\int\tanh u\,du +%= +%-\log\cosh u +%\\ +%& +%&&\Rightarrow& +%\operatorname{cn}(u,1) +%&= +%\frac{1}{\cosh u} +%= +%\operatorname{sech}u. +%\end{align*} +%Die Grenzfunktionen sind in Abbildung~\ref{buch:elliptisch:fig:sncnlimit} +%dargestellt. % -% Die abgeleiteten elliptischen Funktionen +%% +%% Das Argument u +%% +%\subsubsection{Das Argument $u$} +%Die Gleichung +%\begin{equation} +%\frac{d\varphi}{du} +%= +%\operatorname{dn}(u,k) +%\label{buch:elliptisch:eqn:uableitung} +%\end{equation} +%ermöglicht, $\varphi$ in Abhängigkeit von $u$ zu berechnen, ohne jedoch +%die geometrische Bedeutung zu klären. +%Das beginnt bereits damit, dass der Winkel $\varphi$ nicht nicht der +%Polarwinkel des Punktes $P$ in Abbildung~\ref{buch:elliptisch:fig:jacobidef} +%ist, diesen nennen wir $\vartheta$. +%Der Zusammenhang zwischen $\varphi$ und $\vartheta$ ist +%\begin{equation} +%\frac1{a}\tan\varphi = \tan\vartheta +%\label{buch:elliptisch:eqn:phitheta} +%\end{equation} % -\begin{figure} -\centering -\includegraphics[width=\textwidth]{chapters/110-elliptisch/images/jacobi12.pdf} -\caption{Die Verhältnisse der Funktionen -$\operatorname{sn}(u,k)$, -$\operatorname{cn}(u,k)$ -udn -$\operatorname{dn}(u,k)$ -geben Anlass zu neun weitere Funktionen, die sich mit Hilfe -des Strahlensatzes geometrisch interpretieren lassen. -\label{buch:elliptisch:fig:jacobi12}} -\end{figure} -\begin{table} -\centering -\renewcommand{\arraystretch}{2.5} -\begin{tabular}{|>{$\displaystyle}c<{$}|>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}|} -\hline -\cdot & -\frac{1}{1} & -\frac{1}{\operatorname{sn}(u,k)} & -\frac{1}{\operatorname{cn}(u,k)} & -\frac{1}{\operatorname{dn}(u,k)} -\\[5pt] -\hline -1& -&%\operatorname{nn}(u,k)=\frac{1}{1} & -\operatorname{ns}(u,k)=\frac{1}{\operatorname{sn}(u,k)} & -\operatorname{nc}(u,k)=\frac{1}{\operatorname{cn}(u,k)} & -\operatorname{nd}(u,k)=\frac{1}{\operatorname{dn}(u,k)} -\\ -\operatorname{sn}(u,k) & -\operatorname{sn}(u,k)=\frac{\operatorname{sn}(u,k)}{1}& -&%\operatorname{ss}(u,k)=\frac{\operatorname{sn}(u,k)}{\operatorname{sn}(u,k)}& -\operatorname{sc}(u,k)=\frac{\operatorname{sn}(u,k)}{\operatorname{cn}(u,k)}& -\operatorname{sd}(u,k)=\frac{\operatorname{sn}(u,k)}{\operatorname{dn}(u,k)} -\\ -\operatorname{cn}(u,k) & -\operatorname{cn}(u,k)=\frac{\operatorname{cn}(u,k)}{1} & -\operatorname{cs}(u,k)=\frac{\operatorname{cn}(u,k)}{\operatorname{sn}(u,k)}& -&%\operatorname{cc}(u,k)=\frac{\operatorname{cn}(u,k)}{\operatorname{cn}(u,k)}& -\operatorname{cd}(u,k)=\frac{\operatorname{cn}(u,k)}{\operatorname{dn}(u,k)} -\\ -\operatorname{dn}(u,k) & -\operatorname{dn}(u,k)=\frac{\operatorname{dn}(u,k)}{1} & -\operatorname{ds}(u,k)=\frac{\operatorname{dn}(u,k)}{\operatorname{sn}(u,k)}& -\operatorname{dc}(u,k)=\frac{\operatorname{dn}(u,k)}{\operatorname{cn}(u,k)}& -%\operatorname{dd}(u,k)=\frac{\operatorname{dn}(u,k)}{\operatorname{dn}(u,k)} -\\[5pt] -\hline -\end{tabular} -\caption{Zusammenstellung der abgeleiteten Jacobischen elliptischen -Funktionen in hinteren drei Spalten als Quotienten der grundlegenden -Jacobischen elliptischen Funktionen. -Die erste Spalte zum Nenner $1$ enthält die grundlegenden -Jacobischen elliptischen Funktionen. -\label{buch:elliptisch:table:abgeleitetjacobi}} -\end{table} -\subsubsection{Die abgeleiteten elliptischen Funktionen} -Zusätzlich zu den grundlegenden Jacobischen elliptischen Funktioenn -lassen sich weitere elliptische Funktionen bilden, die unglücklicherweise -die {\em abgeleiteten elliptischen Funktionen} genannt werden. -Ähnlich wie die trigonometrischen Funktionen $\tan\alpha$, $\cot\alpha$, -$\sec\alpha$ und $\csc\alpha$ als Quotienten von $\sin\alpha$ und -$\cos\alpha$ definiert sind, sind die abgeleiteten elliptischen Funktionen -die in Tabelle~\ref{buch:elliptisch:table:abgeleitetjacobi} zusammengestellten -Quotienten der grundlegenden Jacobischen elliptischen Funktionen. -Die Bezeichnungskonvention ist, dass die Funktion $\operatorname{pq}(u,k)$ -ein Quotient ist, dessen Zähler durch den Buchstaben p bestimmt ist, -der Nenner durch den Buchstaben q. -Der Buchstabe n steht für eine $1$, die Buchstaben s, c und d stehen für -die Anfangsbuchstaben der grundlegenden Jacobischen elliptischen -Funktionen. -Meint man irgend eine der Jacobischen elliptischen Funktionen, schreibt -man manchmal auch $\operatorname{zn}(u,k)$. - -In Abbildung~\ref{buch:elliptisch:fig:jacobi12} sind die Quotienten auch -geometrisch interpretiert. -Der Wert der Funktion $\operatorname{nq}(u,k)$ ist die auf dem Strahl -mit Polarwinkel $\varphi$ abgetragene Länge bis zu den vertikalen -Geraden, die den verschiedenen möglichen Nennern entsprechen. -Entsprechend ist der Wert der Funktion $\operatorname{dq}(u,k)$ die -Länge auf dem Strahl mit Polarwinkel $\vartheta$. - -Die Relationen~\ref{buch:elliptisch:eqn:jacobi-relationen} -ermöglichen, jede Funktion $\operatorname{zn}(u,k)$ durch jede -andere auszudrücken. -Die schiere Anzahl solcher Beziehungen macht es unmöglich, sie -übersichtlich in einer Tabelle zusammenzustellen, daher soll hier -nur an einem Beispiel das Vorgehen gezeigt werden: - -\begin{beispiel} -Die Funktion $\operatorname{sc}(u,k)$ soll durch $\operatorname{cd}(u,k)$ -ausgedrückt werden. -Zunächst ist -\[ -\operatorname{sc}(u,k) -= -\frac{\operatorname{sn}(u,k)}{\operatorname{cn}(u,k)} -\] -nach Definition. -Im Resultat sollen nur noch $\operatorname{cn}(u,k)$ und -$\operatorname{dn}(u,k)$ vorkommen. -Daher eliminieren wir zunächst die Funktion $\operatorname{sn}(u,k)$ -mit Hilfe von \eqref{buch:elliptisch:eqn:jacobi-relationen} und erhalten -\begin{equation} -\operatorname{sc}(u,k) -= -\frac{\sqrt{1-\operatorname{cn}^2(u,k)}}{\operatorname{cn}(u,k)}. -\label{buch:elliptisch:eqn:allgausdruecken} -\end{equation} -Nun genügt es, die Funktion $\operatorname{cn}(u,k)$ durch -$\operatorname{cd}(u,k)$ auszudrücken. -Aus der Definition und der -dritten Relation in \eqref{buch:elliptisch:eqn:jacobi-relationen} -erhält man -\begin{align*} -\operatorname{cd}^2(u,k) -&= -\frac{\operatorname{cn}^2(u,k)}{\operatorname{dn}^2(u,k)} -= -\frac{\operatorname{cn}^2(u,k)}{k^{\prime2}+k^2\operatorname{cn}^2(u,k)} -\\ -\Rightarrow -\qquad -k^{\prime 2} -\operatorname{cd}^2(u,k) -+ -k^2\operatorname{cd}^2(u,k)\operatorname{cn}^2(u,k) -&= -\operatorname{cn}^2(u,k) -\\ -\operatorname{cn}^2(u,k) -- -k^2\operatorname{cd}^2(u,k)\operatorname{cn}^2(u,k) -&= -k^{\prime 2} -\operatorname{cd}^2(u,k) -\\ -\operatorname{cn}^2(u,k) -&= -\frac{ -k^{\prime 2} -\operatorname{cd}^2(u,k) -}{ -1 - k^2\operatorname{cd}^2(u,k) -} -\end{align*} -Für den Zähler brauchen wir $1-\operatorname{cn}^2(u,k)$, also -\[ -1-\operatorname{cn}^2(u,k) -= -\frac{ -1 -- -k^2\operatorname{cd}^2(u,k) -- -k^{\prime 2} -\operatorname{cd}^2(u,k) -}{ -1 -- -k^2\operatorname{cd}^2(u,k) -} -= -\frac{1-\operatorname{cd}^2(u,k)}{1-k^2\operatorname{cd}^2(u,k)} -\] -Einsetzen in~\eqref{buch:elliptisch:eqn:allgausdruecken} gibt -\begin{align*} -\operatorname{sc}(u,k) -&= -\frac{ -\sqrt{1-\operatorname{cd}^2(u,k)} -}{\sqrt{1-k^2\operatorname{cd}^2(u,k)}} -\cdot -\frac{ -\sqrt{1 - k^2\operatorname{cd}^2(u,k)} -}{ -k' -\operatorname{cd}(u,k) -} -= -\frac{ -\sqrt{1-\operatorname{cd}^2(u,k)} -}{ -k' -\operatorname{cd}(u,k) -}. -\qedhere -\end{align*} -\end{beispiel} - -\subsubsection{Ableitung der abgeleiteten elliptischen Funktionen} -Aus den Ableitungen der grundlegenden Jacobischen elliptischen Funktionen -können mit der Quotientenregel nun auch beliebige Ableitungen der -abgeleiteten Jacobischen elliptischen Funktionen gefunden werden. -Als Beispiel berechnen wir die Ableitung von $\operatorname{sc}(u,k)$. -Sie ist -\begin{align*} -\frac{d}{du} -\operatorname{sc}(u,k) -&= -\frac{d}{du} -\frac{\operatorname{sn}(u,k)}{\operatorname{cn}(u,k)} -= -\frac{ -\operatorname{sn}'(u,k)\operatorname{cn}(u,k) -- -\operatorname{sn}(u,k)\operatorname{cn}'(u,k)}{ -\operatorname{cn}^2(u,k) -} -\\ -&= -\frac{ -\operatorname{cn}^2(u,k)\operatorname{dn}(u,k) -+ -\operatorname{sn}^2(u,k)\operatorname{dn}(u,k) -}{ -\operatorname{cn}^2(u,k) -} -= -\frac{( -\operatorname{sn}^2(u,k) -+ -\operatorname{cn}^2(u,k) -)\operatorname{dn}(u,k)}{ -\operatorname{cn}^2(u,k) -} -\\ -&= -\frac{1}{\operatorname{cn}(u,k)} -\cdot -\frac{\operatorname{dn}(u,k)}{\operatorname{cn}(u,k)} -= -\operatorname{nc}(u,k) -\operatorname{dc}(u,k). -\end{align*} -Man beachte, dass das Quadrat der Nennerfunktion im Resultat -der Quotientenregel zur Folge hat, dass die -beiden Funktionen im Resultat beide den gleichen Nenner haben wie -die Funktion, die abgeleitet wird. - -Mit etwas Fleiss kann man nach diesem Muster alle Ableitungen -\begin{equation} -%\small -\begin{aligned} -\operatorname{sn}'(u,k) -&= -\phantom{-} -\operatorname{cn}(u,k)\,\operatorname{dn}(u,k) -&&\qquad& -\operatorname{ns}'(u,k) -&= -- -\operatorname{cs}(u,k)\,\operatorname{ds}(u,k) -\\ -\operatorname{cn}'(u,k) -&= -- -\operatorname{sn}(u,k)\,\operatorname{dn}(u,k) -&&& -\operatorname{nc}'(u,k) -&= -\phantom{-} -\operatorname{sc}(u,k)\,\operatorname{dc}(u,k) -\\ -\operatorname{dn}'(u,k) -&= --k^2 -\operatorname{sn}(u,k)\,\operatorname{cn}(u,k) -&&& -\operatorname{nd}'(u,k) -&= -\phantom{-} -k^2 -\operatorname{sd}(u,k)\,\operatorname{cd}(u,k) -\\ -\operatorname{sc}'(u,k) -&= -\phantom{-} -\operatorname{dc}(u,k)\,\operatorname{nc}(u,k) -&&& -\operatorname{cs}'(u,k) -&= -- -\operatorname{ds}(u,k)\,\operatorname{ns}(u,k) -\\ -\operatorname{cd}'(u,k) -&= --k^{\prime2} -\operatorname{sd}(u,k)\,\operatorname{nd}(u,k) -&&& -\operatorname{dc}'(u,k) -&= -\phantom{-} -k^{\prime2} -\operatorname{dc}(u,k)\,\operatorname{nc}(u,k) -\\ -\operatorname{ds}'(d,k) -&= -- -\operatorname{cs}(u,k)\,\operatorname{ns}(u,k) -&&& -\operatorname{sd}'(d,k) -&= -\phantom{-} -\operatorname{cd}(u,k)\,\operatorname{nd}(u,k) -\end{aligned} -\label{buch:elliptisch:eqn:alleableitungen} -\end{equation} -finden. -Man beachte, dass in jeder Identität alle Funktionen den gleichen -zweiten Buchstaben haben. - -\subsubsection{TODO} -XXX algebraische Beziehungen \\ -XXX Additionstheoreme \\ -XXX Perioden -% use https://math.stackexchange.com/questions/3013692/how-to-show-that-jacobi-sine-function-is-doubly-periodic - - -XXX Ableitungen \\ -XXX Werte \\ - +%Um die geometrische Bedeutung besser zu verstehen, nehmen wir jetzt an, +%dass die Ellipse mit einem Parameter $t$ parametrisiert ist, dass also +%$\varphi(t)$, $\vartheta(t)$ und $u(t)$ Funktionen von $t$ sind. +%Die Ableitung von~\eqref{buch:elliptisch:eqn:phitheta} ist +%\[ +%\frac1{a}\cdot \frac{1}{\cos^2\varphi}\cdot \dot{\varphi} +%= +%\frac{1}{\cos^2\vartheta}\cdot \dot{\vartheta}. +%\] +%Daraus kann die Ableitung von $\vartheta$ nach $\varphi$ bestimmt +%werden, sie ist +%\[ +%\frac{d\vartheta}{d\varphi} +%= +%\frac{\dot{\vartheta}}{\dot{\varphi}} +%= +%\frac{1}{a} +%\cdot +%\frac{\cos^2\vartheta}{\cos^2\varphi} +%= +%\frac{1}{a} +%\cdot +%\frac{(x/r)^2}{(x/a)^2} +%= +%\frac{1}{a}\cdot +%\frac{a^2}{r^2} +%= +%\frac{1}{a}\cdot\frac{1}{\operatorname{dn}^2(u,k)}. +%\] +%Damit kann man jetzt mit Hilfe von~\eqref{buch:elliptisch:eqn:uableitung} +%Die Ableitung von $\vartheta$ nach $u$ ermitteln, sie ist +%\[ +%\frac{d\vartheta}{du} +%= +%\frac{d\vartheta}{d\varphi} +%\cdot +%\frac{d\varphi}{du} +%= +%\frac{1}{a}\cdot\frac{1}{\operatorname{dn}^2(u,k)} +%\cdot +%\operatorname{dn}(u,k) +%= +%\frac{1}{a} +%\cdot +%\frac{1}{\operatorname{dn}(u,k)} +%= +%\frac{1}{a} +%\cdot\frac{a}{r} +%= +%\frac{1}{r}, +%\] +%wobei wir auch die Definition der Funktion $\operatorname{dn}(u,k)$ +%verwendet haben. % -% Lösung von Differentialgleichungen +%In der Parametrisierung mit dem Parameter $t$ kann man jetzt die Ableitung +%von $u$ nach $t$ berechnen als +%\[ +%\frac{du}{dt} +%= +%\frac{du}{d\vartheta} +%\frac{d\vartheta}{dt} +%= +%r +%\dot{\vartheta}. +%\] +%Darin ist $\dot{\vartheta}$ die Winkelgeschwindigkeit des Punktes um +%das Zentrum $O$ und $r$ ist die aktuelle Entfernung des Punktes $P$ +%von $O$. +%$r\dot{\vartheta}$ ist also die Geschwindigkeitskomponenten des Punktes +%$P$ senkrecht auf den aktuellen Radiusvektor. +%Der Parameter $u$, der zum Punkt $P$ gehört, ist also das Integral +%\[ +%u(P) = \int_0^P r\,d\vartheta. +%\] +%Für einen Kreis ist die Geschwindigkeit von $P$ immer senkrecht +%auf dem Radiusvektor und der Radius ist konstant, so dass +%$u(P)=\vartheta(P)$ ist. % -\subsection{Lösungen von Differentialgleichungen} -Die elliptischen Funktionen ermöglichen die Lösung gewisser nichtlinearer -Differentialgleichungen in geschlossener Form. -Ziel dieses Abschnitts ist, Differentialgleichungen der Form -\( -\ddot{x}(t) -= -p(x(t)) -\) -mit einem Polynom dritten Grades als rechter Seite lösen zu können. - +%% +%% Die abgeleiteten elliptischen Funktionen +%% +%\begin{figure} +%\centering +%\includegraphics[width=\textwidth]{chapters/110-elliptisch/images/jacobi12.pdf} +%\caption{Die Verhältnisse der Funktionen +%$\operatorname{sn}(u,k)$, +%$\operatorname{cn}(u,k)$ +%udn +%$\operatorname{dn}(u,k)$ +%geben Anlass zu neun weitere Funktionen, die sich mit Hilfe +%des Strahlensatzes geometrisch interpretieren lassen. +%\label{buch:elliptisch:fig:jacobi12}} +%\end{figure} +%\begin{table} +%\centering +%\renewcommand{\arraystretch}{2.5} +%\begin{tabular}{|>{$\displaystyle}c<{$}|>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}|} +%\hline +%\cdot & +%\frac{1}{1} & +%\frac{1}{\operatorname{sn}(u,k)} & +%\frac{1}{\operatorname{cn}(u,k)} & +%\frac{1}{\operatorname{dn}(u,k)} +%\\[5pt] +%\hline +%1& +%&%\operatorname{nn}(u,k)=\frac{1}{1} & +%\operatorname{ns}(u,k)=\frac{1}{\operatorname{sn}(u,k)} & +%\operatorname{nc}(u,k)=\frac{1}{\operatorname{cn}(u,k)} & +%\operatorname{nd}(u,k)=\frac{1}{\operatorname{dn}(u,k)} +%\\ +%\operatorname{sn}(u,k) & +%\operatorname{sn}(u,k)=\frac{\operatorname{sn}(u,k)}{1}& +%&%\operatorname{ss}(u,k)=\frac{\operatorname{sn}(u,k)}{\operatorname{sn}(u,k)}& +%\operatorname{sc}(u,k)=\frac{\operatorname{sn}(u,k)}{\operatorname{cn}(u,k)}& +%\operatorname{sd}(u,k)=\frac{\operatorname{sn}(u,k)}{\operatorname{dn}(u,k)} +%\\ +%\operatorname{cn}(u,k) & +%\operatorname{cn}(u,k)=\frac{\operatorname{cn}(u,k)}{1} & +%\operatorname{cs}(u,k)=\frac{\operatorname{cn}(u,k)}{\operatorname{sn}(u,k)}& +%&%\operatorname{cc}(u,k)=\frac{\operatorname{cn}(u,k)}{\operatorname{cn}(u,k)}& +%\operatorname{cd}(u,k)=\frac{\operatorname{cn}(u,k)}{\operatorname{dn}(u,k)} +%\\ +%\operatorname{dn}(u,k) & +%\operatorname{dn}(u,k)=\frac{\operatorname{dn}(u,k)}{1} & +%\operatorname{ds}(u,k)=\frac{\operatorname{dn}(u,k)}{\operatorname{sn}(u,k)}& +%\operatorname{dc}(u,k)=\frac{\operatorname{dn}(u,k)}{\operatorname{cn}(u,k)}& +%%\operatorname{dd}(u,k)=\frac{\operatorname{dn}(u,k)}{\operatorname{dn}(u,k)} +%\\[5pt] +%\hline +%\end{tabular} +%\caption{Zusammenstellung der abgeleiteten Jacobischen elliptischen +%Funktionen in hinteren drei Spalten als Quotienten der grundlegenden +%Jacobischen elliptischen Funktionen. +%Die erste Spalte zum Nenner $1$ enthält die grundlegenden +%Jacobischen elliptischen Funktionen. +%\label{buch:elliptisch:table:abgeleitetjacobi}} +%\end{table} +%\subsubsection{Die abgeleiteten elliptischen Funktionen} +%Zusätzlich zu den grundlegenden Jacobischen elliptischen Funktioenn +%lassen sich weitere elliptische Funktionen bilden, die unglücklicherweise +%die {\em abgeleiteten elliptischen Funktionen} genannt werden. +%Ähnlich wie die trigonometrischen Funktionen $\tan\alpha$, $\cot\alpha$, +%$\sec\alpha$ und $\csc\alpha$ als Quotienten von $\sin\alpha$ und +%$\cos\alpha$ definiert sind, sind die abgeleiteten elliptischen Funktionen +%die in Tabelle~\ref{buch:elliptisch:table:abgeleitetjacobi} zusammengestellten +%Quotienten der grundlegenden Jacobischen elliptischen Funktionen. +%Die Bezeichnungskonvention ist, dass die Funktion $\operatorname{pq}(u,k)$ +%ein Quotient ist, dessen Zähler durch den Buchstaben p bestimmt ist, +%der Nenner durch den Buchstaben q. +%Der Buchstabe n steht für eine $1$, die Buchstaben s, c und d stehen für +%die Anfangsbuchstaben der grundlegenden Jacobischen elliptischen +%Funktionen. +%Meint man irgend eine der Jacobischen elliptischen Funktionen, schreibt +%man manchmal auch $\operatorname{zn}(u,k)$. % -% Die Differentialgleichung der elliptischen Funktionen +%In Abbildung~\ref{buch:elliptisch:fig:jacobi12} sind die Quotienten auch +%geometrisch interpretiert. +%Der Wert der Funktion $\operatorname{nq}(u,k)$ ist die auf dem Strahl +%mit Polarwinkel $\varphi$ abgetragene Länge bis zu den vertikalen +%Geraden, die den verschiedenen möglichen Nennern entsprechen. +%Entsprechend ist der Wert der Funktion $\operatorname{dq}(u,k)$ die +%Länge auf dem Strahl mit Polarwinkel $\vartheta$. % -\subsubsection{Die Differentialgleichungen der elliptischen Funktionen} -Um Differentialgleichungen mit elliptischen Funktion lösen zu -können, muss man als erstes die Differentialgleichungen derselben -finden. -Quadriert man die Ableitungsregel für $\operatorname{sn}(u,k)$, erhält -man -\[ -\biggl(\frac{d}{du}\operatorname{sn}(u,k)\biggr)^2 -= -\operatorname{cn}(u,k)^2 \operatorname{dn}(u,k)^2. -\] -Die Funktionen auf der rechten Seite können durch $\operatorname{sn}(u,k)$ -ausgedrückt werden. -\begin{align*} -\biggl(\frac{d}{du}\operatorname{sn}(u,k)\biggr)^2 -&= -\biggl( -1-\operatorname{sn}(u,k)^2 -\biggr) -\biggl( -1-k^2 \operatorname{sn}(u,k)^2 -\biggr) -\\ -&= -k^2\operatorname{sn}(u,k)^4 --(1+k^2) -\operatorname{sn}(u,k)^2 -+1. -\end{align*} -Für die Funktion $\operatorname{cn}(u,k)$ ergibt analoge Rechnung -\begin{align*} -\frac{d}{du}\operatorname{cn}(u,k) -&= --\operatorname{sn}(u,k) \operatorname{dn}(u,k) -\\ -\biggl(\frac{d}{du}\operatorname{cn}(u,k)\biggr)^2 -&= -\operatorname{sn}(u,k)^2 \operatorname{dn}(u,k)^2 -\\ -&= -\biggl(1-\operatorname{cn}(u,k)^2\biggr) -\biggl(1-k^2+k^2 \operatorname{cn}(u,k)^2\biggr) -\\ -&= --k^2\operatorname{cn}(u,k)^4 -- -(1-k^2-k^2)\operatorname{cn}(u,k)^2 -+ -(1-k^2) -\\ -\frac{d}{du}\operatorname{dn}(u,k) -&= --k^2\operatorname{sn}(u,k)\operatorname{cn}(u,k) -\\ -\biggl( -\frac{d}{du}\operatorname{dn}(u,k) -\biggr)^2 -&= -\bigl(k^2 \operatorname{sn}(u,k)^2\bigr) -\bigl(k^2 \operatorname{cn}(u,k)^2\bigr) -\\ -&= -\biggl( -1-\operatorname{dn}(u,k)^2 -\biggr) -\biggl( -\operatorname{dn}(u,k)^2-k^2+1 -\biggr) -\\ -&= --\operatorname{dn}(u,k)^4 -- -2\operatorname{dn}(u,k)^2 --k^2+1. -\end{align*} -\begin{table} -\centering -\renewcommand{\arraystretch}{2} -\begin{tabular}{|>{$}l<{$}|>{$}l<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}|} -\hline -\text{Funktion $y=$}&\text{Differentialgleichung}&\alpha&\beta&\gamma&\multicolumn{3}{c|}{Signatur}\\ -\hline -\operatorname{sn}(u,k) - & y'^2 = \phantom{-}(1-y^2)(1-k^2y^2) - &k^2&1&1 &+&+&+ -\\ -\operatorname{cn}(u,k) - &y'^2 = \phantom{-}(1-y^2)(1-k^2+k^2y^2) - &-k^2 &2k^2-1&1-k^2 &-&&+ -\\ -\operatorname{dn}(u,k) - & y'^2 = -(1-y^2)(1-k^2-y^2) - &1 &1-k^2 &-(1-k^2)&+&+&- -\\ -\hline -\end{tabular} -\caption{Elliptische Funktionen als Lösungsfunktionen für verschiedene -nichtlineare Differentialgleichungen der Art -\eqref{buch:elliptisch:eqn:1storderdglell}. -Die Vorzeichen der Koeffizienten $\alpha$, $\beta$ und $\gamma$ -entscheidet darüber, welche Funktion für die Lösung verwendet werden -muss. -\label{buch:elliptisch:tabelle:loesungsfunktionen}} -\end{table} - -Die elliptischen Funktionen genügen also alle einer nichtlinearen -Differentialgleichung erster Ordnung der selben Art. -Das Quadrat der Ableitung ist ein Polynom vierten Grades der Funktion. -Um dies besser einzufangen, schreiben wir $\operatorname{zn}(u,k)$, -wenn wir eine beliebige der drei Funktionen -$\operatorname{sn}(u,k)$, -$\operatorname{cn}(u,k)$ -oder -$\operatorname{dn}(u,k)$ -meinen. -Die Funktion $\operatorname{zn}(u,k)$ ist also Lösung der -Differentialgleichung -\begin{equation} -\operatorname{zn}'(u,k)^2 -= -\alpha \operatorname{zn}(u,k)^4 + \beta \operatorname{zn}(u,)^2 + \gamma, -\label{buch:elliptisch:eqn:1storderdglell} -\end{equation} -wobei wir mit $\operatorname{zn}'(u,k)$ die Ableitung von -$\operatorname{zn}(u,k)$ nach dem ersten Argument meinen. -Die Koeffizienten $\alpha$, $\beta$ und $\gamma$ hängen von $k$ ab, -vor allem aber haben Sie verschiedene Vorzeichen. -Je nach Vorzeichen sind also eine andere elliptische Funktion als -Lösung zu verwenden. - +%Die Relationen~\ref{buch:elliptisch:eqn:jacobi-relationen} +%ermöglichen, jede Funktion $\operatorname{zn}(u,k)$ durch jede +%andere auszudrücken. +%Die schiere Anzahl solcher Beziehungen macht es unmöglich, sie +%übersichtlich in einer Tabelle zusammenzustellen, daher soll hier +%nur an einem Beispiel das Vorgehen gezeigt werden: % -% Jacobischen elliptische Funktionen und elliptische Integrale +%\begin{beispiel} +%Die Funktion $\operatorname{sc}(u,k)$ soll durch $\operatorname{cd}(u,k)$ +%ausgedrückt werden. +%Zunächst ist +%\[ +%\operatorname{sc}(u,k) +%= +%\frac{\operatorname{sn}(u,k)}{\operatorname{cn}(u,k)} +%\] +%nach Definition. +%Im Resultat sollen nur noch $\operatorname{cn}(u,k)$ und +%$\operatorname{dn}(u,k)$ vorkommen. +%Daher eliminieren wir zunächst die Funktion $\operatorname{sn}(u,k)$ +%mit Hilfe von \eqref{buch:elliptisch:eqn:jacobi-relationen} und erhalten +%\begin{equation} +%\operatorname{sc}(u,k) +%= +%\frac{\sqrt{1-\operatorname{cn}^2(u,k)}}{\operatorname{cn}(u,k)}. +%\label{buch:elliptisch:eqn:allgausdruecken} +%\end{equation} +%Nun genügt es, die Funktion $\operatorname{cn}(u,k)$ durch +%$\operatorname{cd}(u,k)$ auszudrücken. +%Aus der Definition und der +%dritten Relation in \eqref{buch:elliptisch:eqn:jacobi-relationen} +%erhält man +%\begin{align*} +%\operatorname{cd}^2(u,k) +%&= +%\frac{\operatorname{cn}^2(u,k)}{\operatorname{dn}^2(u,k)} +%= +%\frac{\operatorname{cn}^2(u,k)}{k^{\prime2}+k^2\operatorname{cn}^2(u,k)} +%\\ +%\Rightarrow +%\qquad +%k^{\prime 2} +%\operatorname{cd}^2(u,k) +%+ +%k^2\operatorname{cd}^2(u,k)\operatorname{cn}^2(u,k) +%&= +%\operatorname{cn}^2(u,k) +%\\ +%\operatorname{cn}^2(u,k) +%- +%k^2\operatorname{cd}^2(u,k)\operatorname{cn}^2(u,k) +%&= +%k^{\prime 2} +%\operatorname{cd}^2(u,k) +%\\ +%\operatorname{cn}^2(u,k) +%&= +%\frac{ +%k^{\prime 2} +%\operatorname{cd}^2(u,k) +%}{ +%1 - k^2\operatorname{cd}^2(u,k) +%} +%\end{align*} +%Für den Zähler brauchen wir $1-\operatorname{cn}^2(u,k)$, also +%\[ +%1-\operatorname{cn}^2(u,k) +%= +%\frac{ +%1 +%- +%k^2\operatorname{cd}^2(u,k) +%- +%k^{\prime 2} +%\operatorname{cd}^2(u,k) +%}{ +%1 +%- +%k^2\operatorname{cd}^2(u,k) +%} +%= +%\frac{1-\operatorname{cd}^2(u,k)}{1-k^2\operatorname{cd}^2(u,k)} +%\] +%Einsetzen in~\eqref{buch:elliptisch:eqn:allgausdruecken} gibt +%\begin{align*} +%\operatorname{sc}(u,k) +%&= +%\frac{ +%\sqrt{1-\operatorname{cd}^2(u,k)} +%}{\sqrt{1-k^2\operatorname{cd}^2(u,k)}} +%\cdot +%\frac{ +%\sqrt{1 - k^2\operatorname{cd}^2(u,k)} +%}{ +%k' +%\operatorname{cd}(u,k) +%} +%= +%\frac{ +%\sqrt{1-\operatorname{cd}^2(u,k)} +%}{ +%k' +%\operatorname{cd}(u,k) +%}. +%\qedhere +%\end{align*} +%\end{beispiel} % -\subsubsection{Jacobische elliptische Funktionen als elliptische Integrale} -Die in Tabelle~\ref{buch:elliptisch:tabelle:loesungsfunktionen} -zusammengestellten Differentialgleichungen ermöglichen nun, den -Zusammenhang zwischen den Funktionen -$\operatorname{sn}(u,k)$, $\operatorname{cn}(u,k)$ und $\operatorname{dn}(u,k)$ -und den unvollständigen elliptischen Integralen herzustellen. -Die Differentialgleichungen sind alle von der Form -\begin{equation} -\biggl( -\frac{d y}{d u} -\biggr)^2 -= -p(u), -\label{buch:elliptisch:eqn:allgdgl} -\end{equation} -wobei $p(u)$ ein Polynom vierten Grades in $y$ ist. -Diese Differentialgleichung lässt sich mit Separation lösen. -Dazu zieht man aus~\eqref{buch:elliptisch:eqn:allgdgl} die -Wurzel -\begin{align} -\frac{dy}{du} -= -\sqrt{p(y)} -\notag -\intertext{und trennt die Variablen. Man erhält} -\int\frac{dy}{\sqrt{p(y)}} = u+C. -\label{buch:elliptisch:eqn:yintegral} -\end{align} -Solange $p(y)>0$ ist, ist der Integrand auf der linken Seite -von~\eqref{buch:elliptisch:eqn:yintegral} ebenfalls positiv und -das Integral ist eine monoton wachsende Funktion $F(y)$. -Insbesondere ist $F(y)$ invertierbar. -Die Lösung $y(u)$ der Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} -ist daher -\[ -y(u) = F^{-1}(u+C). -\] -Die Jacobischen elliptischen Funktionen sind daher inverse Funktionen -der unvollständigen elliptischen Integrale. - -\subsubsection{Differentialgleichung zweiter Ordnung} -Leitet die Differentialgleichung ~\eqref{buch:elliptisch:eqn:1storderdglell} -man dies nochmals nach $u$ ab, erhält man die Differentialgleichung -\[ -2\operatorname{zn}''(u,k)\operatorname{zn}'(u,k) -= -4\alpha \operatorname{zn}(u,k)^3\operatorname{zn}'(u,k) + 2\beta \operatorname{zn}'(u,k)\operatorname{zn}(u,k). -\] -Teilt man auf beiden Seiten durch $2\operatorname{zn}'(u,k)$, -bleibt die nichtlineare -Differentialgleichung -\[ -\frac{d^2\operatorname{zn}}{du^2} -= -\beta \operatorname{zn} + 2\alpha \operatorname{zn}^3. -\] -Dies ist die Gleichung eines harmonischen Oszillators mit einer -Anharmonizität der Form $2\alpha z^3$. - +%\subsubsection{Ableitung der abgeleiteten elliptischen Funktionen} +%Aus den Ableitungen der grundlegenden Jacobischen elliptischen Funktionen +%können mit der Quotientenregel nun auch beliebige Ableitungen der +%abgeleiteten Jacobischen elliptischen Funktionen gefunden werden. +%Als Beispiel berechnen wir die Ableitung von $\operatorname{sc}(u,k)$. +%Sie ist +%\begin{align*} +%\frac{d}{du} +%\operatorname{sc}(u,k) +%&= +%\frac{d}{du} +%\frac{\operatorname{sn}(u,k)}{\operatorname{cn}(u,k)} +%= +%\frac{ +%\operatorname{sn}'(u,k)\operatorname{cn}(u,k) +%- +%\operatorname{sn}(u,k)\operatorname{cn}'(u,k)}{ +%\operatorname{cn}^2(u,k) +%} +%\\ +%&= +%\frac{ +%\operatorname{cn}^2(u,k)\operatorname{dn}(u,k) +%+ +%\operatorname{sn}^2(u,k)\operatorname{dn}(u,k) +%}{ +%\operatorname{cn}^2(u,k) +%} +%= +%\frac{( +%\operatorname{sn}^2(u,k) +%+ +%\operatorname{cn}^2(u,k) +%)\operatorname{dn}(u,k)}{ +%\operatorname{cn}^2(u,k) +%} +%\\ +%&= +%\frac{1}{\operatorname{cn}(u,k)} +%\cdot +%\frac{\operatorname{dn}(u,k)}{\operatorname{cn}(u,k)} +%= +%\operatorname{nc}(u,k) +%\operatorname{dc}(u,k). +%\end{align*} +%Man beachte, dass das Quadrat der Nennerfunktion im Resultat +%der Quotientenregel zur Folge hat, dass die +%beiden Funktionen im Resultat beide den gleichen Nenner haben wie +%die Funktion, die abgeleitet wird. % -% Differentialgleichung des anharmonischen Oszillators +%Mit etwas Fleiss kann man nach diesem Muster alle Ableitungen +%\begin{equation} +%%\small +%\begin{aligned} +%\operatorname{sn}'(u,k) +%&= +%\phantom{-} +%\operatorname{cn}(u,k)\,\operatorname{dn}(u,k) +%&&\qquad& +%\operatorname{ns}'(u,k) +%&= +%- +%\operatorname{cs}(u,k)\,\operatorname{ds}(u,k) +%\\ +%\operatorname{cn}'(u,k) +%&= +%- +%\operatorname{sn}(u,k)\,\operatorname{dn}(u,k) +%&&& +%\operatorname{nc}'(u,k) +%&= +%\phantom{-} +%\operatorname{sc}(u,k)\,\operatorname{dc}(u,k) +%\\ +%\operatorname{dn}'(u,k) +%&= +%-k^2 +%\operatorname{sn}(u,k)\,\operatorname{cn}(u,k) +%&&& +%\operatorname{nd}'(u,k) +%&= +%\phantom{-} +%k^2 +%\operatorname{sd}(u,k)\,\operatorname{cd}(u,k) +%\\ +%\operatorname{sc}'(u,k) +%&= +%\phantom{-} +%\operatorname{dc}(u,k)\,\operatorname{nc}(u,k) +%&&& +%\operatorname{cs}'(u,k) +%&= +%- +%\operatorname{ds}(u,k)\,\operatorname{ns}(u,k) +%\\ +%\operatorname{cd}'(u,k) +%&= +%-k^{\prime2} +%\operatorname{sd}(u,k)\,\operatorname{nd}(u,k) +%&&& +%\operatorname{dc}'(u,k) +%&= +%\phantom{-} +%k^{\prime2} +%\operatorname{dc}(u,k)\,\operatorname{nc}(u,k) +%\\ +%\operatorname{ds}'(d,k) +%&= +%- +%\operatorname{cs}(u,k)\,\operatorname{ns}(u,k) +%&&& +%\operatorname{sd}'(d,k) +%&= +%\phantom{-} +%\operatorname{cd}(u,k)\,\operatorname{nd}(u,k) +%\end{aligned} +%\label{buch:elliptisch:eqn:alleableitungen} +%\end{equation} +%finden. +%Man beachte, dass in jeder Identität alle Funktionen den gleichen +%zweiten Buchstaben haben. % -\subsubsection{Differentialgleichung des anharmonischen Oszillators} -Wir möchten die nichtlineare Differentialgleichung -\begin{equation} -\biggl( -\frac{dx}{dt} -\biggr)^2 -= -Ax^4+Bx^2 + C -\label{buch:elliptisch:eqn:allgdgl} -\end{equation} -mit Hilfe elliptischer Funktionen lösen. -Wir nehmen also an, dass die gesuchte Lösung eine Funktion der Form -\begin{equation} -x(t) = a\operatorname{zn}(bt,k) -\label{buch:elliptisch:eqn:loesungsansatz} -\end{equation} -ist. -Die erste Ableitung von $x(t)$ ist -\[ -\dot{x}(t) -= -a\operatorname{zn}'(bt,k). -\] - -Indem wir diesen Lösungsansatz in die -Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} -einsetzen, erhalten wir -\begin{equation} -a^2b^2 \operatorname{zn}'(bt,k)^2 -= -a^4A\operatorname{zn}(bt,k)^4 -+ -a^2B\operatorname{zn}(bt,k)^2 -+C -\label{buch:elliptisch:eqn:dglx} -\end{equation} -Andererseits wissen wir, dass $\operatorname{zn}(u,k)$ einer -Differentilgleichung der Form~\eqref{buch:elliptisch:eqn:1storderdglell} -erfüllt. -Wenn wir \eqref{buch:elliptisch:eqn:dglx} durch $a^2b^2$ teilen, können wir -die rechte Seite von \eqref{buch:elliptisch:eqn:dglx} mit der rechten -Seite von \eqref{buch:elliptisch:eqn:1storderdglell} vergleichen: -\[ -\frac{a^2A}{b^2}\operatorname{zn}(bt,k)^4 -+ -\frac{B}{b^2}\operatorname{zn}(bt,k)^2 -+\frac{C}{a^2b^2} -= -\alpha\operatorname{zn}(bt,k)^4 -+ -\beta\operatorname{zn}(bt,k)^2 -+ -\gamma\operatorname{zn}(bt,k). -\] -Daraus ergeben sich die Gleichungen -\begin{align} -\alpha &= \frac{a^2A}{b^2}, -& -\beta &= \frac{B}{b^2} -&&\text{und} -& -\gamma &= \frac{C}{a^2b^2} -\label{buch:elliptisch:eqn:koeffvergl} -\intertext{oder aufgelöst nach den Koeffizienten der ursprünglichen -Differentialgleichung} -A&=\frac{\alpha b^2}{a^2} -& -B&=\beta b^2 -&&\text{und}& -C &= \gamma a^2b^2 -\label{buch:elliptisch:eqn:koeffABC} -\end{align} -für die Koeffizienten der Differentialgleichung der zu verwendenden -Funktion. - -Man beachte, dass nach \eqref{buch:elliptisch:eqn:koeffvergl} die -Koeffizienten $A$, $B$ und $C$ die gleichen Vorzeichen haben wie -$\alpha$, $\beta$ und $\gamma$, da in -\eqref{buch:elliptisch:eqn:koeffvergl} nur mit Quadraten multipliziert -wird, die immer positiv sind. -Diese Vorzeichen bestimmen, welche der Funktionen gewählt werden muss. - -In den Differentialgleichungen für die elliptischen Funktionen gibt -es nur den Parameter $k$, der angepasst werden kann. -Es folgt, dass die Gleichungen -\eqref{buch:elliptisch:eqn:koeffvergl} -auch $a$ und $b$ bestimmen. -Zum Beispiel folgt aus der letzten Gleichung, dass -\[ -b = \pm\sqrt{\frac{B}{\beta}}. -\] -Damit folgt dann aus der zweiten -\[ -a=\pm\sqrt{\frac{\beta C}{\gamma B}}. -\] -Die verbleibende Gleichung legt $k$ fest. -Das folgende Beispiel illustriert das Vorgehen am Beispiel einer -Gleichung, die Lösungsfunktion $\operatorname{sn}(u,k)$ verlangt. - -\begin{beispiel} -Wir nehmen an, dass die Vorzeichen von $A$, $B$ und $C$ gemäss -Tabelle~\ref{buch:elliptische:tabelle:loesungsfunktionen} verlangen, -dass die Funktion $\operatorname{sn}(u,k)$ für die Lösung verwendet -werden muss. -Die Tabelle sagt dann auch, dass -$\alpha=k^2$, $\beta=1$ und $\gamma=1$ gewählt werden müssen. -Aus dem Koeffizientenvergleich~\eqref{buch:elliptisch:eqn:koeffvergl} -folgt dann der Reihe nach -\begin{align*} -b&=\pm \sqrt{B} -\\ -a&=\pm \sqrt{\frac{C}{B}} -\\ -k^2 -&= -\frac{AC}{B^2}. -\end{align*} -Man beachte, dass man $k^2$ durch Einsetzen von -\eqref{buch:elliptisch:eqn:koeffABC} -auch direkt aus den Koeffizienten $\alpha$, $\beta$ und $\gamma$ -erhalten kann, nämlich -\[ -\frac{AC}{B^2} -= -\frac{\frac{\alpha b^2}{a^2} \gamma a^2b^2}{\beta^2 b^4} -= -\frac{\alpha\gamma}{\beta^2}. -\qedhere -\] -\end{beispiel} - -Da alle Parameter im -Lösungsansatz~\eqref{buch:elliptisch:eqn:loesungsansatz} bereits -festgelegt sind stellt sich die Frage, woher man einen weiteren -Parameter nehmen kann, mit dem Anfangsbedingungen erfüllen kann. -Die Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} ist -autonom, die Koeffizienten der rechten Seite der Differentialgleichung -sind nicht von der Zeit abhängig. -Damit ist eine zeitverschobene Funktion $x(t-t_0)$ ebenfalls eine -Lösung der Differentialgleichung. -Die allgmeine Lösung der -Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} hat -also die Form -\[ -x(t) = a\operatorname{zn}(b(t-t_0)), -\] -wobei die Funktion $\operatorname{zn}(u,k)$ auf Grund der Vorzeichen -von $A$, $B$ und $C$ gewählt werden müssen. - +%\subsubsection{TODO} +%XXX algebraische Beziehungen \\ +%XXX Additionstheoreme \\ +%XXX Perioden +%% use https://math.stackexchange.com/questions/3013692/how-to-show-that-jacobi-sine-function-is-doubly-periodic % -% Das mathematische Pendel % -\subsection{Das mathematische Pendel -\label{buch:elliptisch:subsection:mathpendel}} -\begin{figure} -\centering -\includegraphics{chapters/110-elliptisch/images/pendel.pdf} -\caption{Mathematisches Pendel -\label{buch:elliptisch:fig:mathpendel}} -\end{figure} -Das in Abbildung~\ref{buch:elliptisch:fig:mathpendel} dargestellte -Mathematische Pendel besteht aus einem Massepunkt der Masse $m$ -im Punkt $P$, -der über eine masselose Stange der Länge $l$ mit dem Drehpunkt $O$ -verbunden ist. -Das Pendel bewegt sich unter dem Einfluss der Schwerebeschleunigung $g$. - -Das Trägheitsmoment des Massepunktes um den Drehpunkt $O$ ist -\( -I=ml^2 -\). -Das Drehmoment der Schwerkraft ist -\(M=gl\sin\vartheta\). -Die Bewegungsgleichung wird daher -\[ -\begin{aligned} -\frac{d}{dt} I\dot{\vartheta} -&= -M -= -gl\sin\vartheta -\\ -ml^2\ddot{\vartheta} -&= -gl\sin\vartheta -&&\Rightarrow& -\ddot{\vartheta} -&=\frac{g}{l}\sin\vartheta -\end{aligned} -\] -Dies ist eine nichtlineare Differentialgleichung zweiter Ordnung, die -wir nicht unmittelbar mit den Differentialgleichungen erster Ordnung -der elliptischen Funktionen vergleichen können. - -Die Differentialgleichungen erster Ordnung der elliptischen Funktionen -enthalten das Quadrat der ersten Ableitung. -In unserem Fall entspricht das einer Gleichung, die $\dot{\vartheta}^2$ -enthält. -Der Energieerhaltungssatz kann uns eine solche Gleichung geben. -Die Summe von kinetischer und potentieller Energie muss konstant sein. -Dies führt auf -\[ -E_{\text{kinetisch}} -+ -E_{\text{potentiell}} -= -\frac12I\dot{\vartheta}^2 -+ -mgl(1-\cos\vartheta) -= -\frac12ml^2\dot{\vartheta}^2 -+ -mgl(1-\cos\vartheta) -= -E -\] -Durch Auflösen nach $\dot{\vartheta}$ kann man jetzt die -Differentialgleichung -\[ -\dot{\vartheta}^2 -= -- -\frac{2g}{l}(1-\cos\vartheta) -+\frac{2E}{ml^2} -\] -finden. -In erster Näherung, d.h. wenn man die rechte Seite bis zu vierten -Potenzen in eine Taylor-Reihe in $\vartheta$ entwickelt, ist dies -tatsächlich eine Differentialgleichung der Art, wie wir sie für -elliptische Funktionen gefunden haben, wir möchten aber eine exakte -Lösung konstruieren. - -Die maximale Energie für eine Bewegung, bei der sich das Pendel gerade -über den höchsten Punkt hinweg zu bewegen vermag, ist -$E=2lmg$. -Falls $E<2mgl$ ist, erwarten wir Schwingungslösungen, bei denen -der Winkel $\vartheta$ immer im offenen Interval $(-\pi,\pi)$ -bleibt. -Für $E>2mgl$ wird sich das Pendel im Kreis bewegen, für sehr grosse -Energie ist die kinetische Energie dominant, die Verlangsamung im -höchsten Punkt wird immer weniger ausgeprägt sein. +%XXX Ableitungen \\ +%XXX Werte \\ +%% +%% Lösung von Differentialgleichungen +%% +%\subsection{Lösungen von Differentialgleichungen +%\label{buch:elliptisch:subsection:differentialgleichungen}} +%Die elliptischen Funktionen ermöglichen die Lösung gewisser nichtlinearer +%Differentialgleichungen in geschlossener Form. +%Ziel dieses Abschnitts ist, Differentialgleichungen der Form +%\( +%\dot{x}(t)^2 +%= +%P(x(t)) +%\) +%mit einem Polynom $P$ vierten Grades oder +%\( +%\ddot{x}(t) +%= +%p(x(t)) +%\) +%mit einem Polynom dritten Grades als rechter Seite lösen zu können. % -% Koordinatentransformation auf elliptische Funktionen +%% +%% Die Differentialgleichung der elliptischen Funktionen +%% +%\subsubsection{Die Differentialgleichungen der elliptischen Funktionen} +%Um Differentialgleichungen mit elliptischen Funktion lösen zu +%können, muss man als erstes die Differentialgleichungen derselben +%finden. +%Quadriert man die Ableitungsregel für $\operatorname{sn}(u,k)$, erhält +%man +%\[ +%\biggl(\frac{d}{du}\operatorname{sn}(u,k)\biggr)^2 +%= +%\operatorname{cn}(u,k)^2 \operatorname{dn}(u,k)^2. +%\] +%Die Funktionen auf der rechten Seite können durch $\operatorname{sn}(u,k)$ +%ausgedrückt werden, dies führt auf die Differentialgleichung +%\begin{align*} +%\biggl(\frac{d}{du}\operatorname{sn}(u,k)\biggr)^2 +%&= +%\bigl( +%1-\operatorname{sn}(u,k)^2 +%\bigr) +%\bigl( +%1-k^2 \operatorname{sn}(u,k)^2 +%\bigr) +%\\ +%&= +%k^2\operatorname{sn}(u,k)^4 +%-(1+k^2) +%\operatorname{sn}(u,k)^2 +%+1. +%\end{align*} +%Für die Funktion $\operatorname{cn}(u,k)$ ergibt die analoge Rechnung +%\begin{align*} +%\frac{d}{du}\operatorname{cn}(u,k) +%&= +%-\operatorname{sn}(u,k) \operatorname{dn}(u,k) +%\\ +%\biggl(\frac{d}{du}\operatorname{cn}(u,k)\biggr)^2 +%&= +%\operatorname{sn}(u,k)^2 \operatorname{dn}(u,k)^2 +%\\ +%&= +%\bigl(1-\operatorname{cn}(u,k)^2\bigr) +%\bigl(k^{\prime 2}+k^2 \operatorname{cn}(u,k)^2\bigr) +%\\ +%&= +%-k^2\operatorname{cn}(u,k)^4 +%+ +%(k^2-k^{\prime 2})\operatorname{cn}(u,k)^2 +%+ +%k^{\prime 2} +%\intertext{und weiter für $\operatorname{dn}(u,k)$:} +%\frac{d}{du}\operatorname{dn}(u,k) +%&= +%-k^2\operatorname{sn}(u,k)\operatorname{cn}(u,k) +%\\ +%\biggl( +%\frac{d}{du}\operatorname{dn}(u,k) +%\biggr)^2 +%&= +%\bigl(k^2 \operatorname{sn}(u,k)^2\bigr) +%\bigl(k^2 \operatorname{cn}(u,k)^2\bigr) +%\\ +%&= +%\bigl( +%1-\operatorname{dn}(u,k)^2 +%\bigr) +%\bigl( +%\operatorname{dn}(u,k)^2-k^{\prime 2} +%\bigr) +%\\ +%&= +%-\operatorname{dn}(u,k)^4 +%+ +%(1+k^{\prime 2})\operatorname{dn}(u,k)^2 +%-k^{\prime 2}. +%\end{align*} % -\subsubsection{Koordinatentransformation auf elliptische Funktionen} -Wir verwenden als neue Variable -\[ -y = \sin\frac{\vartheta}2 -\] -mit der Ableitung -\[ -\dot{y}=\frac12\cos\frac{\vartheta}{2}\cdot \dot{\vartheta}. -\] -Man beachte, dass $y$ nicht eine Koordinate in -Abbildung~\ref{buch:elliptisch:fig:mathpendel} ist. - -Aus den Halbwinkelformeln finden wir -\[ -\cos\vartheta -= -1-2\sin^2 \frac{\vartheta}2 -= -1-2y^2. -\] -Dies können wir zusammen mit der -Identität $\cos^2\vartheta/2 = 1-\sin^2\vartheta/2 = 1-y^2$ -in die Energiegleichung einsetzen und erhalten -\[ -\frac12ml^2\dot{\vartheta}^2 + mgly^2 = E -\qquad\Rightarrow\qquad -\frac14 \dot{\vartheta}^2 = \frac{E}{2ml^2} - \frac{g}{2l}y^2. -\] -Der konstante Term auf der rechten Seite ist grösser oder kleiner als -$1$ je nachdem, ob das Pendel sich im Kreis bewegt oder nicht. - -Durch Multiplizieren mit $\cos^2\frac{\vartheta}{2}=1-y^2$ -erhalten wir auf der linken Seite einen Ausdruck, den wir -als Funktion von $\dot{y}$ ausdrücken können. -Wir erhalten -\begin{align*} -\frac14 -\cos^2\frac{\vartheta}2 -\cdot -\dot{\vartheta}^2 -&= -\frac14 -(1-y^2) -\biggl(\frac{E}{2ml^2} -\frac{g}{2l}y^2\biggr) -\\ -\dot{y}^2 -&= -\frac{1}{4} -(1-y^2) -\biggl(\frac{E}{2ml^2} -\frac{g}{2l}y^2\biggr) -\end{align*} -Die letzte Gleichung hat die Form einer Differentialgleichung -für elliptische Funktionen. -Welche Funktion verwendet werden muss, hängt von der Grösse der -Koeffizienten in der zweiten Klammer ab. -Die Tabelle~\ref{buch:elliptisch:tabelle:loesungsfunktionen} -zeigt, dass in der zweiten Klammer jeweils einer der Terme -$1$ sein muss. - +%\begin{table} +%\centering +%\renewcommand{\arraystretch}{1.7} +%\begin{tabular}{|>{$}l<{$}|>{$}l<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} +%\hline +%\text{Funktion $y=$}&\text{Differentialgleichung}&\alpha&\beta&\gamma\\ +%\hline +%\operatorname{sn}(u,k) +% & y'^2 = \phantom{-}(1-y^2)(1-k^2y^2) +% &k^2&1+k^2&1 +%\\ +%\operatorname{cn}(u,k) &y'^2 = \phantom{-}(1-y^2)(k^{\prime2}+k^2y^2) +% &-k^2 &k^2-k^{\prime 2}=2k^2-1&k^{\prime2} +%\\ +%\operatorname{dn}(u,k) +% & y'^2 = -(1-y^2)(k^{\prime 2}-y^2) +% &-1 &1+k^{\prime 2}=2-k^2 &-k^{\prime2} +%\\ +%\hline +%\end{tabular} +%\caption{Elliptische Funktionen als Lösungsfunktionen für verschiedene +%nichtlineare Differentialgleichungen der Art +%\eqref{buch:elliptisch:eqn:1storderdglell}. +%Die Vorzeichen der Koeffizienten $\alpha$, $\beta$ und $\gamma$ +%entscheidet darüber, welche Funktion für die Lösung verwendet werden +%muss. +%\label{buch:elliptisch:tabelle:loesungsfunktionen}} +%\end{table} % -% Der Fall E < 2mgl +%Die drei grundlegenden Jacobischen elliptischen Funktionen genügen also alle +%einer nichtlinearen Differentialgleichung erster Ordnung der selben Art. +%Das Quadrat der Ableitung ist ein Polynom vierten Grades der Funktion. +%Die Differentialgleichungen sind in der +%Tabelle~\ref{buch:elliptisch:tabelle:loesungsfunktionen} zusammengefasst. % -\subsubsection{Der Fall $E<2mgl$} -\begin{figure} -\centering -\includegraphics[width=\textwidth]{chapters/110-elliptisch/images/jacobiplots.pdf} -\caption{% -Abhängigkeit der elliptischen Funktionen von $u$ für -verschiedene Werte von $k^2=m$. -Für $m=0$ ist $\operatorname{sn}(u,0)=\sin u$, -$\operatorname{cn}(u,0)=\cos u$ und $\operatorname{dn}(u,0)=1$, diese -sind in allen Plots in einer helleren Farbe eingezeichnet. -Für kleine Werte von $m$ weichen die elliptischen Funktionen nur wenig -von den trigonometrischen Funktionen ab, -es ist aber klar erkennbar, dass die anharmonischen Terme in der -Differentialgleichung die Periode mit steigender Amplitude verlängern. -Sehr grosse Werte von $m$ nahe bei $1$ entsprechen der Situation, dass -die Energie des Pendels fast ausreicht, dass es den höchsten Punkt -erreichen kann, was es für $m$ macht. -\label{buch:elliptisch:fig:jacobiplots}} -\end{figure} - - -Wir verwenden als neue Variable -\[ -y = \sin\frac{\vartheta}2 -\] -mit der Ableitung -\[ -\dot{y}=\frac12\cos\frac{\vartheta}{2}\cdot \dot{\vartheta}. -\] -Man beachte, dass $y$ nicht eine Koordinate in -Abbildung~\ref{buch:elliptisch:fig:mathpendel} ist. - -Aus den Halbwinkelformeln finden wir -\[ -\cos\vartheta -= -1-2\sin^2 \frac{\vartheta}2 -= -1-2y^2. -\] -Dies können wir zusammen mit der -Identität $\cos^2\vartheta/2 = 1-\sin^2\vartheta/2 = 1-y^2$ -in die Energiegleichung einsetzen und erhalten -\[ -\frac12ml^2\dot{\vartheta}^2 + mgly^2 = E. -\] -Durch Multiplizieren mit $\cos^2\frac{\vartheta}{2}=1-y^2$ -erhalten wir auf der linken Seite einen Ausdruck, den wir -als Funktion von $\dot{y}$ ausdrücken können. -Wir erhalten -\begin{align*} -\frac12ml^2 -\cos^2\frac{\vartheta}2 -\dot{\vartheta}^2 -&= -(1-y^2) -(E -mgly^2) -\\ -\frac{1}{4}\cos^2\frac{\vartheta}{2}\dot{\vartheta}^2 -&= -\frac{1}{2} -(1-y^2) -\biggl(\frac{E}{ml^2} -\frac{g}{l}y^2\biggr) -\\ -\dot{y}^2 -&= -\frac{E}{2ml^2} -(1-y^2)\biggl( -1-\frac{2gml}{E}y^2 -\biggr). -\end{align*} -Dies ist genau die Form der Differentialgleichung für die elliptische -Funktion $\operatorname{sn}(u,k)$ -mit $k^2 = 2gml/E< 1$. - +%% +%% Differentialgleichung der abgeleiteten elliptischen Funktionen +%% +%\subsubsection{Die Differentialgleichung der abgeleiteten elliptischen +%Funktionen} +%Da auch die Ableitungen der abgeleiteten Jacobischen elliptischen +%Funktionen Produkte von genau zwei Funktionen sind, die sich wieder +%durch die ursprüngliche Funktion ausdrücken lassen, darf man erwarten, +%dass alle elliptischen Funktionen einer ähnlichen Differentialgleichung +%genügen. +%Um dies besser einzufangen, schreiben wir $\operatorname{pq}(u,k)$, +%wenn wir eine beliebige abgeleitete Jacobische elliptische Funktion. +%Für +%$\operatorname{pq}=\operatorname{sn}$ +%$\operatorname{pq}=\operatorname{cn}$ +%und +%$\operatorname{pq}=\operatorname{dn}$ +%wissen wir bereits und erwarten für jede andere Funktion dass +%$\operatorname{pq}(u,k)$ auch, dass sie Lösung einer Differentialgleichung +%der Form +%\begin{equation} +%\operatorname{pq}'(u,k)^2 +%= +%\alpha \operatorname{pq}(u,k)^4 + \beta \operatorname{pq}(u,k)^2 + \gamma +%\label{buch:elliptisch:eqn:1storderdglell} +%\end{equation} +%erfüllt, +%wobei wir mit $\operatorname{pq}'(u,k)$ die Ableitung von +%$\operatorname{pq}(u,k)$ nach dem ersten Argument meinen. +%Die Koeffizienten $\alpha$, $\beta$ und $\gamma$ hängen von $k$ ab, +%ihre Werte für die grundlegenden Jacobischen elliptischen +%sind in Tabelle~\ref{buch:elliptisch:table:differentialgleichungen} +%zusammengestellt. % -% Der Fall E > 2mgl +%Die Koeffizienten müssen nicht für jede Funktion wieder neu bestimmt +%werden, denn für den Kehrwert einer Funktion lässt sich die +%Differentialgleichung aus der Differentialgleichung der ursprünglichen +%Funktion ermitteln. % -\subsection{Der Fall $E > 2mgl$} -In diesem Fall hat das Pendel im höchsten Punkte immer noch genügend -kinetische Energie, so dass es sich im Kreise dreht. -Indem wir die Gleichung - -XXX Differentialgleichung \\ -XXX Mathematisches Pendel \\ - -\subsection{Soliton-Lösungen der Sinus-Gordon-Gleichung} +%% +%% Differentialgleichung der Kehrwertfunktion +%% +%\subsubsection{Differentialgleichung für den Kehrwert einer elliptischen Funktion} +%Aus der Differentialgleichung~\eqref{buch:elliptisch:eqn:1storderdglell} +%für die Funktion $\operatorname{pq}(u,k)$ kann auch eine +%Differentialgleichung für den Kehrwert +%$\operatorname{qp}(u,k)=\operatorname{pq}(u,k)^{-1}$ +%ableiten. +%Dazu rechnet man +%\[ +%\operatorname{qp}'(u,k) +%= +%\frac{d}{du}\frac{1}{\operatorname{pq}(u,k)} +%= +%\frac{\operatorname{pq}'(u,k)}{\operatorname{pq}(u,k)^2} +%\qquad\Rightarrow\qquad +%\left\{ +%\quad +%\begin{aligned} +%\operatorname{pq}(u,k) +%&= +%\frac{1}{\operatorname{qp}(u,k)} +%\\ +%\operatorname{pq}'(u,k) +%&= +%\frac{\operatorname{qp}'(u,k)}{\operatorname{qp}(u,k)^2} +%\end{aligned} +%\right. +%\] +%und setzt in die Differentialgleichung ein: +%\begin{align*} +%\biggl( +%\frac{ +%\operatorname{qp}'(u,k) +%}{ +%\operatorname{qp}(u,k) +%} +%\biggr)^2 +%&= +%\alpha \frac{1}{\operatorname{qp}(u,k)^4} +%+ +%\beta \frac{1}{\operatorname{qp}(u,k)^2} +%+ +%\gamma. +%\end{align*} +%Nach Multiplikation mit $\operatorname{qp}(u,k)^4$ erhält man den +%folgenden Satz. +% +%\begin{satz} +%Wenn die Jacobische elliptische Funktion $\operatorname{pq}(u,k)$ +%der Differentialgleichung genügt, dann genügt der Kehrwert +%$\operatorname{qp}(u,k) = 1/\operatorname{pq}(u,k)$ der Differentialgleichung +%\begin{equation} +%(\operatorname{qp}'(u,k))^2 +%= +%\gamma \operatorname{qp}(u,k)^4 +%+ +%\beta \operatorname{qp}(u,k)^2 +%+ +%\alpha +%\label{buch:elliptisch:eqn:kehrwertdgl} +%\end{equation} +%\end{satz} +% +%\begin{table} +%\centering +%\def\lfn#1{\multicolumn{1}{|l|}{#1}} +%\def\rfn#1{\multicolumn{1}{r|}{#1}} +%\renewcommand{\arraystretch}{1.3} +%\begin{tabular}{l|>{$}c<{$}>{$}c<{$}>{$}c<{$}|r} +%\cline{1-4} +%\lfn{Funktion} +% & \alpha & \beta & \gamma &\\ +%\hline +%\lfn{sn}& k^2 & -(1+k^2) & 1 &\rfn{ns}\\ +%\lfn{cn}& -k^2 & -(1-2k^2) & 1-k^2 &\rfn{nc}\\ +%\lfn{dn}& 1 & 2-k^2 & -(1-k^2) &\rfn{nd}\\ +%\hline +%\lfn{sc}& 1-k^2 & 2-k^2 & 1 &\rfn{cs}\\ +%\lfn{sd}&-k^2(1-k^2)&-(1-2k^2) & 1 &\rfn{ds}\\ +%\lfn{cd}& k^2 &-(1+k^2) & 1 &\rfn{dc}\\ +%\hline +% & \gamma & \beta & \alpha &\rfn{Reziproke}\\ +%\cline{2-5} +%\end{tabular} +%\caption{Koeffizienten der Differentialgleichungen für die Jacobischen +%elliptischen Funktionen. +%Der Kehrwert einer Funktion hat jeweils die Differentialgleichung der +%ursprünglichen Funktion, in der die Koeffizienten $\alpha$ und $\gamma$ +%vertauscht worden sind. +%\label{buch:elliptisch:table:differentialgleichungen}} +%\end{table} +% +%% +%% Differentialgleichung zweiter Ordnung +%% +%\subsubsection{Differentialgleichung zweiter Ordnung} +%Leitet die Differentialgleichung~\eqref{buch:elliptisch:eqn:1storderdglell} +%man dies nochmals nach $u$ ab, erhält man die Differentialgleichung +%\[ +%2\operatorname{pq}''(u,k)\operatorname{pq}'(u,k) +%= +%4\alpha \operatorname{pq}(u,k)^3\operatorname{pq}'(u,k) + 2\beta \operatorname{pq}'(u,k)\operatorname{pq}(u,k). +%\] +%Teilt man auf beiden Seiten durch $2\operatorname{pq}'(u,k)$, +%bleibt die nichtlineare +%Differentialgleichung +%\[ +%\frac{d^2\operatorname{pq}}{du^2} +%= +%\beta \operatorname{pq} + 2\alpha \operatorname{pq}^3. +%\] +%Dies ist die Gleichung eines harmonischen Oszillators mit einer +%Anharmonizität der Form $2\alpha z^3$. +% +% +% +%% +%% Jacobischen elliptische Funktionen und elliptische Integrale +%% +%\subsubsection{Jacobische elliptische Funktionen als elliptische Integrale} +%Die in Tabelle~\ref{buch:elliptisch:tabelle:loesungsfunktionen} +%zusammengestellten Differentialgleichungen ermöglichen nun, den +%Zusammenhang zwischen den Funktionen +%$\operatorname{sn}(u,k)$, $\operatorname{cn}(u,k)$ und $\operatorname{dn}(u,k)$ +%und den unvollständigen elliptischen Integralen herzustellen. +%Die Differentialgleichungen sind alle von der Form +%\begin{equation} +%\biggl( +%\frac{d y}{d u} +%\biggr)^2 +%= +%p(u), +%\label{buch:elliptisch:eqn:allgdgl} +%\end{equation} +%wobei $p(u)$ ein Polynom vierten Grades in $y$ ist. +%Diese Differentialgleichung lässt sich mit Separation lösen. +%Dazu zieht man aus~\eqref{buch:elliptisch:eqn:allgdgl} die +%Wurzel +%\begin{align} +%\frac{dy}{du} +%= +%\sqrt{p(y)} +%\notag +%\intertext{und trennt die Variablen. Man erhält} +%\int\frac{dy}{\sqrt{p(y)}} = u+C. +%\label{buch:elliptisch:eqn:yintegral} +%\end{align} +%Solange $p(y)>0$ ist, ist der Integrand auf der linken Seite +%von~\eqref{buch:elliptisch:eqn:yintegral} ebenfalls positiv und +%das Integral ist eine monoton wachsende Funktion $F(y)$. +%Insbesondere ist $F(y)$ invertierbar. +%Die Lösung $y(u)$ der Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} +%ist daher +%\[ +%y(u) = F^{-1}(u+C). +%\] +%Die Jacobischen elliptischen Funktionen sind daher inverse Funktionen +%der unvollständigen elliptischen Integrale. +% +% +%% +%% Differentialgleichung des anharmonischen Oszillators +%% +%\subsubsection{Differentialgleichung des anharmonischen Oszillators} +%Wir möchten die nichtlineare Differentialgleichung +%\begin{equation} +%\biggl( +%\frac{dx}{dt} +%\biggr)^2 +%= +%Ax^4+Bx^2 + C +%\label{buch:elliptisch:eqn:allgdgl} +%\end{equation} +%mit Hilfe elliptischer Funktionen lösen. +%Wir nehmen also an, dass die gesuchte Lösung eine Funktion der Form +%\begin{equation} +%x(t) = a\operatorname{zn}(bt,k) +%\label{buch:elliptisch:eqn:loesungsansatz} +%\end{equation} +%ist. +%Die erste Ableitung von $x(t)$ ist +%\[ +%\dot{x}(t) +%= +%a\operatorname{zn}'(bt,k). +%\] +% +%Indem wir diesen Lösungsansatz in die +%Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} +%einsetzen, erhalten wir +%\begin{equation} +%a^2b^2 \operatorname{zn}'(bt,k)^2 +%= +%a^4A\operatorname{zn}(bt,k)^4 +%+ +%a^2B\operatorname{zn}(bt,k)^2 +%+C +%\label{buch:elliptisch:eqn:dglx} +%\end{equation} +%Andererseits wissen wir, dass $\operatorname{zn}(u,k)$ einer +%Differentilgleichung der Form~\eqref{buch:elliptisch:eqn:1storderdglell} +%erfüllt. +%Wenn wir \eqref{buch:elliptisch:eqn:dglx} durch $a^2b^2$ teilen, können wir +%die rechte Seite von \eqref{buch:elliptisch:eqn:dglx} mit der rechten +%Seite von \eqref{buch:elliptisch:eqn:1storderdglell} vergleichen: +%\[ +%\frac{a^2A}{b^2}\operatorname{zn}(bt,k)^4 +%+ +%\frac{B}{b^2}\operatorname{zn}(bt,k)^2 +%+\frac{C}{a^2b^2} +%= +%\alpha\operatorname{zn}(bt,k)^4 +%+ +%\beta\operatorname{zn}(bt,k)^2 +%+ +%\gamma\operatorname{zn}(bt,k). +%\] +%Daraus ergeben sich die Gleichungen +%\begin{align} +%\alpha &= \frac{a^2A}{b^2}, +%& +%\beta &= \frac{B}{b^2} +%&&\text{und} +%& +%\gamma &= \frac{C}{a^2b^2} +%\label{buch:elliptisch:eqn:koeffvergl} +%\intertext{oder aufgelöst nach den Koeffizienten der ursprünglichen +%Differentialgleichung} +%A&=\frac{\alpha b^2}{a^2} +%& +%B&=\beta b^2 +%&&\text{und}& +%C &= \gamma a^2b^2 +%\label{buch:elliptisch:eqn:koeffABC} +%\end{align} +%für die Koeffizienten der Differentialgleichung der zu verwendenden +%Funktion. +% +%Man beachte, dass nach \eqref{buch:elliptisch:eqn:koeffvergl} die +%Koeffizienten $A$, $B$ und $C$ die gleichen Vorzeichen haben wie +%$\alpha$, $\beta$ und $\gamma$, da in +%\eqref{buch:elliptisch:eqn:koeffvergl} nur mit Quadraten multipliziert +%wird, die immer positiv sind. +%Diese Vorzeichen bestimmen, welche der Funktionen gewählt werden muss. +% +%In den Differentialgleichungen für die elliptischen Funktionen gibt +%es nur den Parameter $k$, der angepasst werden kann. +%Es folgt, dass die Gleichungen +%\eqref{buch:elliptisch:eqn:koeffvergl} +%auch $a$ und $b$ bestimmen. +%Zum Beispiel folgt aus der letzten Gleichung, dass +%\[ +%b = \pm\sqrt{\frac{B}{\beta}}. +%\] +%Damit folgt dann aus der zweiten +%\[ +%a=\pm\sqrt{\frac{\beta C}{\gamma B}}. +%\] +%Die verbleibende Gleichung legt $k$ fest. +%Das folgende Beispiel illustriert das Vorgehen am Beispiel einer +%Gleichung, die Lösungsfunktion $\operatorname{sn}(u,k)$ verlangt. +% +%\begin{beispiel} +%Wir nehmen an, dass die Vorzeichen von $A$, $B$ und $C$ gemäss +%Tabelle~\ref{buch:elliptische:tabelle:loesungsfunktionen} verlangen, +%dass die Funktion $\operatorname{sn}(u,k)$ für die Lösung verwendet +%werden muss. +%Die Tabelle sagt dann auch, dass +%$\alpha=k^2$, $\beta=1$ und $\gamma=1$ gewählt werden müssen. +%Aus dem Koeffizientenvergleich~\eqref{buch:elliptisch:eqn:koeffvergl} +%folgt dann der Reihe nach +%\begin{align*} +%b&=\pm \sqrt{B} +%\\ +%a&=\pm \sqrt{\frac{C}{B}} +%\\ +%k^2 +%&= +%\frac{AC}{B^2}. +%\end{align*} +%Man beachte, dass man $k^2$ durch Einsetzen von +%\eqref{buch:elliptisch:eqn:koeffABC} +%auch direkt aus den Koeffizienten $\alpha$, $\beta$ und $\gamma$ +%erhalten kann, nämlich +%\[ +%\frac{AC}{B^2} +%= +%\frac{\frac{\alpha b^2}{a^2} \gamma a^2b^2}{\beta^2 b^4} +%= +%\frac{\alpha\gamma}{\beta^2}. +%\qedhere +%\] +%\end{beispiel} +% +%Da alle Parameter im +%Lösungsansatz~\eqref{buch:elliptisch:eqn:loesungsansatz} bereits +%festgelegt sind stellt sich die Frage, woher man einen weiteren +%Parameter nehmen kann, mit dem Anfangsbedingungen erfüllen kann. +%Die Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} ist +%autonom, die Koeffizienten der rechten Seite der Differentialgleichung +%sind nicht von der Zeit abhängig. +%Damit ist eine zeitverschobene Funktion $x(t-t_0)$ ebenfalls eine +%Lösung der Differentialgleichung. +%Die allgmeine Lösung der +%Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} hat +%also die Form +%\[ +%x(t) = a\operatorname{zn}(b(t-t_0)), +%\] +%wobei die Funktion $\operatorname{zn}(u,k)$ auf Grund der Vorzeichen +%von $A$, $B$ und $C$ gewählt werden müssen. -\subsection{Nichtlineare Differentialgleichung vierter Ordnung} -XXX Möbius-Transformation \\ -XXX Reduktion auf die Differentialgleichung elliptischer Funktionen +%% +%% Das mathematische Pendel +%% +%\subsection{Das mathematische Pendel +%\label{buch:elliptisch:subsection:mathpendel}} +%\begin{figure} +%\centering +%\includegraphics{chapters/110-elliptisch/images/pendel.pdf} +%\caption{Mathematisches Pendel +%\label{buch:elliptisch:fig:mathpendel}} +%\end{figure} +%Das in Abbildung~\ref{buch:elliptisch:fig:mathpendel} dargestellte +%Mathematische Pendel besteht aus einem Massepunkt der Masse $m$ +%im Punkt $P$, +%der über eine masselose Stange der Länge $l$ mit dem Drehpunkt $O$ +%verbunden ist. +%Das Pendel bewegt sich unter dem Einfluss der Schwerebeschleunigung $g$. +% +%Das Trägheitsmoment des Massepunktes um den Drehpunkt $O$ ist +%\( +%I=ml^2 +%\). +%Das Drehmoment der Schwerkraft ist +%\(M=gl\sin\vartheta\). +%Die Bewegungsgleichung wird daher +%\[ +%\begin{aligned} +%\frac{d}{dt} I\dot{\vartheta} +%&= +%M +%= +%gl\sin\vartheta +%\\ +%ml^2\ddot{\vartheta} +%&= +%gl\sin\vartheta +%&&\Rightarrow& +%\ddot{\vartheta} +%&=\frac{g}{l}\sin\vartheta +%\end{aligned} +%\] +%Dies ist eine nichtlineare Differentialgleichung zweiter Ordnung, die +%wir nicht unmittelbar mit den Differentialgleichungen erster Ordnung +%der elliptischen Funktionen vergleichen können. +% +%Die Differentialgleichungen erster Ordnung der elliptischen Funktionen +%enthalten das Quadrat der ersten Ableitung. +%In unserem Fall entspricht das einer Gleichung, die $\dot{\vartheta}^2$ +%enthält. +%Der Energieerhaltungssatz kann uns eine solche Gleichung geben. +%Die Summe von kinetischer und potentieller Energie muss konstant sein. +%Dies führt auf +%\[ +%E_{\text{kinetisch}} +%+ +%E_{\text{potentiell}} +%= +%\frac12I\dot{\vartheta}^2 +%+ +%mgl(1-\cos\vartheta) +%= +%\frac12ml^2\dot{\vartheta}^2 +%+ +%mgl(1-\cos\vartheta) +%= +%E +%\] +%Durch Auflösen nach $\dot{\vartheta}$ kann man jetzt die +%Differentialgleichung +%\[ +%\dot{\vartheta}^2 +%= +%- +%\frac{2g}{l}(1-\cos\vartheta) +%+\frac{2E}{ml^2} +%\] +%finden. +%In erster Näherung, d.h. wenn man die rechte Seite bis zu vierten +%Potenzen in eine Taylor-Reihe in $\vartheta$ entwickelt, ist dies +%tatsächlich eine Differentialgleichung der Art, wie wir sie für +%elliptische Funktionen gefunden haben, wir möchten aber eine exakte +%Lösung konstruieren. +% +%Die maximale Energie für eine Bewegung, bei der sich das Pendel gerade +%über den höchsten Punkt hinweg zu bewegen vermag, ist +%$E=2lmg$. +%Falls $E<2mgl$ ist, erwarten wir Schwingungslösungen, bei denen +%der Winkel $\vartheta$ immer im offenen Interval $(-\pi,\pi)$ +%bleibt. +%Für $E>2mgl$ wird sich das Pendel im Kreis bewegen, für sehr grosse +%Energie ist die kinetische Energie dominant, die Verlangsamung im +%höchsten Punkt wird immer weniger ausgeprägt sein. +% +%% +%% Koordinatentransformation auf elliptische Funktionen +%% +%\subsubsection{Koordinatentransformation auf elliptische Funktionen} +%Wir verwenden als neue Variable +%\[ +%y = \sin\frac{\vartheta}2 +%\] +%mit der Ableitung +%\[ +%\dot{y}=\frac12\cos\frac{\vartheta}{2}\cdot \dot{\vartheta}. +%\] +%Man beachte, dass $y$ nicht eine Koordinate in +%Abbildung~\ref{buch:elliptisch:fig:mathpendel} ist. +% +%Aus den Halbwinkelformeln finden wir +%\[ +%\cos\vartheta +%= +%1-2\sin^2 \frac{\vartheta}2 +%= +%1-2y^2. +%\] +%Dies können wir zusammen mit der +%Identität $\cos^2\vartheta/2 = 1-\sin^2\vartheta/2 = 1-y^2$ +%in die Energiegleichung einsetzen und erhalten +%\[ +%\frac12ml^2\dot{\vartheta}^2 + mgly^2 = E +%\qquad\Rightarrow\qquad +%\frac14 \dot{\vartheta}^2 = \frac{E}{2ml^2} - \frac{g}{2l}y^2. +%\] +%Der konstante Term auf der rechten Seite ist grösser oder kleiner als +%$1$ je nachdem, ob das Pendel sich im Kreis bewegt oder nicht. +% +%Durch Multiplizieren mit $\cos^2\frac{\vartheta}{2}=1-y^2$ +%erhalten wir auf der linken Seite einen Ausdruck, den wir +%als Funktion von $\dot{y}$ ausdrücken können. +%Wir erhalten +%\begin{align*} +%\frac14 +%\cos^2\frac{\vartheta}2 +%\cdot +%\dot{\vartheta}^2 +%&= +%\frac14 +%(1-y^2) +%\biggl(\frac{E}{2ml^2} -\frac{g}{2l}y^2\biggr) +%\\ +%\dot{y}^2 +%&= +%\frac{1}{4} +%(1-y^2) +%\biggl(\frac{E}{2ml^2} -\frac{g}{2l}y^2\biggr) +%\end{align*} +%Die letzte Gleichung hat die Form einer Differentialgleichung +%für elliptische Funktionen. +%Welche Funktion verwendet werden muss, hängt von der Grösse der +%Koeffizienten in der zweiten Klammer ab. +%Die Tabelle~\ref{buch:elliptisch:tabelle:loesungsfunktionen} +%zeigt, dass in der zweiten Klammer jeweils einer der Terme +%$1$ sein muss. +% +%% +%% Der Fall E < 2mgl +%% +%\subsubsection{Der Fall $E<2mgl$} +%\begin{figure} +%\centering +%\includegraphics[width=\textwidth]{chapters/110-elliptisch/images/jacobiplots.pdf} +%\caption{% +%Abhängigkeit der elliptischen Funktionen von $u$ für +%verschiedene Werte von $k^2=m$. +%Für $m=0$ ist $\operatorname{sn}(u,0)=\sin u$, +%$\operatorname{cn}(u,0)=\cos u$ und $\operatorname{dn}(u,0)=1$, diese +%sind in allen Plots in einer helleren Farbe eingezeichnet. +%Für kleine Werte von $m$ weichen die elliptischen Funktionen nur wenig +%von den trigonometrischen Funktionen ab, +%es ist aber klar erkennbar, dass die anharmonischen Terme in der +%Differentialgleichung die Periode mit steigender Amplitude verlängern. +%Sehr grosse Werte von $m$ nahe bei $1$ entsprechen der Situation, dass +%die Energie des Pendels fast ausreicht, dass es den höchsten Punkt +%erreichen kann, was es für $m$ macht. +%\label{buch:elliptisch:fig:jacobiplots}} +%\end{figure} +% +% +%Wir verwenden als neue Variable +%\[ +%y = \sin\frac{\vartheta}2 +%\] +%mit der Ableitung +%\[ +%\dot{y}=\frac12\cos\frac{\vartheta}{2}\cdot \dot{\vartheta}. +%\] +%Man beachte, dass $y$ nicht eine Koordinate in +%Abbildung~\ref{buch:elliptisch:fig:mathpendel} ist. +% +%Aus den Halbwinkelformeln finden wir +%\[ +%\cos\vartheta +%= +%1-2\sin^2 \frac{\vartheta}2 +%= +%1-2y^2. +%\] +%Dies können wir zusammen mit der +%Identität $\cos^2\vartheta/2 = 1-\sin^2\vartheta/2 = 1-y^2$ +%in die Energiegleichung einsetzen und erhalten +%\[ +%\frac12ml^2\dot{\vartheta}^2 + mgly^2 = E. +%\] +%Durch Multiplizieren mit $\cos^2\frac{\vartheta}{2}=1-y^2$ +%erhalten wir auf der linken Seite einen Ausdruck, den wir +%als Funktion von $\dot{y}$ ausdrücken können. +%Wir erhalten +%\begin{align*} +%\frac12ml^2 +%\cos^2\frac{\vartheta}2 +%\dot{\vartheta}^2 +%&= +%(1-y^2) +%(E -mgly^2) +%\\ +%\frac{1}{4}\cos^2\frac{\vartheta}{2}\dot{\vartheta}^2 +%&= +%\frac{1}{2} +%(1-y^2) +%\biggl(\frac{E}{ml^2} -\frac{g}{l}y^2\biggr) +%\\ +%\dot{y}^2 +%&= +%\frac{E}{2ml^2} +%(1-y^2)\biggl( +%1-\frac{2gml}{E}y^2 +%\biggr). +%\end{align*} +%Dies ist genau die Form der Differentialgleichung für die elliptische +%Funktion $\operatorname{sn}(u,k)$ +%mit $k^2 = 2gml/E< 1$. +% +%%% +%%% Der Fall E > 2mgl +%%% +%%\subsection{Der Fall $E > 2mgl$} +%%In diesem Fall hat das Pendel im höchsten Punkte immer noch genügend +%%kinetische Energie, so dass es sich im Kreise dreht. +%%Indem wir die Gleichung +% +% +%%\subsection{Soliton-Lösungen der Sinus-Gordon-Gleichung} +% +%%\subsection{Nichtlineare Differentialgleichung vierter Ordnung} +%%XXX Möbius-Transformation \\ +%%XXX Reduktion auf die Differentialgleichung elliptischer Funktionen diff --git a/buch/chapters/110-elliptisch/lemniskate.tex b/buch/chapters/110-elliptisch/lemniskate.tex index 7083b63..e766779 100644 --- a/buch/chapters/110-elliptisch/lemniskate.tex +++ b/buch/chapters/110-elliptisch/lemniskate.tex @@ -22,23 +22,46 @@ elliptischen Funktionen hergestellt werden. \end{figure} Die Lemniskate von Bernoulli ist die Kurve vierten Grades mit der Gleichung \begin{equation} -(x^2+y^2)^2 = 2a^2(x^2-y^2). +(X^2+Y^2)^2 = 2a^2(X^2-Y^2). \label{buch:elliptisch:eqn:lemniskate} \end{equation} Sie ist in Abbildung~\ref{buch:elliptisch:fig:lemniskate} dargestellt. -Die beiden Scheitel der Lemniskate befinden sich bei $x=\pm a/\sqrt{2}$. +Die beiden Scheitel der Lemniskate befinden sich bei $X_s=\pm a\sqrt{2}$. +Dividiert man die Gleichung der Lemniskate durch $X_s^2=4a^4$ entsteht +\begin{equation} +\biggl( +\biggl(\frac{X}{a\sqrt{2}}\biggr)^2 ++ +\biggl(\frac{Y}{a\sqrt{2}}\biggr)^2 +\biggr)^2 += +2\frac{a^2}{2a^2}\biggl( +\biggl(\frac{X}{a\sqrt{2}}\biggr)^2 +- +\biggl(\frac{Y}{a\sqrt{2}}\biggr)^2 +\biggr). +\qquad +\Leftrightarrow +\qquad +(x^2+y^2)^2 = x^2-y^2, +\label{buch:elliptisch:eqn:lemniskatenormiert} +\end{equation} +wobei wir $x=X/a\sqrt{2}$ und $y=Y/a\sqrt{2}$ gesetzt haben. +In dieser Normierung liegen die Scheitel bei $\pm 1$. +Dies ist die Skalierung, die für die Definition des lemniskatischen +Sinus und Kosinus verwendet werden soll. In Polarkoordinaten $x=r\cos\varphi$ und $y=r\sin\varphi$ -gilt nach Einsetzen in \eqref{buch:elliptisch:eqn:lemniskate} +gilt nach Einsetzen in \eqref{buch:elliptisch:eqn:lemniskatenormiert} \begin{equation} r^4 = -2a^2r^2(\cos^2\varphi-\sin^2\varphi) +r^2(\cos^2\varphi-\sin^2\varphi) = -2a^2r^2\cos2\varphi +r^2\cos2\varphi \qquad\Rightarrow\qquad -r^2 = 2a^2\cos 2\varphi +r^2 = \cos 2\varphi \label{buch:elliptisch:eqn:lemniskatepolar} \end{equation} als Darstellung der Lemniskate in Polardarstellung. @@ -46,15 +69,7 @@ Sie gilt für Winkel $\varphi\in[-\frac{\pi}4,\frac{\pi}4]$ für das rechte Blatt und $\varphi\in[\frac{3\pi}4,\frac{5\pi}4]$ für das linke Blatt der Lemniskate. -Für die Definition des lemniskatischen Sinus wird eine Skalierung -verwendet, die den rechten Scheitel im Punkt $(1,0)$. -Dies ist der Fall für $a=1/\sqrt{2}$, die Gleichung der Lemniskate -wird dann zu -\[ -(x^2+y^2)^2 = 2(x^2-y^2). -\] - -\subsubsection{Bogelänge} +\subsection{Bogenlänge} Die Funktionen \begin{equation} x(r) = \frac{r}{\sqrt{2}}\sqrt{1+r^2}, @@ -76,7 +91,7 @@ r^4 \end{align*} sie stellen also eine Parametrisierung der Lemniskate dar. -Mit Hilfe der Parametrsierung~\eqref{buch:geometrie:eqn:lemniskateparam} +Mit Hilfe der Parametrisierung~\eqref{buch:geometrie:eqn:lemniskateparam} kann man die Länge $s$ des in Abbildung~\ref{buch:elliptisch:fig:lemniskate} dargestellten Bogens der Lemniskate berechnen. Dazu benötigt man die Ableitungen nach $r$, die man mit der Produkt- und @@ -123,11 +138,16 @@ s(r) \label{buch:elliptisch:eqn:lemniskatebogenlaenge} \end{equation} -\subsubsection{Darstellung als elliptisches Integral} +% +% Als elliptisches Integral +% +\subsection{Darstellung als elliptisches Integral} Das unvollständige elliptische Integral erster Art mit Parameter -$m=-1$ ist +$k^2=-1$ oder $k=i$ ist \[ -K(r,-1) +K(r,i) += +\int_0^x \frac{dt}{\sqrt{(1-t^2)(1-i^2 t^2)}} = \int_0^x \frac{dt}{\sqrt{(1-t^2)(1-(-1)t^2)}} = @@ -136,11 +156,209 @@ K(r,-1) s(r). \] Der lemniskatische Sinus ist also eine Umkehrfunktion des -ellptischen Integrals erster Art für einen speziellen Wert des -Parameters $m$ +elliptischen Integrals erster Art für den speziellen Wert $i$ des +Parameters $k$. + +Die Länge des rechten Blattes der Lemniskate wird mit $\varpi$ bezeichnet +und hat den numerischen Wert +\[ +\varpi += +2\int_0^1\sqrt{\frac{1}{1-t^4}}\,dt += +2.6220575542. +\] +$\varpi$ ist auch als die {\em lemniskatische Konstante} bekannt. +\index{lemniskatische Konstante}% +Der Lemniskatenbogen zwischen dem Nullpunkt und $(1,0)$ hat die Länge +$\varpi/2$. + +% +% Bogenlängenparametrisierung +% +\subsection{Bogenlängenparametrisierung} +Die Lemniskate mit der Gleichung +\[ +(X^2+X^2)^2=2(X^2-X^2) +\] +(der Fall $a=1$ in \eqref{buch:elliptisch:eqn:lemniskate}) +kann mit Jacobischen elliptischen Funktionen +parametrisiert werden. +Dazu schreibt man +\[ +\left. +\begin{aligned} +X(t) +&= +\sqrt{2}\operatorname{cn}(t,k) \operatorname{dn}(t,k) +\\ +Y(t) +&= +\phantom{\sqrt{2}} +\operatorname{cn}(t,k) \operatorname{sn}(t,k) +\end{aligned} +\quad\right\} +\qquad\text{mit $k=\displaystyle\frac{1}{\sqrt{2}}$} +\] +und berechnet die beiden Seiten der definierenden Gleichung der +Lemniskate. +Zunächst ist +\begin{align*} +X(t)^2 +&= +2\operatorname{cn}(t,k)^2 +\operatorname{dn}(t,k)^2 +\\ +Y(t)^2 +&= +\operatorname{cn}(t,k)^2 +\operatorname{sn}(t,k)^2 +\\ +X(t)^2+Y(t)^2 +&= +2\operatorname{cn}(t,k)^2 +\bigl( +\underbrace{ +\operatorname{dn}(t,k)^2 ++{\textstyle\frac12} +\operatorname{sn}(t,k)^2 +}_{\displaystyle =1} +\bigr) +%\\ +%& += +2\operatorname{cn}(t,k)^2 +\\ +X(t)^2-Y(t)^2 +&= +\operatorname{cn}(t,k)^2 +\bigl( +2\operatorname{dn}(t,k)^2 - \operatorname{sn}(t,k)^2 +\bigr) +\\ +&= +\operatorname{cn}(t,k)^2 +\bigl( +2\bigl({\textstyle\frac12}+{\textstyle\frac12}\operatorname{cn}(t,k)^2\bigr) +- +\bigl(1-\operatorname{cn}(t,k)^2\bigr) +\bigr) +\\ +&= +2\operatorname{cn}(t,k)^4 +\\ +\Rightarrow\qquad +(X(t)^2+Y(t)^2)^2 +&= +4\operatorname{cn}(t,k)^4 += +2(X(t)^2-Y(t)^2). +\end{align*} +Wir zeigen jetzt, dass dies tatsächlich eine Bogenlängenparametrisierung +der Lemniskate ist. +Dazu berechnen wir die Ableitungen +\begin{align*} +\dot{X}(t) +&= +\sqrt{2}\operatorname{cn}'(t,k)\operatorname{dn}(t,k) ++ +\sqrt{2}\operatorname{cn}(t,k)\operatorname{dn}'(t,k) +\\ +&= +-\sqrt{2}\operatorname{sn}(t,k)\operatorname{dn}(t,k)^2 +-\frac12\sqrt{2}\operatorname{sn}(t,k)\operatorname{cn}(t,k)^2 +\\ +&= +-\sqrt{2}\operatorname{sn}(t,k)\bigl( +1-{\textstyle\frac12}\operatorname{sn}(t,k)^2 ++{\textstyle\frac12}-{\textstyle\frac12}\operatorname{sn}(u,t)^2 +\bigr) +\\ +&= +\sqrt{2}\operatorname{sn}(t,k) +\bigl( +{\textstyle \frac32}-\operatorname{sn}(t,k)^2 +\bigr) +\\ +\dot{X}(t)^2 +&= +2\operatorname{sn}(t,k)^2 +\bigl( +{\textstyle \frac32}-\operatorname{sn}(t,k)^2 +\bigr)^2 +\\ +&= +{\textstyle\frac{9}{2}}\operatorname{sn}(t,k)^2 +- +6\operatorname{sn}(t,k)^4 ++2\operatorname{sn}(t,k)^6 +\\ +\dot{Y}(t) +&= +\operatorname{cn}'(t,k)\operatorname{sn}(t,k) ++ +\operatorname{cn}(t,k)\operatorname{sn}'(t,k) +\\ +&= +-\operatorname{sn}(t,k)^2 +\operatorname{dn}(t,k) ++\operatorname{cn}(t,k)^2 +\operatorname{dn}(t,k) +\\ +&= +\operatorname{dn}(t,k)\bigl(1-2\operatorname{sn}(t,k)^2\bigr) +\\ +\dot{Y}(t)^2 +&= +\bigl(1-{\textstyle\frac12}\operatorname{sn}(t,k)^2\bigr) +\bigl(1-2\operatorname|{sn}(t,k)^2\bigr)^2 +\\ +&= +1-{\textstyle\frac{9}{2}}\operatorname{sn}(t,k)^2 ++6\operatorname{sn}(t,k)^4 +-2\operatorname{sn}(t,k)^6 +\\ +\dot{X}(t)^2 + \dot{Y}(t)^2 +&= +1. +\end{align*} +Dies bedeutet, dass die Bogenlänge zwischen den Parameterwerten $0$ und $s$ +\[ +\int_0^s +\sqrt{\dot{X}(t)^2 + \dot{Y}(t)^2} +\,dt += +\int_0^s\,dt += +s, +\] +der Parameter $t$ ist also ein Bogenlängenparameter. + +Die mit dem Faktor $1/\sqrt{2}$ skalierte Standard-Lemniskate mit der +Gleichung +\[ +(x^2+y^2)^2 = x^2-y^2 +\] +hat daher eine Bogenlängenparametrisierung mit +\begin{equation} +\begin{aligned} +x(t) +&= +\phantom{\frac{1}{\sqrt{2}}} +\operatorname{cn}(\sqrt{2}t,k)\operatorname{dn}(\sqrt{2}t,k) +\\ +y(t) +&= +\frac{1}{\sqrt{2}}\operatorname{cn}(\sqrt{2}t,k)\operatorname{sn}(\sqrt{2}t,k) +\end{aligned} +\label{buch:elliptisch:lemniskate:bogenlaenge} +\end{equation} + +\subsection{Der lemniskatische Sinus und Kosinus} +Der Sinus Berechnet die Gegenkathete zu einer gegebenen Bogenlänge des +Kreises, er ist die Umkehrfunktion der Funktion, die der Gegenkathete +die Bogenlänge zuordnet. -\subsubsection{Der lemniskatische Sinus und Kosinus} -Berechnet die Gegenkathete zu einer gegebenen Bogenlänge des Kreises. Daher ist es naheliegend, die Umkehrfunktion von $s(r)$ in \eqref{buch:elliptisch:eqn:lemniskatebogenlaenge} den {\em lemniskatischen Sinus} zu nennen mit der Bezeichnung @@ -150,22 +368,29 @@ Der Kosinus ist der Sinus des komplementären Winkels. Auch für die lemniskatische Bogenlänge $s(r)$ lässt sich eine komplementäre Bogenlänge definieren, nämlich die Bogenlänge zwischen dem Punkt $(x(r), y(r))$ und $(1,0)$. -Die Länge des rechten Blattes der Lemniskate wird mit $\varpi$ bezeichnet -und hat den numerischen Wert + +Da die Parametrisierung~\eqref{buch:elliptisch:lemniskate:bogenlaenge} +eine Bogenlängenparametrisierung ist, darf man $t=s$ schreiben. +Dann kann man aber auch $r(s)$ daraus berechnen, +es ist \[ -\varphi +r(s)^2 = -2\int_0^1\sqrt{\frac{1}{1-t^4}}\,dt +x(s)^2 + y(s)^2 = -2.6220575542. +\operatorname{cn}(s\sqrt{2},k)^2 +\qquad\Rightarrow\qquad +r(s) += +\operatorname{cn}(s\sqrt{2},k) \] -Lemniskatenbogens zwischen dem Nullpunkt und $(1,0)$ hat die Länge -$\varpi/2$. - -Der {\em lemniskatische Kosinus} von $s$ ist derjenige Radiuswert $r$, -für den der Lemniskatenbogen zwischen $(x(r), y(r))$ und $(1,0)$ -die Länge $s$ hat. - -XXX Algebraische Beziehungen \\ -XXX Ableitungen \\ +\begin{figure} +\centering +\includegraphics{chapters/110-elliptisch/images/slcl.pdf} +\caption{ +Lemniskatischer Sinus und Kosinus sowie Sinus und Kosinus +mit derart skaliertem Argument, dass die Funktionen die gleichen Nullstellen +haben. +\label{buch:elliptisch:figure:slcl}} +\end{figure} -- cgit v1.2.1 From 4666311b63fb00a3f90d1c9858218e24b14360bc Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 20 Apr 2022 10:31:44 +0200 Subject: add missing files --- buch/chapters/110-elliptisch/dglsol.tex | 494 +++++++++++++ buch/chapters/110-elliptisch/elltrigo.tex | 1012 +++++++++++++++++++++++++++ buch/chapters/110-elliptisch/mathpendel.tex | 250 +++++++ 3 files changed, 1756 insertions(+) create mode 100644 buch/chapters/110-elliptisch/dglsol.tex create mode 100644 buch/chapters/110-elliptisch/elltrigo.tex create mode 100644 buch/chapters/110-elliptisch/mathpendel.tex (limited to 'buch') diff --git a/buch/chapters/110-elliptisch/dglsol.tex b/buch/chapters/110-elliptisch/dglsol.tex new file mode 100644 index 0000000..7eaab38 --- /dev/null +++ b/buch/chapters/110-elliptisch/dglsol.tex @@ -0,0 +1,494 @@ +% +% dglsol.tex -- Lösung von Differentialgleichungen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% + +% +% Lösung von Differentialgleichungen +% +\subsection{Lösungen von Differentialgleichungen +\label{buch:elliptisch:subsection:differentialgleichungen}} +Die elliptischen Funktionen ermöglichen die Lösung gewisser nichtlinearer +Differentialgleichungen in geschlossener Form. +Ziel dieses Abschnitts ist, Differentialgleichungen der Form +\( +\dot{x}(t)^2 += +P(x(t)) +\) +mit einem Polynom $P$ vierten Grades oder +\( +\ddot{x}(t) += +p(x(t)) +\) +mit einem Polynom dritten Grades als rechter Seite lösen zu können. + +% +% Die Differentialgleichung der elliptischen Funktionen +% +\subsubsection{Die Differentialgleichungen der elliptischen Funktionen} +Um Differentialgleichungen mit elliptischen Funktion lösen zu +können, muss man als erstes die Differentialgleichungen derselben +finden. +Quadriert man die Ableitungsregel für $\operatorname{sn}(u,k)$, erhält +man +\[ +\biggl(\frac{d}{du}\operatorname{sn}(u,k)\biggr)^2 += +\operatorname{cn}(u,k)^2 \operatorname{dn}(u,k)^2. +\] +Die Funktionen auf der rechten Seite können durch $\operatorname{sn}(u,k)$ +ausgedrückt werden, dies führt auf die Differentialgleichung +\begin{align*} +\biggl(\frac{d}{du}\operatorname{sn}(u,k)\biggr)^2 +&= +\bigl( +1-\operatorname{sn}(u,k)^2 +\bigr) +\bigl( +1-k^2 \operatorname{sn}(u,k)^2 +\bigr) +\\ +&= +k^2\operatorname{sn}(u,k)^4 +-(1+k^2) +\operatorname{sn}(u,k)^2 ++1. +\end{align*} +Für die Funktion $\operatorname{cn}(u,k)$ ergibt die analoge Rechnung +\begin{align*} +\frac{d}{du}\operatorname{cn}(u,k) +&= +-\operatorname{sn}(u,k) \operatorname{dn}(u,k) +\\ +\biggl(\frac{d}{du}\operatorname{cn}(u,k)\biggr)^2 +&= +\operatorname{sn}(u,k)^2 \operatorname{dn}(u,k)^2 +\\ +&= +\bigl(1-\operatorname{cn}(u,k)^2\bigr) +\bigl(k^{\prime 2}+k^2 \operatorname{cn}(u,k)^2\bigr) +\\ +&= +-k^2\operatorname{cn}(u,k)^4 ++ +(k^2-k^{\prime 2})\operatorname{cn}(u,k)^2 ++ +k^{\prime 2} +\intertext{und weiter für $\operatorname{dn}(u,k)$:} +\frac{d}{du}\operatorname{dn}(u,k) +&= +-k^2\operatorname{sn}(u,k)\operatorname{cn}(u,k) +\\ +\biggl( +\frac{d}{du}\operatorname{dn}(u,k) +\biggr)^2 +&= +\bigl(k^2 \operatorname{sn}(u,k)^2\bigr) +\bigl(k^2 \operatorname{cn}(u,k)^2\bigr) +\\ +&= +\bigl( +1-\operatorname{dn}(u,k)^2 +\bigr) +\bigl( +\operatorname{dn}(u,k)^2-k^{\prime 2} +\bigr) +\\ +&= +-\operatorname{dn}(u,k)^4 ++ +(1+k^{\prime 2})\operatorname{dn}(u,k)^2 +-k^{\prime 2}. +\end{align*} + +\begin{table} +\centering +\renewcommand{\arraystretch}{1.7} +\begin{tabular}{|>{$}l<{$}|>{$}l<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} +\hline +\text{Funktion $y=$}&\text{Differentialgleichung}&\alpha&\beta&\gamma\\ +\hline +\operatorname{sn}(u,k) + & y'^2 = \phantom{-}(1-y^2)(1-k^2y^2) + &k^2&1+k^2&1 +\\ +\operatorname{cn}(u,k) &y'^2 = \phantom{-}(1-y^2)(k^{\prime2}+k^2y^2) + &-k^2 &k^2-k^{\prime 2}=2k^2-1&k^{\prime2} +\\ +\operatorname{dn}(u,k) + & y'^2 = -(1-y^2)(k^{\prime 2}-y^2) + &-1 &1+k^{\prime 2}=2-k^2 &-k^{\prime2} +\\ +\hline +\end{tabular} +\caption{Elliptische Funktionen als Lösungsfunktionen für verschiedene +nichtlineare Differentialgleichungen der Art +\eqref{buch:elliptisch:eqn:1storderdglell}. +Die Vorzeichen der Koeffizienten $\alpha$, $\beta$ und $\gamma$ +entscheidet darüber, welche Funktion für die Lösung verwendet werden +muss. +\label{buch:elliptisch:tabelle:loesungsfunktionen}} +\end{table} + +Die drei grundlegenden Jacobischen elliptischen Funktionen genügen also alle +einer nichtlinearen Differentialgleichung erster Ordnung der selben Art. +Das Quadrat der Ableitung ist ein Polynom vierten Grades der Funktion. +Die Differentialgleichungen sind in der +Tabelle~\ref{buch:elliptisch:tabelle:loesungsfunktionen} zusammengefasst. + +% +% Differentialgleichung der abgeleiteten elliptischen Funktionen +% +\subsubsection{Die Differentialgleichung der abgeleiteten elliptischen +Funktionen} +Da auch die Ableitungen der abgeleiteten Jacobischen elliptischen +Funktionen Produkte von genau zwei Funktionen sind, die sich wieder +durch die ursprüngliche Funktion ausdrücken lassen, darf man erwarten, +dass alle elliptischen Funktionen einer ähnlichen Differentialgleichung +genügen. +Um dies besser einzufangen, schreiben wir $\operatorname{pq}(u,k)$, +wenn wir eine beliebige abgeleitete Jacobische elliptische Funktion. +Für +$\operatorname{pq}=\operatorname{sn}$ +$\operatorname{pq}=\operatorname{cn}$ +und +$\operatorname{pq}=\operatorname{dn}$ +wissen wir bereits und erwarten für jede andere Funktion dass +$\operatorname{pq}(u,k)$ auch, dass sie Lösung einer Differentialgleichung +der Form +\begin{equation} +\operatorname{pq}'(u,k)^2 += +\alpha \operatorname{pq}(u,k)^4 + \beta \operatorname{pq}(u,k)^2 + \gamma +\label{buch:elliptisch:eqn:1storderdglell} +\end{equation} +erfüllt, +wobei wir mit $\operatorname{pq}'(u,k)$ die Ableitung von +$\operatorname{pq}(u,k)$ nach dem ersten Argument meinen. +Die Koeffizienten $\alpha$, $\beta$ und $\gamma$ hängen von $k$ ab, +ihre Werte für die grundlegenden Jacobischen elliptischen +sind in Tabelle~\ref{buch:elliptisch:table:differentialgleichungen} +zusammengestellt. + +Die Koeffizienten müssen nicht für jede Funktion wieder neu bestimmt +werden, denn für den Kehrwert einer Funktion lässt sich die +Differentialgleichung aus der Differentialgleichung der ursprünglichen +Funktion ermitteln. + +% +% Differentialgleichung der Kehrwertfunktion +% +\subsubsection{Differentialgleichung für den Kehrwert einer elliptischen Funktion} +Aus der Differentialgleichung~\eqref{buch:elliptisch:eqn:1storderdglell} +für die Funktion $\operatorname{pq}(u,k)$ kann auch eine +Differentialgleichung für den Kehrwert +$\operatorname{qp}(u,k)=\operatorname{pq}(u,k)^{-1}$ +ableiten. +Dazu rechnet man +\[ +\operatorname{qp}'(u,k) += +\frac{d}{du}\frac{1}{\operatorname{pq}(u,k)} += +\frac{\operatorname{pq}'(u,k)}{\operatorname{pq}(u,k)^2} +\qquad\Rightarrow\qquad +\left\{ +\quad +\begin{aligned} +\operatorname{pq}(u,k) +&= +\frac{1}{\operatorname{qp}(u,k)} +\\ +\operatorname{pq}'(u,k) +&= +\frac{\operatorname{qp}'(u,k)}{\operatorname{qp}(u,k)^2} +\end{aligned} +\right. +\] +und setzt in die Differentialgleichung ein: +\begin{align*} +\biggl( +\frac{ +\operatorname{qp}'(u,k) +}{ +\operatorname{qp}(u,k) +} +\biggr)^2 +&= +\alpha \frac{1}{\operatorname{qp}(u,k)^4} ++ +\beta \frac{1}{\operatorname{qp}(u,k)^2} ++ +\gamma. +\end{align*} +Nach Multiplikation mit $\operatorname{qp}(u,k)^4$ erhält man den +folgenden Satz. + +\begin{satz} +Wenn die Jacobische elliptische Funktion $\operatorname{pq}(u,k)$ +der Differentialgleichung genügt, dann genügt der Kehrwert +$\operatorname{qp}(u,k) = 1/\operatorname{pq}(u,k)$ der Differentialgleichung +\begin{equation} +(\operatorname{qp}'(u,k))^2 += +\gamma \operatorname{qp}(u,k)^4 ++ +\beta \operatorname{qp}(u,k)^2 ++ +\alpha +\label{buch:elliptisch:eqn:kehrwertdgl} +\end{equation} +\end{satz} + +\begin{table} +\centering +\def\lfn#1{\multicolumn{1}{|l|}{#1}} +\def\rfn#1{\multicolumn{1}{r|}{#1}} +\renewcommand{\arraystretch}{1.3} +\begin{tabular}{l|>{$}c<{$}>{$}c<{$}>{$}c<{$}|r} +\cline{1-4} +\lfn{Funktion} + & \alpha & \beta & \gamma &\\ +\hline +\lfn{sn}& k^2 & -(1+k^2) & 1 &\rfn{ns}\\ +\lfn{cn}& -k^2 & -(1-2k^2) & 1-k^2 &\rfn{nc}\\ +\lfn{dn}& 1 & 2-k^2 & -(1-k^2) &\rfn{nd}\\ +\hline +\lfn{sc}& 1-k^2 & 2-k^2 & 1 &\rfn{cs}\\ +\lfn{sd}&-k^2(1-k^2)&-(1-2k^2) & 1 &\rfn{ds}\\ +\lfn{cd}& k^2 &-(1+k^2) & 1 &\rfn{dc}\\ +\hline + & \gamma & \beta & \alpha &\rfn{Reziproke}\\ +\cline{2-5} +\end{tabular} +\caption{Koeffizienten der Differentialgleichungen für die Jacobischen +elliptischen Funktionen. +Der Kehrwert einer Funktion hat jeweils die Differentialgleichung der +ursprünglichen Funktion, in der die Koeffizienten $\alpha$ und $\gamma$ +vertauscht worden sind. +\label{buch:elliptisch:table:differentialgleichungen}} +\end{table} + +% +% Differentialgleichung zweiter Ordnung +% +\subsubsection{Differentialgleichung zweiter Ordnung} +Leitet die Differentialgleichung~\eqref{buch:elliptisch:eqn:1storderdglell} +man dies nochmals nach $u$ ab, erhält man die Differentialgleichung +\[ +2\operatorname{pq}''(u,k)\operatorname{pq}'(u,k) += +4\alpha \operatorname{pq}(u,k)^3\operatorname{pq}'(u,k) + 2\beta \operatorname{pq}'(u,k)\operatorname{pq}(u,k). +\] +Teilt man auf beiden Seiten durch $2\operatorname{pq}'(u,k)$, +bleibt die nichtlineare +Differentialgleichung +\[ +\frac{d^2\operatorname{pq}}{du^2} += +\beta \operatorname{pq} + 2\alpha \operatorname{pq}^3. +\] +Dies ist die Gleichung eines harmonischen Oszillators mit einer +Anharmonizität der Form $2\alpha z^3$. + + + +% +% Jacobischen elliptische Funktionen und elliptische Integrale +% +\subsubsection{Jacobische elliptische Funktionen als elliptische Integrale} +Die in Tabelle~\ref{buch:elliptisch:tabelle:loesungsfunktionen} +zusammengestellten Differentialgleichungen ermöglichen nun, den +Zusammenhang zwischen den Funktionen +$\operatorname{sn}(u,k)$, $\operatorname{cn}(u,k)$ und $\operatorname{dn}(u,k)$ +und den unvollständigen elliptischen Integralen herzustellen. +Die Differentialgleichungen sind alle von der Form +\begin{equation} +\biggl( +\frac{d y}{d u} +\biggr)^2 += +p(u), +\label{buch:elliptisch:eqn:allgdgl} +\end{equation} +wobei $p(u)$ ein Polynom vierten Grades in $y$ ist. +Diese Differentialgleichung lässt sich mit Separation lösen. +Dazu zieht man aus~\eqref{buch:elliptisch:eqn:allgdgl} die +Wurzel +\begin{align} +\frac{dy}{du} += +\sqrt{p(y)} +\notag +\intertext{und trennt die Variablen. Man erhält} +\int\frac{dy}{\sqrt{p(y)}} = u+C. +\label{buch:elliptisch:eqn:yintegral} +\end{align} +Solange $p(y)>0$ ist, ist der Integrand auf der linken Seite +von~\eqref{buch:elliptisch:eqn:yintegral} ebenfalls positiv und +das Integral ist eine monoton wachsende Funktion $F(y)$. +Insbesondere ist $F(y)$ invertierbar. +Die Lösung $y(u)$ der Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} +ist daher +\[ +y(u) = F^{-1}(u+C). +\] +Die Jacobischen elliptischen Funktionen sind daher inverse Funktionen +der unvollständigen elliptischen Integrale. + + +% +% Differentialgleichung des anharmonischen Oszillators +% +\subsubsection{Differentialgleichung des anharmonischen Oszillators} +Wir möchten die nichtlineare Differentialgleichung +\begin{equation} +\biggl( +\frac{dx}{dt} +\biggr)^2 += +Ax^4+Bx^2 + C +\label{buch:elliptisch:eqn:allgdgl} +\end{equation} +mit Hilfe elliptischer Funktionen lösen. +Wir nehmen also an, dass die gesuchte Lösung eine Funktion der Form +\begin{equation} +x(t) = a\operatorname{zn}(bt,k) +\label{buch:elliptisch:eqn:loesungsansatz} +\end{equation} +ist. +Die erste Ableitung von $x(t)$ ist +\[ +\dot{x}(t) += +a\operatorname{zn}'(bt,k). +\] + +Indem wir diesen Lösungsansatz in die +Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} +einsetzen, erhalten wir +\begin{equation} +a^2b^2 \operatorname{zn}'(bt,k)^2 += +a^4A\operatorname{zn}(bt,k)^4 ++ +a^2B\operatorname{zn}(bt,k)^2 ++C +\label{buch:elliptisch:eqn:dglx} +\end{equation} +Andererseits wissen wir, dass $\operatorname{zn}(u,k)$ einer +Differentilgleichung der Form~\eqref{buch:elliptisch:eqn:1storderdglell} +erfüllt. +Wenn wir \eqref{buch:elliptisch:eqn:dglx} durch $a^2b^2$ teilen, können wir +die rechte Seite von \eqref{buch:elliptisch:eqn:dglx} mit der rechten +Seite von \eqref{buch:elliptisch:eqn:1storderdglell} vergleichen: +\[ +\frac{a^2A}{b^2}\operatorname{zn}(bt,k)^4 ++ +\frac{B}{b^2}\operatorname{zn}(bt,k)^2 ++\frac{C}{a^2b^2} += +\alpha\operatorname{zn}(bt,k)^4 ++ +\beta\operatorname{zn}(bt,k)^2 ++ +\gamma\operatorname{zn}(bt,k). +\] +Daraus ergeben sich die Gleichungen +\begin{align} +\alpha &= \frac{a^2A}{b^2}, +& +\beta &= \frac{B}{b^2} +&&\text{und} +& +\gamma &= \frac{C}{a^2b^2} +\label{buch:elliptisch:eqn:koeffvergl} +\intertext{oder aufgelöst nach den Koeffizienten der ursprünglichen +Differentialgleichung} +A&=\frac{\alpha b^2}{a^2} +& +B&=\beta b^2 +&&\text{und}& +C &= \gamma a^2b^2 +\label{buch:elliptisch:eqn:koeffABC} +\end{align} +für die Koeffizienten der Differentialgleichung der zu verwendenden +Funktion. + +Man beachte, dass nach \eqref{buch:elliptisch:eqn:koeffvergl} die +Koeffizienten $A$, $B$ und $C$ die gleichen Vorzeichen haben wie +$\alpha$, $\beta$ und $\gamma$, da in +\eqref{buch:elliptisch:eqn:koeffvergl} nur mit Quadraten multipliziert +wird, die immer positiv sind. +Diese Vorzeichen bestimmen, welche der Funktionen gewählt werden muss. + +In den Differentialgleichungen für die elliptischen Funktionen gibt +es nur den Parameter $k$, der angepasst werden kann. +Es folgt, dass die Gleichungen +\eqref{buch:elliptisch:eqn:koeffvergl} +auch $a$ und $b$ bestimmen. +Zum Beispiel folgt aus der letzten Gleichung, dass +\[ +b = \pm\sqrt{\frac{B}{\beta}}. +\] +Damit folgt dann aus der zweiten +\[ +a=\pm\sqrt{\frac{\beta C}{\gamma B}}. +\] +Die verbleibende Gleichung legt $k$ fest. +Das folgende Beispiel illustriert das Vorgehen am Beispiel einer +Gleichung, die Lösungsfunktion $\operatorname{sn}(u,k)$ verlangt. + +\begin{beispiel} +Wir nehmen an, dass die Vorzeichen von $A$, $B$ und $C$ gemäss +Tabelle~\ref{buch:elliptische:tabelle:loesungsfunktionen} verlangen, +dass die Funktion $\operatorname{sn}(u,k)$ für die Lösung verwendet +werden muss. +Die Tabelle sagt dann auch, dass +$\alpha=k^2$, $\beta=1$ und $\gamma=1$ gewählt werden müssen. +Aus dem Koeffizientenvergleich~\eqref{buch:elliptisch:eqn:koeffvergl} +folgt dann der Reihe nach +\begin{align*} +b&=\pm \sqrt{B} +\\ +a&=\pm \sqrt{\frac{C}{B}} +\\ +k^2 +&= +\frac{AC}{B^2}. +\end{align*} +Man beachte, dass man $k^2$ durch Einsetzen von +\eqref{buch:elliptisch:eqn:koeffABC} +auch direkt aus den Koeffizienten $\alpha$, $\beta$ und $\gamma$ +erhalten kann, nämlich +\[ +\frac{AC}{B^2} += +\frac{\frac{\alpha b^2}{a^2} \gamma a^2b^2}{\beta^2 b^4} += +\frac{\alpha\gamma}{\beta^2}. +\qedhere +\] +\end{beispiel} + +Da alle Parameter im +Lösungsansatz~\eqref{buch:elliptisch:eqn:loesungsansatz} bereits +festgelegt sind stellt sich die Frage, woher man einen weiteren +Parameter nehmen kann, mit dem Anfangsbedingungen erfüllen kann. +Die Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} ist +autonom, die Koeffizienten der rechten Seite der Differentialgleichung +sind nicht von der Zeit abhängig. +Damit ist eine zeitverschobene Funktion $x(t-t_0)$ ebenfalls eine +Lösung der Differentialgleichung. +Die allgmeine Lösung der +Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} hat +also die Form +\[ +x(t) = a\operatorname{zn}(b(t-t_0)), +\] +wobei die Funktion $\operatorname{zn}(u,k)$ auf Grund der Vorzeichen +von $A$, $B$ und $C$ gewählt werden müssen. + diff --git a/buch/chapters/110-elliptisch/elltrigo.tex b/buch/chapters/110-elliptisch/elltrigo.tex new file mode 100644 index 0000000..d600243 --- /dev/null +++ b/buch/chapters/110-elliptisch/elltrigo.tex @@ -0,0 +1,1012 @@ +% +% elltrigo.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% + +% +% elliptische Funktionen als Trigonometrie +% +\subsection{Elliptische Funktionen als Trigonometrie} +\begin{figure} +\centering +\includegraphics{chapters/110-elliptisch/images/ellipse.pdf} +\caption{Kreis und Ellipse zum Vergleich und zur Herleitung der +elliptischen Funktionen von Jacobi als ``trigonometrische'' Funktionen +auf einer Ellipse. +\label{buch:elliptisch:fig:ellipse}} +\end{figure} +% based on Willliam Schwalm, Elliptic functions and elliptic integrals +% https://youtu.be/DCXItCajCyo + +% +% Geometrie einer Ellipse +% +\subsubsection{Geometrie einer Ellipse} +Eine {\em Ellipse} ist die Menge der Punkte der Ebene, für die die Summe +\index{Ellipse}% +der Entfernungen von zwei festen Punkten $F_1$ und $F_2$, +den {\em Brennpunkten}, konstant ist. +\index{Brennpunkt}% +In Abbildung~\ref{buch:elliptisch:fig:ellipse} eine Ellipse +mit Brennpunkten in $F_1=(-e,0)$ und $F_2=(e,0)$ dargestellt, +die durch die Punkte $(\pm a,0)$ und $(0,\pm b)$ auf den Achsen geht. +Der Punkt $(a,0)$ hat die Entfernungen $a+e$ und $a-e$ von den beiden +Brennpunkten, also die Entfernungssumme $a+e+a-e=2a$. +Jeder andere Punkt auf der Ellipse muss ebenfalls diese Entfernungssumme +haben, insbesondere auch der Punkt $(0,b)$. +Seine Entfernung zu jedem Brennpunkt muss aus Symmetriegründen gleich gross, +also $a$ sein. +Aus dem Satz von Pythagoras liest man daher ab, dass +\[ +b^2+e^2=a^2 +\qquad\Rightarrow\qquad +e^2 = a^2-b^2 +\] +sein muss. +Die Strecke $e$ heisst auch {\em (lineare) Exzentrizität} der Ellipse. +Das Verhältnis $\varepsilon= e/a$ heisst die {\em numerische Exzentrizität} +der Ellipse. + +% +% Die Ellipsengleichung +% +\subsubsection{Ellipsengleichung} +Der Punkt $P=(x,y)$ auf der Ellipse hat die Entfernungen +\begin{equation} +\begin{aligned} +\overline{PF_1}^2 +&= +y^2 + (x+e)^2 +\\ +\overline{PF_2}^2 +&= +y^2 + (x-e)^2 +\end{aligned} +\label{buch:elliptisch:eqn:wurzelausdruecke} +\end{equation} +von den Brennpunkten, für die +\begin{equation} +\overline{PF_1}+\overline{PF_2} += +2a +\label{buch:elliptisch:eqn:pf1pf2a} +\end{equation} +gelten muss. +Man kann nachrechnen, dass ein Punkt $P$, der die Gleichung +\[ +\frac{x^2}{a^2} + \frac{y^2}{b^2}=1 +\] +erfüllt, auch die Eigenschaft~\eqref{buch:elliptisch:eqn:pf1pf2a} +erfüllt. +Zur Vereinfachung setzen wir $l_1=\overline{PF_1}$ und $l_2=\overline{PF_2}$. +$l_1$ und $l_2$ sind Wurzeln aus der rechten Seite von +\eqref{buch:elliptisch:eqn:wurzelausdruecke}. +Das Quadrat von $l_1+l_2$ ist +\[ +l_1^2 + 2l_1l_2 + l_2^2 = 4a^2. +\] +Um die Wurzeln ganz zu eliminieren, bringt man das Produkt $l_1l_2$ alleine +auf die rechte Seite und quadriert. +Man muss also verifizieren, dass +\[ +(l_1^2 + l_2^2 -4a^2)^2 = 4l_1^2l_2^2. +\] +In den entstehenden Ausdrücken muss man ausserdem $e=\sqrt{a^2-b^2}$ und +\[ +y=b\sqrt{1-\frac{x^2}{a^2}} +\] +substituieren. +Diese Rechnung führt man am einfachsten mit Hilfe eines +Computeralgebraprogramms durch, welches obige Behauptung bestätigt. + +% +% Normierung +% +\subsubsection{Normierung} +Die trigonometrischen Funktionen sind definiert als Verhältnisse +von Seiten rechtwinkliger Dreiecke. +Dadurch, dass man den die Hypothenuse auf Länge $1$ normiert, +kann man die Sinus- und Kosinus-Funktion als Koordinaten eines +Punktes auf dem Einheitskreis interpretieren. + +Für die Koordinaten eines Punktes auf der Ellipse ist dies nicht so einfach, +weil es nicht nur eine Ellipse gibt, sondern für jede numerische Exzentrizität +mindestens eine mit Halbeachse $1$. +Wir wählen die Ellipsen so, dass $a$ die grosse Halbachse ist, also $a>b$. +Als Normierungsbedingung verwenden wir, dass $b=1$ sein soll, wie in +Abbildung~\ref{buch:elliptisch:fig:jacobidef}. +Dann ist $a=1/\varepsilon>1$. +In dieser Normierung haben Punkte $(x,y)$ auf der Ellipse $y$-Koordinaten +zwischen $-1$ und $1$ und $x$-Koordinaten zwischen $-a$ und $a$. + +Im Zusammenhang mit elliptischen Funktionen wird die numerische Exzentrizität +$\varepsilon$ auch mit +\[ +k += +\varepsilon += +\frac{e}{a} += +\frac{\sqrt{a^2-b^2}}{a} += +\frac{\sqrt{a^2-1}}{a}, +\] +die Zahl $k$ heisst auch der {\em Modulus}. +Man kann $a$ auch durch $k$ ausdrücken, durch Quadrieren und Umstellen +findet man +\[ +k^2a^2 = a^2-1 +\quad\Rightarrow\quad +1=a^2(k^2-1) +\quad\Rightarrow\quad +a=\frac{1}{\sqrt{k^2-1}}. +\] + +Die Gleichung der ``Einheitsellipse'' zu diesem Modulus ist +\[ +\frac{x^2}{a^2}+y^2=1 +\qquad\text{oder}\qquad +x^2(k^2-1) + y^2 = 1. +\] + +% +% Definition der elliptischen Funktionen +% +\begin{figure} +\centering +\includegraphics{chapters/110-elliptisch/images/jacobidef.pdf} +\caption{Definition der elliptischen Funktionen als Trigonometrie +an einer Ellipse mit Halbachsen $a$ und $1$. +\label{buch:elliptisch:fig:jacobidef}} +\end{figure} +\subsubsection{Definition der elliptischen Funktionen} +Die elliptischen Funktionen für einen Punkt $P$ auf der Ellipse mit Modulus $k$ +können jetzt als Verhältnisse der Koordinaten des Punktes definieren. +Es stellt sich aber die Frage, was man als Argument verwenden soll. +Es soll so etwas wie den Winkel $\varphi$ zwischen der $x$-Achse und dem +Radiusvektor zum Punkt $P$ +darstellen, aber wir haben hier noch eine Wahlfreiheit, die wir später +ausnützen möchten. +Im Moment müssen wir die Frage noch nicht beantworten und nennen das +noch unbestimmte Argument $u$. +Wir kümmern uns später um die Frage, wie $u$ von $\varphi$ abhängt. + +Die Funktionen, die wir definieren wollen, hängen ausserdem auch +vom Modulus ab. +Falls der verwendete Modulus aus dem Zusammenhang klar ist, lassen +wir das $k$-Argument weg. + +Die Punkte auf dem Einheitskreis haben alle den gleichen Abstand vom +Nullpunkt, dies ist gleichzeitig die definierende Gleichung $r^2=x^2+y^2=1$ +des Kreises. +Die Punkte auf der Ellipse erfüllen die Gleichung $x^2/a^2+y^2=1$, +die Entfernung der Punkte $r=\sqrt{x^2+y^2}$ vom Nullpunkt variert aber. + +In Analogie zu den trigonometrischen Funktionen setzen wir jetzt für +die Funktionen +\[ +\begin{aligned} +&\text{sinus amplitudinis:}& +{\color{red}\operatorname{sn}(u,k)}&= y \\ +&\text{cosinus amplitudinis:}& +{\color{blue}\operatorname{cn}(u,k)}&= \frac{x}{a} \\ +&\text{delta amplitudinis:}& +{\color{darkgreen}\operatorname{dn}(u,k)}&=\frac{r}{a}, +\end{aligned} +\] +die auch in Abbildung~\ref{buch:elliptisch:fig:jacobidef} +dargestellt sind. +Aus der Gleichung der Ellipse folgt sofort, dass +\[ +\operatorname{sn}(u,k)^2 + \operatorname{cn}(u,k)^2 = 1 +\] +ist. +Der Satz von Pythagoras kann verwendet werden, um die Entfernung zu +berechnen, also gilt +\begin{equation} +r^2 += +a^2 \operatorname{dn}(u,k)^2 += +x^2 + y^2 += +a^2\operatorname{cn}(u,k)^2 + \operatorname{sn}(u,k)^2 +\quad +\Rightarrow +\quad +a^2 \operatorname{dn}(u,k)^2 += +a^2\operatorname{cn}(u,k)^2 + \operatorname{sn}(u,k)^2. +\label{buch:elliptisch:eqn:sncndnrelation} +\end{equation} +Ersetzt man +$ +a^2\operatorname{cn}(u,k)^2 += +a^2-a^2\operatorname{sn}(u,k)^2 +$, ergibt sich +\[ +a^2 \operatorname{dn}(u,k)^2 += +a^2-a^2\operatorname{sn}(u,k)^2 ++ +\operatorname{sn}(u,k)^2 +\quad +\Rightarrow +\quad +\operatorname{dn}(u,k)^2 ++ +\frac{a^2-1}{a^2}\operatorname{sn}(u,k)^2 += +1, +\] +woraus sich die Identität +\[ +\operatorname{dn}(u,k)^2 + k^2 \operatorname{sn}(u,k)^2 = 1 +\] +ergibt. +Ebenso kann man aus~\eqref{buch:elliptisch:eqn:sncndnrelation} +die Funktion $\operatorname{cn}(u,k)$ eliminieren, was auf +\[ +a^2\operatorname{dn}(u,k)^2 += +a^2\operatorname{cn}(u,k)^2 ++1-\operatorname{cn}(u,k)^2 += +(a^2-1)\operatorname{cn}(u,k)^2 ++1. +\] +Nach Division durch $a^2$ ergibt sich +\begin{align*} +\operatorname{dn}(u,k)^2 +- +k^2\operatorname{cn}(u,k)^2 +&= +\frac{1}{a^2} += +\frac{a^2-a^2+1}{a^2} += +1-k^2 =: k^{\prime 2}. +\end{align*} +Wir stellen die hiermit gefundenen Relationen zwischen den grundlegenden +Jacobischen elliptischen Funktionen für später zusammen in den Formeln +\begin{equation} +\begin{aligned} +\operatorname{sn}^2(u,k) ++ +\operatorname{cn}^2(u,k) +&= +1 +\\ +\operatorname{dn}^2(u,k) + k^2\operatorname{sn}^2(u,k) +&= +1 +\\ +\operatorname{dn}^2(u,k) -k^2\operatorname{cn}^2(u,k) +&= +k^{\prime 2}. +\end{aligned} +\label{buch:elliptisch:eqn:jacobi-relationen} +\end{equation} +zusammen. +So wie es möglich ist, $\sin\alpha$ durch $\cos\alpha$ auszudrücken, +ist es mit +\eqref{buch:elliptisch:eqn:jacobi-relationen} +jetzt auch möglich jede grundlegende elliptische Funktion durch +jede anderen auszudrücken. +Die Resultate sind in der Tabelle~\ref{buch:elliptisch:fig:jacobi-relationen} +zusammengestellt. + +\begin{table} +\centering +\renewcommand{\arraystretch}{2.1} +\begin{tabular}{|>{$\displaystyle}c<{$}|>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}|} +\hline +&\operatorname{sn}(u,k) +&\operatorname{cn}(u,k) +&\operatorname{dn}(u,k)\\ +\hline +\operatorname{sn}(u,k) +&\operatorname{sn}(u,k) +&\sqrt{1-\operatorname{cn}^2(u,k)} +&\frac1k\sqrt{1-\operatorname{dn}^2(u,k)} +\\ +\operatorname{cn}(u,k) +&\sqrt{1-\operatorname{sn}^2(u,k)} +&\operatorname{cn}(u,k) +&\frac{1}{k}\sqrt{\operatorname{dn}^2(u,k)-k^{\prime2}} +\\ +\operatorname{dn}(u,k) +&\sqrt{1-k^2\operatorname{sn}^2(u,k)} +&\sqrt{k^{\prime2}+k^2\operatorname{cn}^2(u,k)} +&\operatorname{dn}(u,k) +\\ +\hline +\end{tabular} +\caption{Jede der Jacobischen elliptischen Funktionen lässt sich +unter Verwendung der Relationen~\eqref{buch:elliptisch:eqn:jacobi-relationen} +durch jede andere ausdrücken. +\label{buch:elliptisch:fig:jacobi-relationen}} +\end{table} + +% +% Ableitungen der Jacobi-ellpitischen Funktionen +% +\subsubsection{Ableitung} +Die trigonometrischen Funktionen sind deshalb so besonders nützlich +für die Lösung von Schwingungsdifferentialgleichungen, weil sie die +Beziehungen +\[ +\frac{d}{d\varphi} \cos\varphi = -\sin\varphi +\qquad\text{und}\qquad +\frac{d}{d\varphi} \sin\varphi = \cos\varphi +\] +erfüllen. +So einfach können die Beziehungen natürlich nicht sein, sonst würde sich +durch Integration ja wieder nur die trigonometrischen Funktionen ergeben. +Durch geschickte Wahl des Arguments $u$ kann man aber erreichen, dass +sie ähnlich nützliche Beziehungen zwischen den Ableitungen ergeben. + +Gesucht ist jetzt also eine Wahl für das Argument $u$ zum Beispiel in +Abhängigkeit von $\varphi$, dass sich einfache und nützliche +Ableitungsformeln ergeben. +Wir setzen daher $u(\varphi)$ voraus und beachten, dass $x$ und $y$ +ebenfalls von $\varphi$ abhängen, es ist +$y=\sin\varphi$ und $x=a\cos\varphi$. +Die Ableitungen von $x$ und $y$ nach $\varphi$ sind +\begin{align*} +\frac{dy}{d\varphi} +&= +\cos\varphi += +\frac{1}{a} x += +\operatorname{cn}(u,k) +\\ +\frac{dx}{d\varphi} +&= +-a\sin\varphi += +-a y += +-a\operatorname{sn}(u,k). +\end{align*} +Daraus kann man jetzt die folgenden Ausdrücke für die Ableitungen der +elliptischen Funktionen nach $\varphi$ ableiten: +\begin{align*} +\frac{d}{d\varphi} \operatorname{sn}(u,z) +&= +\frac{d}{d\varphi} y(\varphi) += +\cos\varphi += +\frac{x}{a} += +\operatorname{cn}(u,k) +&&\Rightarrow& +\frac{d}{du} +\operatorname{sn}(u,k) +&= +\operatorname{cn}(u,k) \frac{d\varphi}{du} +\\ +\frac{d}{d\varphi} \operatorname{cn}(u,z) +&= +\frac{d}{d\varphi} \frac{x(\varphi)}{a} += +-\sin\varphi += +-\operatorname{sn}(u,k) +&&\Rightarrow& +\frac{d}{du}\operatorname{cn}(u,k) +&= +-\operatorname{sn}(u,k) \frac{d\varphi}{du} +\\ +\frac{d}{d\varphi} \operatorname{dn}(u,z) +&= +\frac{1}{a}\frac{dr}{d\varphi} += +\frac{1}{a}\frac{d\sqrt{x^2+y^2}}{d\varphi} +%\\ +%& +\rlap{$\displaystyle\mathstrut += +\frac{x}{ar} \frac{dx}{d\varphi} ++ +\frac{y}{ar} \frac{dy}{d\varphi} +%\\ +%& += +\frac{x}{ar} (-a\operatorname{sn}(u,k)) ++ +\frac{y}{ar} \operatorname{cn}(u,k) +$} +\\ +& +\rlap{$\displaystyle\mathstrut += +\frac{x}{ar}(-ay) ++ +\frac{y}{ar} \frac{x}{a} +%\rlap{$\displaystyle += +\frac{xy(-1+\frac{1}{a^2})}{r} +%$} +%\\ +%& += +-\frac{xy(a^2-1)}{a^2r} +$} +\\ +&= +-\frac{a^2-1}{ar} +\operatorname{cn}(u,k) \operatorname{sn}(u,k) +%\\ +%& +\rlap{$\displaystyle\mathstrut += +-k^2 +\frac{a}{r} +\operatorname{cn}(u,k) \operatorname{sn}(u,k) +$} +\\ +&= +-k^2\frac{\operatorname{cn}(u,k)\operatorname{sn}(u,k)}{\operatorname{dn}(u,k)} +&&\Rightarrow& +\frac{d}{du} \operatorname{dn}(u,k) +&= +-k^2\frac{\operatorname{cn}(u,k) +\operatorname{sn}(u,k)}{\operatorname{dn}(u,k)} +\frac{d\varphi}{du}. +\end{align*} +Die einfachsten Beziehungen ergeben sich offenbar, wenn man $u$ so +wählt, dass +\[ +\frac{d\varphi}{du} += +\operatorname{dn}(u,k) += +\frac{r}{a}. +\] +Damit haben wir die grundlegenden Ableitungsregeln + +\begin{satz} +\label{buch:elliptisch:satz:ableitungen} +Die Jacobischen elliptischen Funktionen haben die Ableitungen +\begin{equation} +\begin{aligned} +\frac{d}{du}\operatorname{sn}(u,k) +&= +\phantom{-}\operatorname{cn}(u,k)\operatorname{dn}(u,k) +\\ +\frac{d}{du}\operatorname{cn}(u,k) +&= +-\operatorname{sn}(u,k)\operatorname{dn}(u,k) +\\ +\frac{d}{du}\operatorname{dn}(u,k) +&= +-k^2\operatorname{sn}(u,k)\operatorname{cn}(u,k). +\end{aligned} +\label{buch:elliptisch:eqn:ableitungsregeln} +\end{equation} +\end{satz} + +% +% Der Grenzfall $k=1$ +% +\subsubsection{Der Grenzwert $k\to1$} +\begin{figure} +\centering +\includegraphics{chapters/110-elliptisch/images/sncnlimit.pdf} +\caption{Grenzfälle der Jacobischen elliptischen Funktionen +für die Werte $0$ und $1$ des Parameters $k$. +\label{buch:elliptisch:fig:sncnlimit}} +\end{figure} +Für $k=1$ ist $k^{\prime2}=1-k^2=$ und es folgt aus den +Relationen~\eqref{buch:elliptisch:eqn:jacobi-relationen} +\[ +\operatorname{cn}^2(u,k) +- +k^2 +\operatorname{dn}^2(u,k) += +k^{\prime2} += +0 +\qquad\Rightarrow\qquad +\operatorname{cn}^2(u,1) += +\operatorname{dn}^2(u,1), +\] +die beiden Funktionen +$\operatorname{cn}(u,k)$ +und +$\operatorname{dn}(u,k)$ +fallen also zusammen. +Die Ableitungsregeln werden dadurch vereinfacht: +\begin{align*} +\operatorname{sn}'(u,1) +&= +\operatorname{cn}(u,1) +\operatorname{dn}(u,1) += +\operatorname{cn}^2(u,1) += +1-\operatorname{sn}^2(u,1) +&&\Rightarrow& y'&=1-y^2 +\\ +\operatorname{cn}'(u,1) +&= +- +\operatorname{sn}(u,1) +\operatorname{dn}(u,1) += +- +\operatorname{sn}(u,1)\operatorname{cn}(u,1) +&&\Rightarrow& +\frac{z'}{z}&=(\log z)' = -y +\end{align*} +Die erste Differentialgleichung für $y$ lässt sich separieren, man findet +die Lösung +\[ +\frac{y'}{1-y^2} += +1 +\quad\Rightarrow\quad +\int \frac{dy}{1-y^2} = \int \,du +\quad\Rightarrow\quad +\operatorname{artanh}(y) = u +\quad\Rightarrow\quad +\operatorname{sn}(u,1)=\tanh u. +\] +Damit kann man jetzt auch $z$ berechnen: +\begin{align*} +(\log \operatorname{cn}(u,1))' +&= +\tanh u +&&\Rightarrow& +\log\operatorname{cn}(u,1) +&= +-\int\tanh u\,du += +-\log\cosh u +\\ +& +&&\Rightarrow& +\operatorname{cn}(u,1) +&= +\frac{1}{\cosh u} += +\operatorname{sech}u. +\end{align*} +Die Grenzfunktionen sind in Abbildung~\ref{buch:elliptisch:fig:sncnlimit} +dargestellt. + +% +% Das Argument u +% +\subsubsection{Das Argument $u$} +Die Gleichung +\begin{equation} +\frac{d\varphi}{du} += +\operatorname{dn}(u,k) +\label{buch:elliptisch:eqn:uableitung} +\end{equation} +ermöglicht, $\varphi$ in Abhängigkeit von $u$ zu berechnen, ohne jedoch +die geometrische Bedeutung zu klären. +Das beginnt bereits damit, dass der Winkel $\varphi$ nicht nicht der +Polarwinkel des Punktes $P$ in Abbildung~\ref{buch:elliptisch:fig:jacobidef} +ist, diesen nennen wir $\vartheta$. +Der Zusammenhang zwischen $\varphi$ und $\vartheta$ ist +\begin{equation} +\frac1{a}\tan\varphi = \tan\vartheta +\label{buch:elliptisch:eqn:phitheta} +\end{equation} + +Um die geometrische Bedeutung besser zu verstehen, nehmen wir jetzt an, +dass die Ellipse mit einem Parameter $t$ parametrisiert ist, dass also +$\varphi(t)$, $\vartheta(t)$ und $u(t)$ Funktionen von $t$ sind. +Die Ableitung von~\eqref{buch:elliptisch:eqn:phitheta} ist +\[ +\frac1{a}\cdot \frac{1}{\cos^2\varphi}\cdot \dot{\varphi} += +\frac{1}{\cos^2\vartheta}\cdot \dot{\vartheta}. +\] +Daraus kann die Ableitung von $\vartheta$ nach $\varphi$ bestimmt +werden, sie ist +\[ +\frac{d\vartheta}{d\varphi} += +\frac{\dot{\vartheta}}{\dot{\varphi}} += +\frac{1}{a} +\cdot +\frac{\cos^2\vartheta}{\cos^2\varphi} += +\frac{1}{a} +\cdot +\frac{(x/r)^2}{(x/a)^2} += +\frac{1}{a}\cdot +\frac{a^2}{r^2} += +\frac{1}{a}\cdot\frac{1}{\operatorname{dn}^2(u,k)}. +\] +Damit kann man jetzt mit Hilfe von~\eqref{buch:elliptisch:eqn:uableitung} +Die Ableitung von $\vartheta$ nach $u$ ermitteln, sie ist +\[ +\frac{d\vartheta}{du} += +\frac{d\vartheta}{d\varphi} +\cdot +\frac{d\varphi}{du} += +\frac{1}{a}\cdot\frac{1}{\operatorname{dn}^2(u,k)} +\cdot +\operatorname{dn}(u,k) += +\frac{1}{a} +\cdot +\frac{1}{\operatorname{dn}(u,k)} += +\frac{1}{a} +\cdot\frac{a}{r} += +\frac{1}{r}, +\] +wobei wir auch die Definition der Funktion $\operatorname{dn}(u,k)$ +verwendet haben. + +In der Parametrisierung mit dem Parameter $t$ kann man jetzt die Ableitung +von $u$ nach $t$ berechnen als +\[ +\frac{du}{dt} += +\frac{du}{d\vartheta} +\frac{d\vartheta}{dt} += +r +\dot{\vartheta}. +\] +Darin ist $\dot{\vartheta}$ die Winkelgeschwindigkeit des Punktes um +das Zentrum $O$ und $r$ ist die aktuelle Entfernung des Punktes $P$ +von $O$. +$r\dot{\vartheta}$ ist also die Geschwindigkeitskomponenten des Punktes +$P$ senkrecht auf den aktuellen Radiusvektor. +Der Parameter $u$, der zum Punkt $P$ gehört, ist also das Integral +\[ +u(P) = \int_0^P r\,d\vartheta. +\] +Für einen Kreis ist die Geschwindigkeit von $P$ immer senkrecht +auf dem Radiusvektor und der Radius ist konstant, so dass +$u(P)=\vartheta(P)$ ist. + +% +% Die abgeleiteten elliptischen Funktionen +% +\begin{figure} +\centering +\includegraphics[width=\textwidth]{chapters/110-elliptisch/images/jacobi12.pdf} +\caption{Die Verhältnisse der Funktionen +$\operatorname{sn}(u,k)$, +$\operatorname{cn}(u,k)$ +udn +$\operatorname{dn}(u,k)$ +geben Anlass zu neun weitere Funktionen, die sich mit Hilfe +des Strahlensatzes geometrisch interpretieren lassen. +\label{buch:elliptisch:fig:jacobi12}} +\end{figure} +\begin{table} +\centering +\renewcommand{\arraystretch}{2.5} +\begin{tabular}{|>{$\displaystyle}c<{$}|>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}|} +\hline +\cdot & +\frac{1}{1} & +\frac{1}{\operatorname{sn}(u,k)} & +\frac{1}{\operatorname{cn}(u,k)} & +\frac{1}{\operatorname{dn}(u,k)} +\\[5pt] +\hline +1& +&%\operatorname{nn}(u,k)=\frac{1}{1} & +\operatorname{ns}(u,k)=\frac{1}{\operatorname{sn}(u,k)} & +\operatorname{nc}(u,k)=\frac{1}{\operatorname{cn}(u,k)} & +\operatorname{nd}(u,k)=\frac{1}{\operatorname{dn}(u,k)} +\\ +\operatorname{sn}(u,k) & +\operatorname{sn}(u,k)=\frac{\operatorname{sn}(u,k)}{1}& +&%\operatorname{ss}(u,k)=\frac{\operatorname{sn}(u,k)}{\operatorname{sn}(u,k)}& +\operatorname{sc}(u,k)=\frac{\operatorname{sn}(u,k)}{\operatorname{cn}(u,k)}& +\operatorname{sd}(u,k)=\frac{\operatorname{sn}(u,k)}{\operatorname{dn}(u,k)} +\\ +\operatorname{cn}(u,k) & +\operatorname{cn}(u,k)=\frac{\operatorname{cn}(u,k)}{1} & +\operatorname{cs}(u,k)=\frac{\operatorname{cn}(u,k)}{\operatorname{sn}(u,k)}& +&%\operatorname{cc}(u,k)=\frac{\operatorname{cn}(u,k)}{\operatorname{cn}(u,k)}& +\operatorname{cd}(u,k)=\frac{\operatorname{cn}(u,k)}{\operatorname{dn}(u,k)} +\\ +\operatorname{dn}(u,k) & +\operatorname{dn}(u,k)=\frac{\operatorname{dn}(u,k)}{1} & +\operatorname{ds}(u,k)=\frac{\operatorname{dn}(u,k)}{\operatorname{sn}(u,k)}& +\operatorname{dc}(u,k)=\frac{\operatorname{dn}(u,k)}{\operatorname{cn}(u,k)}& +%\operatorname{dd}(u,k)=\frac{\operatorname{dn}(u,k)}{\operatorname{dn}(u,k)} +\\[5pt] +\hline +\end{tabular} +\caption{Zusammenstellung der abgeleiteten Jacobischen elliptischen +Funktionen in hinteren drei Spalten als Quotienten der grundlegenden +Jacobischen elliptischen Funktionen. +Die erste Spalte zum Nenner $1$ enthält die grundlegenden +Jacobischen elliptischen Funktionen. +\label{buch:elliptisch:table:abgeleitetjacobi}} +\end{table} + +% +% Die abgeleiteten elliptischen Funktionen +% +\subsubsection{Die abgeleiteten elliptischen Funktionen} +Zusätzlich zu den grundlegenden Jacobischen elliptischen Funktioenn +lassen sich weitere elliptische Funktionen bilden, die unglücklicherweise +die {\em abgeleiteten elliptischen Funktionen} genannt werden. +Ähnlich wie die trigonometrischen Funktionen $\tan\alpha$, $\cot\alpha$, +$\sec\alpha$ und $\csc\alpha$ als Quotienten von $\sin\alpha$ und +$\cos\alpha$ definiert sind, sind die abgeleiteten elliptischen Funktionen +die in Tabelle~\ref{buch:elliptisch:table:abgeleitetjacobi} zusammengestellten +Quotienten der grundlegenden Jacobischen elliptischen Funktionen. +Die Bezeichnungskonvention ist, dass die Funktion $\operatorname{pq}(u,k)$ +ein Quotient ist, dessen Zähler durch den Buchstaben p bestimmt ist, +der Nenner durch den Buchstaben q. +Der Buchstabe n steht für eine $1$, die Buchstaben s, c und d stehen für +die Anfangsbuchstaben der grundlegenden Jacobischen elliptischen +Funktionen. +Meint man irgend eine der Jacobischen elliptischen Funktionen, schreibt +man manchmal auch $\operatorname{zn}(u,k)$. + +In Abbildung~\ref{buch:elliptisch:fig:jacobi12} sind die Quotienten auch +geometrisch interpretiert. +Der Wert der Funktion $\operatorname{nq}(u,k)$ ist die auf dem Strahl +mit Polarwinkel $\varphi$ abgetragene Länge bis zu den vertikalen +Geraden, die den verschiedenen möglichen Nennern entsprechen. +Entsprechend ist der Wert der Funktion $\operatorname{dq}(u,k)$ die +Länge auf dem Strahl mit Polarwinkel $\vartheta$. + +Die Relationen~\ref{buch:elliptisch:eqn:jacobi-relationen} +ermöglichen, jede Funktion $\operatorname{zn}(u,k)$ durch jede +andere auszudrücken. +Die schiere Anzahl solcher Beziehungen macht es unmöglich, sie +übersichtlich in einer Tabelle zusammenzustellen, daher soll hier +nur an einem Beispiel das Vorgehen gezeigt werden: + +\begin{beispiel} +Die Funktion $\operatorname{sc}(u,k)$ soll durch $\operatorname{cd}(u,k)$ +ausgedrückt werden. +Zunächst ist +\[ +\operatorname{sc}(u,k) += +\frac{\operatorname{sn}(u,k)}{\operatorname{cn}(u,k)} +\] +nach Definition. +Im Resultat sollen nur noch $\operatorname{cn}(u,k)$ und +$\operatorname{dn}(u,k)$ vorkommen. +Daher eliminieren wir zunächst die Funktion $\operatorname{sn}(u,k)$ +mit Hilfe von \eqref{buch:elliptisch:eqn:jacobi-relationen} und erhalten +\begin{equation} +\operatorname{sc}(u,k) += +\frac{\sqrt{1-\operatorname{cn}^2(u,k)}}{\operatorname{cn}(u,k)}. +\label{buch:elliptisch:eqn:allgausdruecken} +\end{equation} +Nun genügt es, die Funktion $\operatorname{cn}(u,k)$ durch +$\operatorname{cd}(u,k)$ auszudrücken. +Aus der Definition und der +dritten Relation in \eqref{buch:elliptisch:eqn:jacobi-relationen} +erhält man +\begin{align*} +\operatorname{cd}^2(u,k) +&= +\frac{\operatorname{cn}^2(u,k)}{\operatorname{dn}^2(u,k)} += +\frac{\operatorname{cn}^2(u,k)}{k^{\prime2}+k^2\operatorname{cn}^2(u,k)} +\\ +\Rightarrow +\qquad +k^{\prime 2} +\operatorname{cd}^2(u,k) ++ +k^2\operatorname{cd}^2(u,k)\operatorname{cn}^2(u,k) +&= +\operatorname{cn}^2(u,k) +\\ +\operatorname{cn}^2(u,k) +- +k^2\operatorname{cd}^2(u,k)\operatorname{cn}^2(u,k) +&= +k^{\prime 2} +\operatorname{cd}^2(u,k) +\\ +\operatorname{cn}^2(u,k) +&= +\frac{ +k^{\prime 2} +\operatorname{cd}^2(u,k) +}{ +1 - k^2\operatorname{cd}^2(u,k) +} +\end{align*} +Für den Zähler brauchen wir $1-\operatorname{cn}^2(u,k)$, also +\[ +1-\operatorname{cn}^2(u,k) += +\frac{ +1 +- +k^2\operatorname{cd}^2(u,k) +- +k^{\prime 2} +\operatorname{cd}^2(u,k) +}{ +1 +- +k^2\operatorname{cd}^2(u,k) +} += +\frac{1-\operatorname{cd}^2(u,k)}{1-k^2\operatorname{cd}^2(u,k)} +\] +Einsetzen in~\eqref{buch:elliptisch:eqn:allgausdruecken} gibt +\begin{align*} +\operatorname{sc}(u,k) +&= +\frac{ +\sqrt{1-\operatorname{cd}^2(u,k)} +}{\sqrt{1-k^2\operatorname{cd}^2(u,k)}} +\cdot +\frac{ +\sqrt{1 - k^2\operatorname{cd}^2(u,k)} +}{ +k' +\operatorname{cd}(u,k) +} += +\frac{ +\sqrt{1-\operatorname{cd}^2(u,k)} +}{ +k' +\operatorname{cd}(u,k) +}. +\qedhere +\end{align*} +\end{beispiel} + +\subsubsection{Ableitung der abgeleiteten elliptischen Funktionen} +Aus den Ableitungen der grundlegenden Jacobischen elliptischen Funktionen +können mit der Quotientenregel nun auch beliebige Ableitungen der +abgeleiteten Jacobischen elliptischen Funktionen gefunden werden. +Als Beispiel berechnen wir die Ableitung von $\operatorname{sc}(u,k)$. +Sie ist +\begin{align*} +\frac{d}{du} +\operatorname{sc}(u,k) +&= +\frac{d}{du} +\frac{\operatorname{sn}(u,k)}{\operatorname{cn}(u,k)} += +\frac{ +\operatorname{sn}'(u,k)\operatorname{cn}(u,k) +- +\operatorname{sn}(u,k)\operatorname{cn}'(u,k)}{ +\operatorname{cn}^2(u,k) +} +\\ +&= +\frac{ +\operatorname{cn}^2(u,k)\operatorname{dn}(u,k) ++ +\operatorname{sn}^2(u,k)\operatorname{dn}(u,k) +}{ +\operatorname{cn}^2(u,k) +} += +\frac{( +\operatorname{sn}^2(u,k) ++ +\operatorname{cn}^2(u,k) +)\operatorname{dn}(u,k)}{ +\operatorname{cn}^2(u,k) +} +\\ +&= +\frac{1}{\operatorname{cn}(u,k)} +\cdot +\frac{\operatorname{dn}(u,k)}{\operatorname{cn}(u,k)} += +\operatorname{nc}(u,k) +\operatorname{dc}(u,k). +\end{align*} +Man beachte, dass das Quadrat der Nennerfunktion im Resultat +der Quotientenregel zur Folge hat, dass die +beiden Funktionen im Resultat beide den gleichen Nenner haben wie +die Funktion, die abgeleitet wird. + +Mit etwas Fleiss kann man nach diesem Muster alle Ableitungen +\begin{equation} +%\small +\begin{aligned} +\operatorname{sn}'(u,k) +&= +\phantom{-} +\operatorname{cn}(u,k)\,\operatorname{dn}(u,k) +&&\qquad& +\operatorname{ns}'(u,k) +&= +- +\operatorname{cs}(u,k)\,\operatorname{ds}(u,k) +\\ +\operatorname{cn}'(u,k) +&= +- +\operatorname{sn}(u,k)\,\operatorname{dn}(u,k) +&&& +\operatorname{nc}'(u,k) +&= +\phantom{-} +\operatorname{sc}(u,k)\,\operatorname{dc}(u,k) +\\ +\operatorname{dn}'(u,k) +&= +-k^2 +\operatorname{sn}(u,k)\,\operatorname{cn}(u,k) +&&& +\operatorname{nd}'(u,k) +&= +\phantom{-} +k^2 +\operatorname{sd}(u,k)\,\operatorname{cd}(u,k) +\\ +\operatorname{sc}'(u,k) +&= +\phantom{-} +\operatorname{dc}(u,k)\,\operatorname{nc}(u,k) +&&& +\operatorname{cs}'(u,k) +&= +- +\operatorname{ds}(u,k)\,\operatorname{ns}(u,k) +\\ +\operatorname{cd}'(u,k) +&= +-k^{\prime2} +\operatorname{sd}(u,k)\,\operatorname{nd}(u,k) +&&& +\operatorname{dc}'(u,k) +&= +\phantom{-} +k^{\prime2} +\operatorname{dc}(u,k)\,\operatorname{nc}(u,k) +\\ +\operatorname{ds}'(d,k) +&= +- +\operatorname{cs}(u,k)\,\operatorname{ns}(u,k) +&&& +\operatorname{sd}'(d,k) +&= +\phantom{-} +\operatorname{cd}(u,k)\,\operatorname{nd}(u,k) +\end{aligned} +\label{buch:elliptisch:eqn:alleableitungen} +\end{equation} +finden. +Man beachte, dass in jeder Identität alle Funktionen den gleichen +zweiten Buchstaben haben. + +\subsubsection{TODO} +XXX algebraische Beziehungen \\ +XXX Additionstheoreme \\ +XXX Perioden +% use https://math.stackexchange.com/questions/3013692/how-to-show-that-jacobi-sine-function-is-doubly-periodic + + diff --git a/buch/chapters/110-elliptisch/mathpendel.tex b/buch/chapters/110-elliptisch/mathpendel.tex new file mode 100644 index 0000000..d61bcf6 --- /dev/null +++ b/buch/chapters/110-elliptisch/mathpendel.tex @@ -0,0 +1,250 @@ +% +% mathpendel.tex -- Das mathematische Pendel +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% + +\subsection{Das mathematische Pendel +\label{buch:elliptisch:subsection:mathpendel}} +\begin{figure} +\centering +\includegraphics{chapters/110-elliptisch/images/pendel.pdf} +\caption{Mathematisches Pendel +\label{buch:elliptisch:fig:mathpendel}} +\end{figure} +Das in Abbildung~\ref{buch:elliptisch:fig:mathpendel} dargestellte +Mathematische Pendel besteht aus einem Massepunkt der Masse $m$ +im Punkt $P$, +der über eine masselose Stange der Länge $l$ mit dem Drehpunkt $O$ +verbunden ist. +Das Pendel bewegt sich unter dem Einfluss der Schwerebeschleunigung $g$. + +Das Trägheitsmoment des Massepunktes um den Drehpunkt $O$ ist +\( +I=ml^2 +\). +Das Drehmoment der Schwerkraft ist +\(M=gl\sin\vartheta\). +Die Bewegungsgleichung wird daher +\[ +\begin{aligned} +\frac{d}{dt} I\dot{\vartheta} +&= +M += +gl\sin\vartheta +\\ +ml^2\ddot{\vartheta} +&= +gl\sin\vartheta +&&\Rightarrow& +\ddot{\vartheta} +&=\frac{g}{l}\sin\vartheta +\end{aligned} +\] +Dies ist eine nichtlineare Differentialgleichung zweiter Ordnung, die +wir nicht unmittelbar mit den Differentialgleichungen erster Ordnung +der elliptischen Funktionen vergleichen können. + +Die Differentialgleichungen erster Ordnung der elliptischen Funktionen +enthalten das Quadrat der ersten Ableitung. +In unserem Fall entspricht das einer Gleichung, die $\dot{\vartheta}^2$ +enthält. +Der Energieerhaltungssatz kann uns eine solche Gleichung geben. +Die Summe von kinetischer und potentieller Energie muss konstant sein. +Dies führt auf +\[ +E_{\text{kinetisch}} ++ +E_{\text{potentiell}} += +\frac12I\dot{\vartheta}^2 ++ +mgl(1-\cos\vartheta) += +\frac12ml^2\dot{\vartheta}^2 ++ +mgl(1-\cos\vartheta) += +E +\] +Durch Auflösen nach $\dot{\vartheta}$ kann man jetzt die +Differentialgleichung +\[ +\dot{\vartheta}^2 += +- +\frac{2g}{l}(1-\cos\vartheta) ++\frac{2E}{ml^2} +\] +finden. +In erster Näherung, d.h. wenn man die rechte Seite bis zu vierten +Potenzen in eine Taylor-Reihe in $\vartheta$ entwickelt, ist dies +tatsächlich eine Differentialgleichung der Art, wie wir sie für +elliptische Funktionen gefunden haben, wir möchten aber eine exakte +Lösung konstruieren. + +Die maximale Energie für eine Bewegung, bei der sich das Pendel gerade +über den höchsten Punkt hinweg zu bewegen vermag, ist +$E=2lmg$. +Falls $E<2mgl$ ist, erwarten wir Schwingungslösungen, bei denen +der Winkel $\vartheta$ immer im offenen Interval $(-\pi,\pi)$ +bleibt. +Für $E>2mgl$ wird sich das Pendel im Kreis bewegen, für sehr grosse +Energie ist die kinetische Energie dominant, die Verlangsamung im +höchsten Punkt wird immer weniger ausgeprägt sein. + +% +% Koordinatentransformation auf elliptische Funktionen +% +\subsubsection{Koordinatentransformation auf elliptische Funktionen} +Wir verwenden als neue Variable +\[ +y = \sin\frac{\vartheta}2 +\] +mit der Ableitung +\[ +\dot{y}=\frac12\cos\frac{\vartheta}{2}\cdot \dot{\vartheta}. +\] +Man beachte, dass $y$ nicht eine Koordinate in +Abbildung~\ref{buch:elliptisch:fig:mathpendel} ist. + +Aus den Halbwinkelformeln finden wir +\[ +\cos\vartheta += +1-2\sin^2 \frac{\vartheta}2 += +1-2y^2. +\] +Dies können wir zusammen mit der +Identität $\cos^2\vartheta/2 = 1-\sin^2\vartheta/2 = 1-y^2$ +in die Energiegleichung einsetzen und erhalten +\[ +\frac12ml^2\dot{\vartheta}^2 + mgly^2 = E +\qquad\Rightarrow\qquad +\frac14 \dot{\vartheta}^2 = \frac{E}{2ml^2} - \frac{g}{2l}y^2. +\] +Der konstante Term auf der rechten Seite ist grösser oder kleiner als +$1$ je nachdem, ob das Pendel sich im Kreis bewegt oder nicht. + +Durch Multiplizieren mit $\cos^2\frac{\vartheta}{2}=1-y^2$ +erhalten wir auf der linken Seite einen Ausdruck, den wir +als Funktion von $\dot{y}$ ausdrücken können. +Wir erhalten +\begin{align*} +\frac14 +\cos^2\frac{\vartheta}2 +\cdot +\dot{\vartheta}^2 +&= +\frac14 +(1-y^2) +\biggl(\frac{E}{2ml^2} -\frac{g}{2l}y^2\biggr) +\\ +\dot{y}^2 +&= +\frac{1}{4} +(1-y^2) +\biggl(\frac{E}{2ml^2} -\frac{g}{2l}y^2\biggr) +\end{align*} +Die letzte Gleichung hat die Form einer Differentialgleichung +für elliptische Funktionen. +Welche Funktion verwendet werden muss, hängt von der Grösse der +Koeffizienten in der zweiten Klammer ab. +Die Tabelle~\ref{buch:elliptisch:tabelle:loesungsfunktionen} +zeigt, dass in der zweiten Klammer jeweils einer der Terme +$1$ sein muss. + +% +% Der Fall E < 2mgl +% +\subsubsection{Der Fall $E<2mgl$} +\begin{figure} +\centering +\includegraphics[width=\textwidth]{chapters/110-elliptisch/images/jacobiplots.pdf} +\caption{% +Abhängigkeit der elliptischen Funktionen von $u$ für +verschiedene Werte von $k^2=m$. +Für $m=0$ ist $\operatorname{sn}(u,0)=\sin u$, +$\operatorname{cn}(u,0)=\cos u$ und $\operatorname{dn}(u,0)=1$, diese +sind in allen Plots in einer helleren Farbe eingezeichnet. +Für kleine Werte von $m$ weichen die elliptischen Funktionen nur wenig +von den trigonometrischen Funktionen ab, +es ist aber klar erkennbar, dass die anharmonischen Terme in der +Differentialgleichung die Periode mit steigender Amplitude verlängern. +Sehr grosse Werte von $m$ nahe bei $1$ entsprechen der Situation, dass +die Energie des Pendels fast ausreicht, dass es den höchsten Punkt +erreichen kann, was es für $m$ macht. +\label{buch:elliptisch:fig:jacobiplots}} +\end{figure} + + +Wir verwenden als neue Variable +\[ +y = \sin\frac{\vartheta}2 +\] +mit der Ableitung +\[ +\dot{y}=\frac12\cos\frac{\vartheta}{2}\cdot \dot{\vartheta}. +\] +Man beachte, dass $y$ nicht eine Koordinate in +Abbildung~\ref{buch:elliptisch:fig:mathpendel} ist. + +Aus den Halbwinkelformeln finden wir +\[ +\cos\vartheta += +1-2\sin^2 \frac{\vartheta}2 += +1-2y^2. +\] +Dies können wir zusammen mit der +Identität $\cos^2\vartheta/2 = 1-\sin^2\vartheta/2 = 1-y^2$ +in die Energiegleichung einsetzen und erhalten +\[ +\frac12ml^2\dot{\vartheta}^2 + mgly^2 = E. +\] +Durch Multiplizieren mit $\cos^2\frac{\vartheta}{2}=1-y^2$ +erhalten wir auf der linken Seite einen Ausdruck, den wir +als Funktion von $\dot{y}$ ausdrücken können. +Wir erhalten +\begin{align*} +\frac12ml^2 +\cos^2\frac{\vartheta}2 +\dot{\vartheta}^2 +&= +(1-y^2) +(E -mgly^2) +\\ +\frac{1}{4}\cos^2\frac{\vartheta}{2}\dot{\vartheta}^2 +&= +\frac{1}{2} +(1-y^2) +\biggl(\frac{E}{ml^2} -\frac{g}{l}y^2\biggr) +\\ +\dot{y}^2 +&= +\frac{E}{2ml^2} +(1-y^2)\biggl( +1-\frac{2gml}{E}y^2 +\biggr). +\end{align*} +Dies ist genau die Form der Differentialgleichung für die elliptische +Funktion $\operatorname{sn}(u,k)$ +mit $k^2 = 2gml/E< 1$. + +%% +%% Der Fall E > 2mgl +%% +%\subsection{Der Fall $E > 2mgl$} +%In diesem Fall hat das Pendel im höchsten Punkte immer noch genügend +%kinetische Energie, so dass es sich im Kreise dreht. +%Indem wir die Gleichung + + +%\subsection{Soliton-Lösungen der Sinus-Gordon-Gleichung} + +%\subsection{Nichtlineare Differentialgleichung vierter Ordnung} +%XXX Möbius-Transformation \\ +%XXX Reduktion auf die Differentialgleichung elliptischer Funktionen -- cgit v1.2.1 From e1b65ea3e46bf60fec0d6503b701a84f68138a24 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 21 Apr 2022 22:36:54 +0200 Subject: add lecture notes for session 5 --- buch/chapters/110-elliptisch/images/slcl.pdf | Bin 28269 -> 28233 bytes buch/chapters/110-elliptisch/images/slcl.tex | 20 +- buch/chapters/110-elliptisch/jacobi.tex | 1738 -------------------------- buch/chapters/110-elliptisch/lemniskate.tex | 2 +- 4 files changed, 11 insertions(+), 1749 deletions(-) (limited to 'buch') diff --git a/buch/chapters/110-elliptisch/images/slcl.pdf b/buch/chapters/110-elliptisch/images/slcl.pdf index 493b5fa..c15051b 100644 Binary files a/buch/chapters/110-elliptisch/images/slcl.pdf and b/buch/chapters/110-elliptisch/images/slcl.pdf differ diff --git a/buch/chapters/110-elliptisch/images/slcl.tex b/buch/chapters/110-elliptisch/images/slcl.tex index 08241ac..0af1027 100644 --- a/buch/chapters/110-elliptisch/images/slcl.tex +++ b/buch/chapters/110-elliptisch/images/slcl.tex @@ -47,35 +47,35 @@ -- ({5*\lemniscateconstant*\dx},{1*\dy}); -\draw[color=red!20,line width=1.4pt] +\draw[color=red!40,line width=1.4pt] plot[domain=0:13,samples=200] ({\x},{\dy*sin(\ts*\x)}); -\draw[color=blue!20,line width=1.4pt] +\draw[color=blue!40,line width=1.4pt] plot[domain=0:13,samples=200] ({\x},{\dy*cos(\ts*\x)}); \draw[color=red,line width=1.4pt] \slpath; \draw[color=blue,line width=1.4pt] \clpath; \draw[->] (0,{-1*\dy-0.1}) -- (0,{1*\dy+0.4}) coordinate[label={right:$r$}]; -\draw[->] (-0.1,0) -- (13.7,0) coordinate[label={$s$}]; +\draw[->] (-0.1,0) -- (13.6,0) coordinate[label={$s$}]; \foreach \i in {1,2,3,4,5}{ \draw ({\lemniscateconstant*\i},-0.1) -- ({\lemniscateconstant*\i},0.1); } -\node at ({\lemniscateconstant*\dx},0) [below left] {$ \varpi\mathstrut$}; -\node at ({2*\lemniscateconstant*\dx},0) [below left] {$2\varpi\mathstrut$}; -\node at ({3*\lemniscateconstant*\dx},0) [below right] {$3\varpi\mathstrut$}; -\node at ({4*\lemniscateconstant*\dx},0) [below right] {$4\varpi\mathstrut$}; -\node at ({5*\lemniscateconstant*\dx},0) [below left] {$5\varpi\mathstrut$}; +\node at ({\lemniscateconstant*\dx},0) [below left] {$\frac{\varpi}2\mathstrut$}; +\node at ({2*\lemniscateconstant*\dx},0) [below left] {$\varpi\mathstrut$}; +\node at ({3*\lemniscateconstant*\dx},0) [below right] {$\frac{3\varpi}2\mathstrut$}; +\node at ({4*\lemniscateconstant*\dx},0) [below right] {$2\varpi\mathstrut$}; +\node at ({5*\lemniscateconstant*\dx},0) [below left] {$\frac{5\varpi}2\mathstrut$}; \node[color=red] at ({1.6*\lemniscateconstant*\dx},{0.6*\dy}) [below left] {$\operatorname{sl}(s)$}; \node[color=red!50] at ({1.5*\lemniscateconstant*\dx},{sin(1.5*90)*\dy*0.90}) - [above right] {$\sin \bigl(\frac{\pi}{2\varpi}s\bigr)$}; + [above right] {$\sin \bigl(\frac{\pi}{\varpi}s\bigr)$}; \node[color=blue] at ({1.4*\lemniscateconstant*\dx},{-0.6*\dy}) [above right] {$\operatorname{cl}(s)$}; \node[color=blue!50] at ({1.5*\lemniscateconstant*\dx},{cos(1.5*90)*\dy*0.90}) - [below left] {$\cos\bigl(\frac{\pi}{2\varpi}s\bigr)$}; + [below left] {$\cos\bigl(\frac{\pi}{\varpi}s\bigr)$}; \draw (-0.1,{1*\dy}) -- (0.1,{1*\dy}); \draw (-0.1,{-1*\dy}) -- (0.1,{-1*\dy}); diff --git a/buch/chapters/110-elliptisch/jacobi.tex b/buch/chapters/110-elliptisch/jacobi.tex index e1fbc00..166ea41 100644 --- a/buch/chapters/110-elliptisch/jacobi.tex +++ b/buch/chapters/110-elliptisch/jacobi.tex @@ -22,1743 +22,5 @@ dann muss man die Umkehrfunktionen der elliptischen Integrale dafür ins Auge fassen. -%% -%% elliptische Funktionen als Trigonometrie -%% -%\subsection{Elliptische Funktionen als Trigonometrie} -%\begin{figure} -%\centering -%\includegraphics{chapters/110-elliptisch/images/ellipse.pdf} -%\caption{Kreis und Ellipse zum Vergleich und zur Herleitung der -%elliptischen Funktionen von Jacobi als ``trigonometrische'' Funktionen -%auf einer Ellipse. -%\label{buch:elliptisch:fig:ellipse}} -%\end{figure} -%% based on Willliam Schwalm, Elliptic functions and elliptic integrals -%% https://youtu.be/DCXItCajCyo -% -%% -%% Geometrie einer Ellipse -%% -%\subsubsection{Geometrie einer Ellipse} -%Eine {\em Ellipse} ist die Menge der Punkte der Ebene, für die die Summe -%\index{Ellipse}% -%der Entfernungen von zwei festen Punkten $F_1$ und $F_2$, -%den {\em Brennpunkten}, konstant ist. -%\index{Brennpunkt}% -%In Abbildung~\ref{buch:elliptisch:fig:ellipse} eine Ellipse -%mit Brennpunkten in $F_1=(-e,0)$ und $F_2=(e,0)$ dargestellt, -%die durch die Punkte $(\pm a,0)$ und $(0,\pm b)$ auf den Achsen geht. -%Der Punkt $(a,0)$ hat die Entfernungen $a+e$ und $a-e$ von den beiden -%Brennpunkten, also die Entfernungssumme $a+e+a-e=2a$. -%Jeder andere Punkt auf der Ellipse muss ebenfalls diese Entfernungssumme -%haben, insbesondere auch der Punkt $(0,b)$. -%Seine Entfernung zu jedem Brennpunkt muss aus Symmetriegründen gleich gross, -%also $a$ sein. -%Aus dem Satz von Pythagoras liest man daher ab, dass -%\[ -%b^2+e^2=a^2 -%\qquad\Rightarrow\qquad -%e^2 = a^2-b^2 -%\] -%sein muss. -%Die Strecke $e$ heisst auch {\em (lineare) Exzentrizität} der Ellipse. -%Das Verhältnis $\varepsilon= e/a$ heisst die {\em numerische Exzentrizität} -%der Ellipse. -% -%% -%% Die Ellipsengleichung -%% -%\subsubsection{Ellipsengleichung} -%Der Punkt $P=(x,y)$ auf der Ellipse hat die Entfernungen -%\begin{equation} -%\begin{aligned} -%\overline{PF_1}^2 -%&= -%y^2 + (x+e)^2 -%\\ -%\overline{PF_2}^2 -%&= -%y^2 + (x-e)^2 -%\end{aligned} -%\label{buch:elliptisch:eqn:wurzelausdruecke} -%\end{equation} -%von den Brennpunkten, für die -%\begin{equation} -%\overline{PF_1}+\overline{PF_2} -%= -%2a -%\label{buch:elliptisch:eqn:pf1pf2a} -%\end{equation} -%gelten muss. -%Man kann nachrechnen, dass ein Punkt $P$, der die Gleichung -%\[ -%\frac{x^2}{a^2} + \frac{y^2}{b^2}=1 -%\] -%erfüllt, auch die Eigenschaft~\eqref{buch:elliptisch:eqn:pf1pf2a} -%erfüllt. -%Zur Vereinfachung setzen wir $l_1=\overline{PF_1}$ und $l_2=\overline{PF_2}$. -%$l_1$ und $l_2$ sind Wurzeln aus der rechten Seite von -%\eqref{buch:elliptisch:eqn:wurzelausdruecke}. -%Das Quadrat von $l_1+l_2$ ist -%\[ -%l_1^2 + 2l_1l_2 + l_2^2 = 4a^2. -%\] -%Um die Wurzeln ganz zu eliminieren, bringt man das Produkt $l_1l_2$ alleine -%auf die rechte Seite und quadriert. -%Man muss also verifizieren, dass -%\[ -%(l_1^2 + l_2^2 -4a^2)^2 = 4l_1^2l_2^2. -%\] -%In den entstehenden Ausdrücken muss man ausserdem $e=\sqrt{a^2-b^2}$ und -%\[ -%y=b\sqrt{1-\frac{x^2}{a^2}} -%\] -%substituieren. -%Diese Rechnung führt man am einfachsten mit Hilfe eines -%Computeralgebraprogramms durch, welches obige Behauptung bestätigt. -% -%% -%% Normierung -%% -%\subsubsection{Normierung} -%Die trigonometrischen Funktionen sind definiert als Verhältnisse -%von Seiten rechtwinkliger Dreiecke. -%Dadurch, dass man den die Hypothenuse auf Länge $1$ normiert, -%kann man die Sinus- und Kosinus-Funktion als Koordinaten eines -%Punktes auf dem Einheitskreis interpretieren. -% -%Für die Koordinaten eines Punktes auf der Ellipse ist dies nicht so einfach, -%weil es nicht nur eine Ellipse gibt, sondern für jede numerische Exzentrizität -%mindestens eine mit Halbeachse $1$. -%Wir wählen die Ellipsen so, dass $a$ die grosse Halbachse ist, also $a>b$. -%Als Normierungsbedingung verwenden wir, dass $b=1$ sein soll, wie in -%Abbildung~\ref{buch:elliptisch:fig:jacobidef}. -%Dann ist $a=1/\varepsilon>1$. -%In dieser Normierung haben Punkte $(x,y)$ auf der Ellipse $y$-Koordinaten -%zwischen $-1$ und $1$ und $x$-Koordinaten zwischen $-a$ und $a$. -% -%Im Zusammenhang mit elliptischen Funktionen wird die numerische Exzentrizität -%$\varepsilon$ auch mit -%\[ -%k -%= -%\varepsilon -%= -%\frac{e}{a} -%= -%\frac{\sqrt{a^2-b^2}}{a} -%= -%\frac{\sqrt{a^2-1}}{a}, -%\] -%die Zahl $k$ heisst auch der {\em Modulus}. -%Man kann $a$ auch durch $k$ ausdrücken, durch Quadrieren und Umstellen -%findet man -%\[ -%k^2a^2 = a^2-1 -%\quad\Rightarrow\quad -%1=a^2(k^2-1) -%\quad\Rightarrow\quad -%a=\frac{1}{\sqrt{k^2-1}}. -%\] -% -%Die Gleichung der ``Einheitsellipse'' zu diesem Modulus ist -%\[ -%\frac{x^2}{a^2}+y^2=1 -%\qquad\text{oder}\qquad -%x^2(k^2-1) + y^2 = 1. -%\] -% -%% -%% Definition der elliptischen Funktionen -%% -%\begin{figure} -%\centering -%\includegraphics{chapters/110-elliptisch/images/jacobidef.pdf} -%\caption{Definition der elliptischen Funktionen als Trigonometrie -%an einer Ellipse mit Halbachsen $a$ und $1$. -%\label{buch:elliptisch:fig:jacobidef}} -%\end{figure} -%\subsubsection{Definition der elliptischen Funktionen} -%Die elliptischen Funktionen für einen Punkt $P$ auf der Ellipse mit Modulus $k$ -%können jetzt als Verhältnisse der Koordinaten des Punktes definieren. -%Es stellt sich aber die Frage, was man als Argument verwenden soll. -%Es soll so etwas wie den Winkel $\varphi$ zwischen der $x$-Achse und dem -%Radiusvektor zum Punkt $P$ -%darstellen, aber wir haben hier noch eine Wahlfreiheit, die wir später -%ausnützen möchten. -%Im Moment müssen wir die Frage noch nicht beantworten und nennen das -%noch unbestimmte Argument $u$. -%Wir kümmern uns später um die Frage, wie $u$ von $\varphi$ abhängt. -% -%Die Funktionen, die wir definieren wollen, hängen ausserdem auch -%vom Modulus ab. -%Falls der verwendete Modulus aus dem Zusammenhang klar ist, lassen -%wir das $k$-Argument weg. -% -%Die Punkte auf dem Einheitskreis haben alle den gleichen Abstand vom -%Nullpunkt, dies ist gleichzeitig die definierende Gleichung $r^2=x^2+y^2=1$ -%des Kreises. -%Die Punkte auf der Ellipse erfüllen die Gleichung $x^2/a^2+y^2=1$, -%die Entfernung der Punkte $r=\sqrt{x^2+y^2}$ vom Nullpunkt variert aber. -% -%In Analogie zu den trigonometrischen Funktionen setzen wir jetzt für -%die Funktionen -%\[ -%\begin{aligned} -%&\text{sinus amplitudinis:}& -%{\color{red}\operatorname{sn}(u,k)}&= y \\ -%&\text{cosinus amplitudinis:}& -%{\color{blue}\operatorname{cn}(u,k)}&= \frac{x}{a} \\ -%&\text{delta amplitudinis:}& -%{\color{darkgreen}\operatorname{dn}(u,k)}&=\frac{r}{a}, -%\end{aligned} -%\] -%die auch in Abbildung~\ref{buch:elliptisch:fig:jacobidef} -%dargestellt sind. -%Aus der Gleichung der Ellipse folgt sofort, dass -%\[ -%\operatorname{sn}(u,k)^2 + \operatorname{cn}(u,k)^2 = 1 -%\] -%ist. -%Der Satz von Pythagoras kann verwendet werden, um die Entfernung zu -%berechnen, also gilt -%\begin{equation} -%r^2 -%= -%a^2 \operatorname{dn}(u,k)^2 -%= -%x^2 + y^2 -%= -%a^2\operatorname{cn}(u,k)^2 + \operatorname{sn}(u,k)^2 -%\quad -%\Rightarrow -%\quad -%a^2 \operatorname{dn}(u,k)^2 -%= -%a^2\operatorname{cn}(u,k)^2 + \operatorname{sn}(u,k)^2. -%\label{buch:elliptisch:eqn:sncndnrelation} -%\end{equation} -%Ersetzt man -%$ -%a^2\operatorname{cn}(u,k)^2 -%= -%a^2-a^2\operatorname{sn}(u,k)^2 -%$, ergibt sich -%\[ -%a^2 \operatorname{dn}(u,k)^2 -%= -%a^2-a^2\operatorname{sn}(u,k)^2 -%+ -%\operatorname{sn}(u,k)^2 -%\quad -%\Rightarrow -%\quad -%\operatorname{dn}(u,k)^2 -%+ -%\frac{a^2-1}{a^2}\operatorname{sn}(u,k)^2 -%= -%1, -%\] -%woraus sich die Identität -%\[ -%\operatorname{dn}(u,k)^2 + k^2 \operatorname{sn}(u,k)^2 = 1 -%\] -%ergibt. -%Ebenso kann man aus~\eqref{buch:elliptisch:eqn:sncndnrelation} -%die Funktion $\operatorname{cn}(u,k)$ eliminieren, was auf -%\[ -%a^2\operatorname{dn}(u,k)^2 -%= -%a^2\operatorname{cn}(u,k)^2 -%+1-\operatorname{cn}(u,k)^2 -%= -%(a^2-1)\operatorname{cn}(u,k)^2 -%+1. -%\] -%Nach Division durch $a^2$ ergibt sich -%\begin{align*} -%\operatorname{dn}(u,k)^2 -%- -%k^2\operatorname{cn}(u,k)^2 -%&= -%\frac{1}{a^2} -%= -%\frac{a^2-a^2+1}{a^2} -%= -%1-k^2 =: k^{\prime 2}. -%\end{align*} -%Wir stellen die hiermit gefundenen Relationen zwischen den grundlegenden -%Jacobischen elliptischen Funktionen für später zusammen in den Formeln -%\begin{equation} -%\begin{aligned} -%\operatorname{sn}^2(u,k) -%+ -%\operatorname{cn}^2(u,k) -%&= -%1 -%\\ -%\operatorname{dn}^2(u,k) + k^2\operatorname{sn}^2(u,k) -%&= -%1 -%\\ -%\operatorname{dn}^2(u,k) -k^2\operatorname{cn}^2(u,k) -%&= -%k^{\prime 2}. -%\end{aligned} -%\label{buch:elliptisch:eqn:jacobi-relationen} -%\end{equation} -%zusammen. -%So wie es möglich ist, $\sin\alpha$ durch $\cos\alpha$ auszudrücken, -%ist es mit -%\eqref{buch:elliptisch:eqn:jacobi-relationen} -%jetzt auch möglich jede grundlegende elliptische Funktion durch -%jede anderen auszudrücken. -%Die Resultate sind in der Tabelle~\ref{buch:elliptisch:fig:jacobi-relationen} -%zusammengestellt. -% -%\begin{table} -%\centering -%\renewcommand{\arraystretch}{2.1} -%\begin{tabular}{|>{$\displaystyle}c<{$}|>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}|} -%\hline -%&\operatorname{sn}(u,k) -%&\operatorname{cn}(u,k) -%&\operatorname{dn}(u,k)\\ -%\hline -%\operatorname{sn}(u,k) -%&\operatorname{sn}(u,k) -%&\sqrt{1-\operatorname{cn}^2(u,k)} -%&\frac1k\sqrt{1-\operatorname{dn}^2(u,k)} -%\\ -%\operatorname{cn}(u,k) -%&\sqrt{1-\operatorname{sn}^2(u,k)} -%&\operatorname{cn}(u,k) -%&\frac{1}{k}\sqrt{\operatorname{dn}^2(u,k)-k^{\prime2}} -%\\ -%\operatorname{dn}(u,k) -%&\sqrt{1-k^2\operatorname{sn}^2(u,k)} -%&\sqrt{k^{\prime2}+k^2\operatorname{cn}^2(u,k)} -%&\operatorname{dn}(u,k) -%\\ -%\hline -%\end{tabular} -%\caption{Jede der Jacobischen elliptischen Funktionen lässt sich -%unter Verwendung der Relationen~\eqref{buch:elliptisch:eqn:jacobi-relationen} -%durch jede andere ausdrücken. -%\label{buch:elliptisch:fig:jacobi-relationen}} -%\end{table} -% -%% -%% Ableitungen der Jacobi-ellpitischen Funktionen -%% -%\subsubsection{Ableitung} -%Die trigonometrischen Funktionen sind deshalb so besonders nützlich -%für die Lösung von Schwingungsdifferentialgleichungen, weil sie die -%Beziehungen -%\[ -%\frac{d}{d\varphi} \cos\varphi = -\sin\varphi -%\qquad\text{und}\qquad -%\frac{d}{d\varphi} \sin\varphi = \cos\varphi -%\] -%erfüllen. -%So einfach können die Beziehungen natürlich nicht sein, sonst würde sich -%durch Integration ja wieder nur die trigonometrischen Funktionen ergeben. -%Durch geschickte Wahl des Arguments $u$ kann man aber erreichen, dass -%sie ähnlich nützliche Beziehungen zwischen den Ableitungen ergeben. -% -%Gesucht ist jetzt also eine Wahl für das Argument $u$ zum Beispiel in -%Abhängigkeit von $\varphi$, dass sich einfache und nützliche -%Ableitungsformeln ergeben. -%Wir setzen daher $u(\varphi)$ voraus und beachten, dass $x$ und $y$ -%ebenfalls von $\varphi$ abhängen, es ist -%$y=\sin\varphi$ und $x=a\cos\varphi$. -%Die Ableitungen von $x$ und $y$ nach $\varphi$ sind -%\begin{align*} -%\frac{dy}{d\varphi} -%&= -%\cos\varphi -%= -%\frac{1}{a} x -%= -%\operatorname{cn}(u,k) -%\\ -%\frac{dx}{d\varphi} -%&= -%-a\sin\varphi -%= -%-a y -%= -%-a\operatorname{sn}(u,k). -%\end{align*} -%Daraus kann man jetzt die folgenden Ausdrücke für die Ableitungen der -%elliptischen Funktionen nach $\varphi$ ableiten: -%\begin{align*} -%\frac{d}{d\varphi} \operatorname{sn}(u,z) -%&= -%\frac{d}{d\varphi} y(\varphi) -%= -%\cos\varphi -%= -%\frac{x}{a} -%= -%\operatorname{cn}(u,k) -%&&\Rightarrow& -%\frac{d}{du} -%\operatorname{sn}(u,k) -%&= -%\operatorname{cn}(u,k) \frac{d\varphi}{du} -%\\ -%\frac{d}{d\varphi} \operatorname{cn}(u,z) -%&= -%\frac{d}{d\varphi} \frac{x(\varphi)}{a} -%= -%-\sin\varphi -%= -%-\operatorname{sn}(u,k) -%&&\Rightarrow& -%\frac{d}{du}\operatorname{cn}(u,k) -%&= -%-\operatorname{sn}(u,k) \frac{d\varphi}{du} -%\\ -%\frac{d}{d\varphi} \operatorname{dn}(u,z) -%&= -%\frac{1}{a}\frac{dr}{d\varphi} -%= -%\frac{1}{a}\frac{d\sqrt{x^2+y^2}}{d\varphi} -%%\\ -%%& -%\rlap{$\displaystyle\mathstrut -%= -%\frac{x}{ar} \frac{dx}{d\varphi} -%+ -%\frac{y}{ar} \frac{dy}{d\varphi} -%%\\ -%%& -%= -%\frac{x}{ar} (-a\operatorname{sn}(u,k)) -%+ -%\frac{y}{ar} \operatorname{cn}(u,k) -%$} -%\\ -%& -%\rlap{$\displaystyle\mathstrut -%= -%\frac{x}{ar}(-ay) -%+ -%\frac{y}{ar} \frac{x}{a} -%%\rlap{$\displaystyle -%= -%\frac{xy(-1+\frac{1}{a^2})}{r} -%%$} -%%\\ -%%& -%= -%-\frac{xy(a^2-1)}{a^2r} -%$} -%\\ -%&= -%-\frac{a^2-1}{ar} -%\operatorname{cn}(u,k) \operatorname{sn}(u,k) -%%\\ -%%& -%\rlap{$\displaystyle\mathstrut -%= -%-k^2 -%\frac{a}{r} -%\operatorname{cn}(u,k) \operatorname{sn}(u,k) -%$} -%\\ -%&= -%-k^2\frac{\operatorname{cn}(u,k)\operatorname{sn}(u,k)}{\operatorname{dn}(u,k)} -%&&\Rightarrow& -%\frac{d}{du} \operatorname{dn}(u,k) -%&= -%-k^2\frac{\operatorname{cn}(u,k) -%\operatorname{sn}(u,k)}{\operatorname{dn}(u,k)} -%\frac{d\varphi}{du}. -%\end{align*} -%Die einfachsten Beziehungen ergeben sich offenbar, wenn man $u$ so -%wählt, dass -%\[ -%\frac{d\varphi}{du} -%= -%\operatorname{dn}(u,k) -%= -%\frac{r}{a}. -%\] -%Damit haben wir die grundlegenden Ableitungsregeln -% -%\begin{satz} -%\label{buch:elliptisch:satz:ableitungen} -%Die Jacobischen elliptischen Funktionen haben die Ableitungen -%\begin{equation} -%\begin{aligned} -%\frac{d}{du}\operatorname{sn}(u,k) -%&= -%\phantom{-}\operatorname{cn}(u,k)\operatorname{dn}(u,k) -%\\ -%\frac{d}{du}\operatorname{cn}(u,k) -%&= -%-\operatorname{sn}(u,k)\operatorname{dn}(u,k) -%\\ -%\frac{d}{du}\operatorname{dn}(u,k) -%&= -%-k^2\operatorname{sn}(u,k)\operatorname{cn}(u,k). -%\end{aligned} -%\label{buch:elliptisch:eqn:ableitungsregeln} -%\end{equation} -%\end{satz} -% -%% -%% Der Grenzfall $k=1$ -%% -%\subsubsection{Der Grenzwert $k\to1$} -%\begin{figure} -%\centering -%\includegraphics{chapters/110-elliptisch/images/sncnlimit.pdf} -%\caption{Grenzfälle der Jacobischen elliptischen Funktionen -%für die Werte $0$ und $1$ des Parameters $k$. -%\label{buch:elliptisch:fig:sncnlimit}} -%\end{figure} -%Für $k=1$ ist $k^{\prime2}=1-k^2=$ und es folgt aus den -%Relationen~\eqref{buch:elliptisch:eqn:jacobi-relationen} -%\[ -%\operatorname{cn}^2(u,k) -%- -%k^2 -%\operatorname{dn}^2(u,k) -%= -%k^{\prime2} -%= -%0 -%\qquad\Rightarrow\qquad -%\operatorname{cn}^2(u,1) -%= -%\operatorname{dn}^2(u,1), -%\] -%die beiden Funktionen -%$\operatorname{cn}(u,k)$ -%und -%$\operatorname{dn}(u,k)$ -%fallen also zusammen. -%Die Ableitungsregeln werden dadurch vereinfacht: -%\begin{align*} -%\operatorname{sn}'(u,1) -%&= -%\operatorname{cn}(u,1) -%\operatorname{dn}(u,1) -%= -%\operatorname{cn}^2(u,1) -%= -%1-\operatorname{sn}^2(u,1) -%&&\Rightarrow& y'&=1-y^2 -%\\ -%\operatorname{cn}'(u,1) -%&= -%- -%\operatorname{sn}(u,1) -%\operatorname{dn}(u,1) -%= -%- -%\operatorname{sn}(u,1)\operatorname{cn}(u,1) -%&&\Rightarrow& -%\frac{z'}{z}&=(\log z)' = -y -%\end{align*} -%Die erste Differentialgleichung für $y$ lässt sich separieren, man findet -%die Lösung -%\[ -%\frac{y'}{1-y^2} -%= -%1 -%\quad\Rightarrow\quad -%\int \frac{dy}{1-y^2} = \int \,du -%\quad\Rightarrow\quad -%\operatorname{artanh}(y) = u -%\quad\Rightarrow\quad -%\operatorname{sn}(u,1)=\tanh u. -%\] -%Damit kann man jetzt auch $z$ berechnen: -%\begin{align*} -%(\log \operatorname{cn}(u,1))' -%&= -%\tanh u -%&&\Rightarrow& -%\log\operatorname{cn}(u,1) -%&= -%-\int\tanh u\,du -%= -%-\log\cosh u -%\\ -%& -%&&\Rightarrow& -%\operatorname{cn}(u,1) -%&= -%\frac{1}{\cosh u} -%= -%\operatorname{sech}u. -%\end{align*} -%Die Grenzfunktionen sind in Abbildung~\ref{buch:elliptisch:fig:sncnlimit} -%dargestellt. -% -%% -%% Das Argument u -%% -%\subsubsection{Das Argument $u$} -%Die Gleichung -%\begin{equation} -%\frac{d\varphi}{du} -%= -%\operatorname{dn}(u,k) -%\label{buch:elliptisch:eqn:uableitung} -%\end{equation} -%ermöglicht, $\varphi$ in Abhängigkeit von $u$ zu berechnen, ohne jedoch -%die geometrische Bedeutung zu klären. -%Das beginnt bereits damit, dass der Winkel $\varphi$ nicht nicht der -%Polarwinkel des Punktes $P$ in Abbildung~\ref{buch:elliptisch:fig:jacobidef} -%ist, diesen nennen wir $\vartheta$. -%Der Zusammenhang zwischen $\varphi$ und $\vartheta$ ist -%\begin{equation} -%\frac1{a}\tan\varphi = \tan\vartheta -%\label{buch:elliptisch:eqn:phitheta} -%\end{equation} -% -%Um die geometrische Bedeutung besser zu verstehen, nehmen wir jetzt an, -%dass die Ellipse mit einem Parameter $t$ parametrisiert ist, dass also -%$\varphi(t)$, $\vartheta(t)$ und $u(t)$ Funktionen von $t$ sind. -%Die Ableitung von~\eqref{buch:elliptisch:eqn:phitheta} ist -%\[ -%\frac1{a}\cdot \frac{1}{\cos^2\varphi}\cdot \dot{\varphi} -%= -%\frac{1}{\cos^2\vartheta}\cdot \dot{\vartheta}. -%\] -%Daraus kann die Ableitung von $\vartheta$ nach $\varphi$ bestimmt -%werden, sie ist -%\[ -%\frac{d\vartheta}{d\varphi} -%= -%\frac{\dot{\vartheta}}{\dot{\varphi}} -%= -%\frac{1}{a} -%\cdot -%\frac{\cos^2\vartheta}{\cos^2\varphi} -%= -%\frac{1}{a} -%\cdot -%\frac{(x/r)^2}{(x/a)^2} -%= -%\frac{1}{a}\cdot -%\frac{a^2}{r^2} -%= -%\frac{1}{a}\cdot\frac{1}{\operatorname{dn}^2(u,k)}. -%\] -%Damit kann man jetzt mit Hilfe von~\eqref{buch:elliptisch:eqn:uableitung} -%Die Ableitung von $\vartheta$ nach $u$ ermitteln, sie ist -%\[ -%\frac{d\vartheta}{du} -%= -%\frac{d\vartheta}{d\varphi} -%\cdot -%\frac{d\varphi}{du} -%= -%\frac{1}{a}\cdot\frac{1}{\operatorname{dn}^2(u,k)} -%\cdot -%\operatorname{dn}(u,k) -%= -%\frac{1}{a} -%\cdot -%\frac{1}{\operatorname{dn}(u,k)} -%= -%\frac{1}{a} -%\cdot\frac{a}{r} -%= -%\frac{1}{r}, -%\] -%wobei wir auch die Definition der Funktion $\operatorname{dn}(u,k)$ -%verwendet haben. -% -%In der Parametrisierung mit dem Parameter $t$ kann man jetzt die Ableitung -%von $u$ nach $t$ berechnen als -%\[ -%\frac{du}{dt} -%= -%\frac{du}{d\vartheta} -%\frac{d\vartheta}{dt} -%= -%r -%\dot{\vartheta}. -%\] -%Darin ist $\dot{\vartheta}$ die Winkelgeschwindigkeit des Punktes um -%das Zentrum $O$ und $r$ ist die aktuelle Entfernung des Punktes $P$ -%von $O$. -%$r\dot{\vartheta}$ ist also die Geschwindigkeitskomponenten des Punktes -%$P$ senkrecht auf den aktuellen Radiusvektor. -%Der Parameter $u$, der zum Punkt $P$ gehört, ist also das Integral -%\[ -%u(P) = \int_0^P r\,d\vartheta. -%\] -%Für einen Kreis ist die Geschwindigkeit von $P$ immer senkrecht -%auf dem Radiusvektor und der Radius ist konstant, so dass -%$u(P)=\vartheta(P)$ ist. -% -%% -%% Die abgeleiteten elliptischen Funktionen -%% -%\begin{figure} -%\centering -%\includegraphics[width=\textwidth]{chapters/110-elliptisch/images/jacobi12.pdf} -%\caption{Die Verhältnisse der Funktionen -%$\operatorname{sn}(u,k)$, -%$\operatorname{cn}(u,k)$ -%udn -%$\operatorname{dn}(u,k)$ -%geben Anlass zu neun weitere Funktionen, die sich mit Hilfe -%des Strahlensatzes geometrisch interpretieren lassen. -%\label{buch:elliptisch:fig:jacobi12}} -%\end{figure} -%\begin{table} -%\centering -%\renewcommand{\arraystretch}{2.5} -%\begin{tabular}{|>{$\displaystyle}c<{$}|>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}>{$\displaystyle}c<{$}|} -%\hline -%\cdot & -%\frac{1}{1} & -%\frac{1}{\operatorname{sn}(u,k)} & -%\frac{1}{\operatorname{cn}(u,k)} & -%\frac{1}{\operatorname{dn}(u,k)} -%\\[5pt] -%\hline -%1& -%&%\operatorname{nn}(u,k)=\frac{1}{1} & -%\operatorname{ns}(u,k)=\frac{1}{\operatorname{sn}(u,k)} & -%\operatorname{nc}(u,k)=\frac{1}{\operatorname{cn}(u,k)} & -%\operatorname{nd}(u,k)=\frac{1}{\operatorname{dn}(u,k)} -%\\ -%\operatorname{sn}(u,k) & -%\operatorname{sn}(u,k)=\frac{\operatorname{sn}(u,k)}{1}& -%&%\operatorname{ss}(u,k)=\frac{\operatorname{sn}(u,k)}{\operatorname{sn}(u,k)}& -%\operatorname{sc}(u,k)=\frac{\operatorname{sn}(u,k)}{\operatorname{cn}(u,k)}& -%\operatorname{sd}(u,k)=\frac{\operatorname{sn}(u,k)}{\operatorname{dn}(u,k)} -%\\ -%\operatorname{cn}(u,k) & -%\operatorname{cn}(u,k)=\frac{\operatorname{cn}(u,k)}{1} & -%\operatorname{cs}(u,k)=\frac{\operatorname{cn}(u,k)}{\operatorname{sn}(u,k)}& -%&%\operatorname{cc}(u,k)=\frac{\operatorname{cn}(u,k)}{\operatorname{cn}(u,k)}& -%\operatorname{cd}(u,k)=\frac{\operatorname{cn}(u,k)}{\operatorname{dn}(u,k)} -%\\ -%\operatorname{dn}(u,k) & -%\operatorname{dn}(u,k)=\frac{\operatorname{dn}(u,k)}{1} & -%\operatorname{ds}(u,k)=\frac{\operatorname{dn}(u,k)}{\operatorname{sn}(u,k)}& -%\operatorname{dc}(u,k)=\frac{\operatorname{dn}(u,k)}{\operatorname{cn}(u,k)}& -%%\operatorname{dd}(u,k)=\frac{\operatorname{dn}(u,k)}{\operatorname{dn}(u,k)} -%\\[5pt] -%\hline -%\end{tabular} -%\caption{Zusammenstellung der abgeleiteten Jacobischen elliptischen -%Funktionen in hinteren drei Spalten als Quotienten der grundlegenden -%Jacobischen elliptischen Funktionen. -%Die erste Spalte zum Nenner $1$ enthält die grundlegenden -%Jacobischen elliptischen Funktionen. -%\label{buch:elliptisch:table:abgeleitetjacobi}} -%\end{table} -%\subsubsection{Die abgeleiteten elliptischen Funktionen} -%Zusätzlich zu den grundlegenden Jacobischen elliptischen Funktioenn -%lassen sich weitere elliptische Funktionen bilden, die unglücklicherweise -%die {\em abgeleiteten elliptischen Funktionen} genannt werden. -%Ähnlich wie die trigonometrischen Funktionen $\tan\alpha$, $\cot\alpha$, -%$\sec\alpha$ und $\csc\alpha$ als Quotienten von $\sin\alpha$ und -%$\cos\alpha$ definiert sind, sind die abgeleiteten elliptischen Funktionen -%die in Tabelle~\ref{buch:elliptisch:table:abgeleitetjacobi} zusammengestellten -%Quotienten der grundlegenden Jacobischen elliptischen Funktionen. -%Die Bezeichnungskonvention ist, dass die Funktion $\operatorname{pq}(u,k)$ -%ein Quotient ist, dessen Zähler durch den Buchstaben p bestimmt ist, -%der Nenner durch den Buchstaben q. -%Der Buchstabe n steht für eine $1$, die Buchstaben s, c und d stehen für -%die Anfangsbuchstaben der grundlegenden Jacobischen elliptischen -%Funktionen. -%Meint man irgend eine der Jacobischen elliptischen Funktionen, schreibt -%man manchmal auch $\operatorname{zn}(u,k)$. -% -%In Abbildung~\ref{buch:elliptisch:fig:jacobi12} sind die Quotienten auch -%geometrisch interpretiert. -%Der Wert der Funktion $\operatorname{nq}(u,k)$ ist die auf dem Strahl -%mit Polarwinkel $\varphi$ abgetragene Länge bis zu den vertikalen -%Geraden, die den verschiedenen möglichen Nennern entsprechen. -%Entsprechend ist der Wert der Funktion $\operatorname{dq}(u,k)$ die -%Länge auf dem Strahl mit Polarwinkel $\vartheta$. -% -%Die Relationen~\ref{buch:elliptisch:eqn:jacobi-relationen} -%ermöglichen, jede Funktion $\operatorname{zn}(u,k)$ durch jede -%andere auszudrücken. -%Die schiere Anzahl solcher Beziehungen macht es unmöglich, sie -%übersichtlich in einer Tabelle zusammenzustellen, daher soll hier -%nur an einem Beispiel das Vorgehen gezeigt werden: -% -%\begin{beispiel} -%Die Funktion $\operatorname{sc}(u,k)$ soll durch $\operatorname{cd}(u,k)$ -%ausgedrückt werden. -%Zunächst ist -%\[ -%\operatorname{sc}(u,k) -%= -%\frac{\operatorname{sn}(u,k)}{\operatorname{cn}(u,k)} -%\] -%nach Definition. -%Im Resultat sollen nur noch $\operatorname{cn}(u,k)$ und -%$\operatorname{dn}(u,k)$ vorkommen. -%Daher eliminieren wir zunächst die Funktion $\operatorname{sn}(u,k)$ -%mit Hilfe von \eqref{buch:elliptisch:eqn:jacobi-relationen} und erhalten -%\begin{equation} -%\operatorname{sc}(u,k) -%= -%\frac{\sqrt{1-\operatorname{cn}^2(u,k)}}{\operatorname{cn}(u,k)}. -%\label{buch:elliptisch:eqn:allgausdruecken} -%\end{equation} -%Nun genügt es, die Funktion $\operatorname{cn}(u,k)$ durch -%$\operatorname{cd}(u,k)$ auszudrücken. -%Aus der Definition und der -%dritten Relation in \eqref{buch:elliptisch:eqn:jacobi-relationen} -%erhält man -%\begin{align*} -%\operatorname{cd}^2(u,k) -%&= -%\frac{\operatorname{cn}^2(u,k)}{\operatorname{dn}^2(u,k)} -%= -%\frac{\operatorname{cn}^2(u,k)}{k^{\prime2}+k^2\operatorname{cn}^2(u,k)} -%\\ -%\Rightarrow -%\qquad -%k^{\prime 2} -%\operatorname{cd}^2(u,k) -%+ -%k^2\operatorname{cd}^2(u,k)\operatorname{cn}^2(u,k) -%&= -%\operatorname{cn}^2(u,k) -%\\ -%\operatorname{cn}^2(u,k) -%- -%k^2\operatorname{cd}^2(u,k)\operatorname{cn}^2(u,k) -%&= -%k^{\prime 2} -%\operatorname{cd}^2(u,k) -%\\ -%\operatorname{cn}^2(u,k) -%&= -%\frac{ -%k^{\prime 2} -%\operatorname{cd}^2(u,k) -%}{ -%1 - k^2\operatorname{cd}^2(u,k) -%} -%\end{align*} -%Für den Zähler brauchen wir $1-\operatorname{cn}^2(u,k)$, also -%\[ -%1-\operatorname{cn}^2(u,k) -%= -%\frac{ -%1 -%- -%k^2\operatorname{cd}^2(u,k) -%- -%k^{\prime 2} -%\operatorname{cd}^2(u,k) -%}{ -%1 -%- -%k^2\operatorname{cd}^2(u,k) -%} -%= -%\frac{1-\operatorname{cd}^2(u,k)}{1-k^2\operatorname{cd}^2(u,k)} -%\] -%Einsetzen in~\eqref{buch:elliptisch:eqn:allgausdruecken} gibt -%\begin{align*} -%\operatorname{sc}(u,k) -%&= -%\frac{ -%\sqrt{1-\operatorname{cd}^2(u,k)} -%}{\sqrt{1-k^2\operatorname{cd}^2(u,k)}} -%\cdot -%\frac{ -%\sqrt{1 - k^2\operatorname{cd}^2(u,k)} -%}{ -%k' -%\operatorname{cd}(u,k) -%} -%= -%\frac{ -%\sqrt{1-\operatorname{cd}^2(u,k)} -%}{ -%k' -%\operatorname{cd}(u,k) -%}. -%\qedhere -%\end{align*} -%\end{beispiel} -% -%\subsubsection{Ableitung der abgeleiteten elliptischen Funktionen} -%Aus den Ableitungen der grundlegenden Jacobischen elliptischen Funktionen -%können mit der Quotientenregel nun auch beliebige Ableitungen der -%abgeleiteten Jacobischen elliptischen Funktionen gefunden werden. -%Als Beispiel berechnen wir die Ableitung von $\operatorname{sc}(u,k)$. -%Sie ist -%\begin{align*} -%\frac{d}{du} -%\operatorname{sc}(u,k) -%&= -%\frac{d}{du} -%\frac{\operatorname{sn}(u,k)}{\operatorname{cn}(u,k)} -%= -%\frac{ -%\operatorname{sn}'(u,k)\operatorname{cn}(u,k) -%- -%\operatorname{sn}(u,k)\operatorname{cn}'(u,k)}{ -%\operatorname{cn}^2(u,k) -%} -%\\ -%&= -%\frac{ -%\operatorname{cn}^2(u,k)\operatorname{dn}(u,k) -%+ -%\operatorname{sn}^2(u,k)\operatorname{dn}(u,k) -%}{ -%\operatorname{cn}^2(u,k) -%} -%= -%\frac{( -%\operatorname{sn}^2(u,k) -%+ -%\operatorname{cn}^2(u,k) -%)\operatorname{dn}(u,k)}{ -%\operatorname{cn}^2(u,k) -%} -%\\ -%&= -%\frac{1}{\operatorname{cn}(u,k)} -%\cdot -%\frac{\operatorname{dn}(u,k)}{\operatorname{cn}(u,k)} -%= -%\operatorname{nc}(u,k) -%\operatorname{dc}(u,k). -%\end{align*} -%Man beachte, dass das Quadrat der Nennerfunktion im Resultat -%der Quotientenregel zur Folge hat, dass die -%beiden Funktionen im Resultat beide den gleichen Nenner haben wie -%die Funktion, die abgeleitet wird. -% -%Mit etwas Fleiss kann man nach diesem Muster alle Ableitungen -%\begin{equation} -%%\small -%\begin{aligned} -%\operatorname{sn}'(u,k) -%&= -%\phantom{-} -%\operatorname{cn}(u,k)\,\operatorname{dn}(u,k) -%&&\qquad& -%\operatorname{ns}'(u,k) -%&= -%- -%\operatorname{cs}(u,k)\,\operatorname{ds}(u,k) -%\\ -%\operatorname{cn}'(u,k) -%&= -%- -%\operatorname{sn}(u,k)\,\operatorname{dn}(u,k) -%&&& -%\operatorname{nc}'(u,k) -%&= -%\phantom{-} -%\operatorname{sc}(u,k)\,\operatorname{dc}(u,k) -%\\ -%\operatorname{dn}'(u,k) -%&= -%-k^2 -%\operatorname{sn}(u,k)\,\operatorname{cn}(u,k) -%&&& -%\operatorname{nd}'(u,k) -%&= -%\phantom{-} -%k^2 -%\operatorname{sd}(u,k)\,\operatorname{cd}(u,k) -%\\ -%\operatorname{sc}'(u,k) -%&= -%\phantom{-} -%\operatorname{dc}(u,k)\,\operatorname{nc}(u,k) -%&&& -%\operatorname{cs}'(u,k) -%&= -%- -%\operatorname{ds}(u,k)\,\operatorname{ns}(u,k) -%\\ -%\operatorname{cd}'(u,k) -%&= -%-k^{\prime2} -%\operatorname{sd}(u,k)\,\operatorname{nd}(u,k) -%&&& -%\operatorname{dc}'(u,k) -%&= -%\phantom{-} -%k^{\prime2} -%\operatorname{dc}(u,k)\,\operatorname{nc}(u,k) -%\\ -%\operatorname{ds}'(d,k) -%&= -%- -%\operatorname{cs}(u,k)\,\operatorname{ns}(u,k) -%&&& -%\operatorname{sd}'(d,k) -%&= -%\phantom{-} -%\operatorname{cd}(u,k)\,\operatorname{nd}(u,k) -%\end{aligned} -%\label{buch:elliptisch:eqn:alleableitungen} -%\end{equation} -%finden. -%Man beachte, dass in jeder Identität alle Funktionen den gleichen -%zweiten Buchstaben haben. -% -%\subsubsection{TODO} -%XXX algebraische Beziehungen \\ -%XXX Additionstheoreme \\ -%XXX Perioden -%% use https://math.stackexchange.com/questions/3013692/how-to-show-that-jacobi-sine-function-is-doubly-periodic -% -% -%XXX Ableitungen \\ -%XXX Werte \\ -%% -%% Lösung von Differentialgleichungen -%% -%\subsection{Lösungen von Differentialgleichungen -%\label{buch:elliptisch:subsection:differentialgleichungen}} -%Die elliptischen Funktionen ermöglichen die Lösung gewisser nichtlinearer -%Differentialgleichungen in geschlossener Form. -%Ziel dieses Abschnitts ist, Differentialgleichungen der Form -%\( -%\dot{x}(t)^2 -%= -%P(x(t)) -%\) -%mit einem Polynom $P$ vierten Grades oder -%\( -%\ddot{x}(t) -%= -%p(x(t)) -%\) -%mit einem Polynom dritten Grades als rechter Seite lösen zu können. -% -%% -%% Die Differentialgleichung der elliptischen Funktionen -%% -%\subsubsection{Die Differentialgleichungen der elliptischen Funktionen} -%Um Differentialgleichungen mit elliptischen Funktion lösen zu -%können, muss man als erstes die Differentialgleichungen derselben -%finden. -%Quadriert man die Ableitungsregel für $\operatorname{sn}(u,k)$, erhält -%man -%\[ -%\biggl(\frac{d}{du}\operatorname{sn}(u,k)\biggr)^2 -%= -%\operatorname{cn}(u,k)^2 \operatorname{dn}(u,k)^2. -%\] -%Die Funktionen auf der rechten Seite können durch $\operatorname{sn}(u,k)$ -%ausgedrückt werden, dies führt auf die Differentialgleichung -%\begin{align*} -%\biggl(\frac{d}{du}\operatorname{sn}(u,k)\biggr)^2 -%&= -%\bigl( -%1-\operatorname{sn}(u,k)^2 -%\bigr) -%\bigl( -%1-k^2 \operatorname{sn}(u,k)^2 -%\bigr) -%\\ -%&= -%k^2\operatorname{sn}(u,k)^4 -%-(1+k^2) -%\operatorname{sn}(u,k)^2 -%+1. -%\end{align*} -%Für die Funktion $\operatorname{cn}(u,k)$ ergibt die analoge Rechnung -%\begin{align*} -%\frac{d}{du}\operatorname{cn}(u,k) -%&= -%-\operatorname{sn}(u,k) \operatorname{dn}(u,k) -%\\ -%\biggl(\frac{d}{du}\operatorname{cn}(u,k)\biggr)^2 -%&= -%\operatorname{sn}(u,k)^2 \operatorname{dn}(u,k)^2 -%\\ -%&= -%\bigl(1-\operatorname{cn}(u,k)^2\bigr) -%\bigl(k^{\prime 2}+k^2 \operatorname{cn}(u,k)^2\bigr) -%\\ -%&= -%-k^2\operatorname{cn}(u,k)^4 -%+ -%(k^2-k^{\prime 2})\operatorname{cn}(u,k)^2 -%+ -%k^{\prime 2} -%\intertext{und weiter für $\operatorname{dn}(u,k)$:} -%\frac{d}{du}\operatorname{dn}(u,k) -%&= -%-k^2\operatorname{sn}(u,k)\operatorname{cn}(u,k) -%\\ -%\biggl( -%\frac{d}{du}\operatorname{dn}(u,k) -%\biggr)^2 -%&= -%\bigl(k^2 \operatorname{sn}(u,k)^2\bigr) -%\bigl(k^2 \operatorname{cn}(u,k)^2\bigr) -%\\ -%&= -%\bigl( -%1-\operatorname{dn}(u,k)^2 -%\bigr) -%\bigl( -%\operatorname{dn}(u,k)^2-k^{\prime 2} -%\bigr) -%\\ -%&= -%-\operatorname{dn}(u,k)^4 -%+ -%(1+k^{\prime 2})\operatorname{dn}(u,k)^2 -%-k^{\prime 2}. -%\end{align*} -% -%\begin{table} -%\centering -%\renewcommand{\arraystretch}{1.7} -%\begin{tabular}{|>{$}l<{$}|>{$}l<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} -%\hline -%\text{Funktion $y=$}&\text{Differentialgleichung}&\alpha&\beta&\gamma\\ -%\hline -%\operatorname{sn}(u,k) -% & y'^2 = \phantom{-}(1-y^2)(1-k^2y^2) -% &k^2&1+k^2&1 -%\\ -%\operatorname{cn}(u,k) &y'^2 = \phantom{-}(1-y^2)(k^{\prime2}+k^2y^2) -% &-k^2 &k^2-k^{\prime 2}=2k^2-1&k^{\prime2} -%\\ -%\operatorname{dn}(u,k) -% & y'^2 = -(1-y^2)(k^{\prime 2}-y^2) -% &-1 &1+k^{\prime 2}=2-k^2 &-k^{\prime2} -%\\ -%\hline -%\end{tabular} -%\caption{Elliptische Funktionen als Lösungsfunktionen für verschiedene -%nichtlineare Differentialgleichungen der Art -%\eqref{buch:elliptisch:eqn:1storderdglell}. -%Die Vorzeichen der Koeffizienten $\alpha$, $\beta$ und $\gamma$ -%entscheidet darüber, welche Funktion für die Lösung verwendet werden -%muss. -%\label{buch:elliptisch:tabelle:loesungsfunktionen}} -%\end{table} -% -%Die drei grundlegenden Jacobischen elliptischen Funktionen genügen also alle -%einer nichtlinearen Differentialgleichung erster Ordnung der selben Art. -%Das Quadrat der Ableitung ist ein Polynom vierten Grades der Funktion. -%Die Differentialgleichungen sind in der -%Tabelle~\ref{buch:elliptisch:tabelle:loesungsfunktionen} zusammengefasst. -% -%% -%% Differentialgleichung der abgeleiteten elliptischen Funktionen -%% -%\subsubsection{Die Differentialgleichung der abgeleiteten elliptischen -%Funktionen} -%Da auch die Ableitungen der abgeleiteten Jacobischen elliptischen -%Funktionen Produkte von genau zwei Funktionen sind, die sich wieder -%durch die ursprüngliche Funktion ausdrücken lassen, darf man erwarten, -%dass alle elliptischen Funktionen einer ähnlichen Differentialgleichung -%genügen. -%Um dies besser einzufangen, schreiben wir $\operatorname{pq}(u,k)$, -%wenn wir eine beliebige abgeleitete Jacobische elliptische Funktion. -%Für -%$\operatorname{pq}=\operatorname{sn}$ -%$\operatorname{pq}=\operatorname{cn}$ -%und -%$\operatorname{pq}=\operatorname{dn}$ -%wissen wir bereits und erwarten für jede andere Funktion dass -%$\operatorname{pq}(u,k)$ auch, dass sie Lösung einer Differentialgleichung -%der Form -%\begin{equation} -%\operatorname{pq}'(u,k)^2 -%= -%\alpha \operatorname{pq}(u,k)^4 + \beta \operatorname{pq}(u,k)^2 + \gamma -%\label{buch:elliptisch:eqn:1storderdglell} -%\end{equation} -%erfüllt, -%wobei wir mit $\operatorname{pq}'(u,k)$ die Ableitung von -%$\operatorname{pq}(u,k)$ nach dem ersten Argument meinen. -%Die Koeffizienten $\alpha$, $\beta$ und $\gamma$ hängen von $k$ ab, -%ihre Werte für die grundlegenden Jacobischen elliptischen -%sind in Tabelle~\ref{buch:elliptisch:table:differentialgleichungen} -%zusammengestellt. -% -%Die Koeffizienten müssen nicht für jede Funktion wieder neu bestimmt -%werden, denn für den Kehrwert einer Funktion lässt sich die -%Differentialgleichung aus der Differentialgleichung der ursprünglichen -%Funktion ermitteln. -% -%% -%% Differentialgleichung der Kehrwertfunktion -%% -%\subsubsection{Differentialgleichung für den Kehrwert einer elliptischen Funktion} -%Aus der Differentialgleichung~\eqref{buch:elliptisch:eqn:1storderdglell} -%für die Funktion $\operatorname{pq}(u,k)$ kann auch eine -%Differentialgleichung für den Kehrwert -%$\operatorname{qp}(u,k)=\operatorname{pq}(u,k)^{-1}$ -%ableiten. -%Dazu rechnet man -%\[ -%\operatorname{qp}'(u,k) -%= -%\frac{d}{du}\frac{1}{\operatorname{pq}(u,k)} -%= -%\frac{\operatorname{pq}'(u,k)}{\operatorname{pq}(u,k)^2} -%\qquad\Rightarrow\qquad -%\left\{ -%\quad -%\begin{aligned} -%\operatorname{pq}(u,k) -%&= -%\frac{1}{\operatorname{qp}(u,k)} -%\\ -%\operatorname{pq}'(u,k) -%&= -%\frac{\operatorname{qp}'(u,k)}{\operatorname{qp}(u,k)^2} -%\end{aligned} -%\right. -%\] -%und setzt in die Differentialgleichung ein: -%\begin{align*} -%\biggl( -%\frac{ -%\operatorname{qp}'(u,k) -%}{ -%\operatorname{qp}(u,k) -%} -%\biggr)^2 -%&= -%\alpha \frac{1}{\operatorname{qp}(u,k)^4} -%+ -%\beta \frac{1}{\operatorname{qp}(u,k)^2} -%+ -%\gamma. -%\end{align*} -%Nach Multiplikation mit $\operatorname{qp}(u,k)^4$ erhält man den -%folgenden Satz. -% -%\begin{satz} -%Wenn die Jacobische elliptische Funktion $\operatorname{pq}(u,k)$ -%der Differentialgleichung genügt, dann genügt der Kehrwert -%$\operatorname{qp}(u,k) = 1/\operatorname{pq}(u,k)$ der Differentialgleichung -%\begin{equation} -%(\operatorname{qp}'(u,k))^2 -%= -%\gamma \operatorname{qp}(u,k)^4 -%+ -%\beta \operatorname{qp}(u,k)^2 -%+ -%\alpha -%\label{buch:elliptisch:eqn:kehrwertdgl} -%\end{equation} -%\end{satz} -% -%\begin{table} -%\centering -%\def\lfn#1{\multicolumn{1}{|l|}{#1}} -%\def\rfn#1{\multicolumn{1}{r|}{#1}} -%\renewcommand{\arraystretch}{1.3} -%\begin{tabular}{l|>{$}c<{$}>{$}c<{$}>{$}c<{$}|r} -%\cline{1-4} -%\lfn{Funktion} -% & \alpha & \beta & \gamma &\\ -%\hline -%\lfn{sn}& k^2 & -(1+k^2) & 1 &\rfn{ns}\\ -%\lfn{cn}& -k^2 & -(1-2k^2) & 1-k^2 &\rfn{nc}\\ -%\lfn{dn}& 1 & 2-k^2 & -(1-k^2) &\rfn{nd}\\ -%\hline -%\lfn{sc}& 1-k^2 & 2-k^2 & 1 &\rfn{cs}\\ -%\lfn{sd}&-k^2(1-k^2)&-(1-2k^2) & 1 &\rfn{ds}\\ -%\lfn{cd}& k^2 &-(1+k^2) & 1 &\rfn{dc}\\ -%\hline -% & \gamma & \beta & \alpha &\rfn{Reziproke}\\ -%\cline{2-5} -%\end{tabular} -%\caption{Koeffizienten der Differentialgleichungen für die Jacobischen -%elliptischen Funktionen. -%Der Kehrwert einer Funktion hat jeweils die Differentialgleichung der -%ursprünglichen Funktion, in der die Koeffizienten $\alpha$ und $\gamma$ -%vertauscht worden sind. -%\label{buch:elliptisch:table:differentialgleichungen}} -%\end{table} -% -%% -%% Differentialgleichung zweiter Ordnung -%% -%\subsubsection{Differentialgleichung zweiter Ordnung} -%Leitet die Differentialgleichung~\eqref{buch:elliptisch:eqn:1storderdglell} -%man dies nochmals nach $u$ ab, erhält man die Differentialgleichung -%\[ -%2\operatorname{pq}''(u,k)\operatorname{pq}'(u,k) -%= -%4\alpha \operatorname{pq}(u,k)^3\operatorname{pq}'(u,k) + 2\beta \operatorname{pq}'(u,k)\operatorname{pq}(u,k). -%\] -%Teilt man auf beiden Seiten durch $2\operatorname{pq}'(u,k)$, -%bleibt die nichtlineare -%Differentialgleichung -%\[ -%\frac{d^2\operatorname{pq}}{du^2} -%= -%\beta \operatorname{pq} + 2\alpha \operatorname{pq}^3. -%\] -%Dies ist die Gleichung eines harmonischen Oszillators mit einer -%Anharmonizität der Form $2\alpha z^3$. -% -% -% -%% -%% Jacobischen elliptische Funktionen und elliptische Integrale -%% -%\subsubsection{Jacobische elliptische Funktionen als elliptische Integrale} -%Die in Tabelle~\ref{buch:elliptisch:tabelle:loesungsfunktionen} -%zusammengestellten Differentialgleichungen ermöglichen nun, den -%Zusammenhang zwischen den Funktionen -%$\operatorname{sn}(u,k)$, $\operatorname{cn}(u,k)$ und $\operatorname{dn}(u,k)$ -%und den unvollständigen elliptischen Integralen herzustellen. -%Die Differentialgleichungen sind alle von der Form -%\begin{equation} -%\biggl( -%\frac{d y}{d u} -%\biggr)^2 -%= -%p(u), -%\label{buch:elliptisch:eqn:allgdgl} -%\end{equation} -%wobei $p(u)$ ein Polynom vierten Grades in $y$ ist. -%Diese Differentialgleichung lässt sich mit Separation lösen. -%Dazu zieht man aus~\eqref{buch:elliptisch:eqn:allgdgl} die -%Wurzel -%\begin{align} -%\frac{dy}{du} -%= -%\sqrt{p(y)} -%\notag -%\intertext{und trennt die Variablen. Man erhält} -%\int\frac{dy}{\sqrt{p(y)}} = u+C. -%\label{buch:elliptisch:eqn:yintegral} -%\end{align} -%Solange $p(y)>0$ ist, ist der Integrand auf der linken Seite -%von~\eqref{buch:elliptisch:eqn:yintegral} ebenfalls positiv und -%das Integral ist eine monoton wachsende Funktion $F(y)$. -%Insbesondere ist $F(y)$ invertierbar. -%Die Lösung $y(u)$ der Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} -%ist daher -%\[ -%y(u) = F^{-1}(u+C). -%\] -%Die Jacobischen elliptischen Funktionen sind daher inverse Funktionen -%der unvollständigen elliptischen Integrale. -% -% -%% -%% Differentialgleichung des anharmonischen Oszillators -%% -%\subsubsection{Differentialgleichung des anharmonischen Oszillators} -%Wir möchten die nichtlineare Differentialgleichung -%\begin{equation} -%\biggl( -%\frac{dx}{dt} -%\biggr)^2 -%= -%Ax^4+Bx^2 + C -%\label{buch:elliptisch:eqn:allgdgl} -%\end{equation} -%mit Hilfe elliptischer Funktionen lösen. -%Wir nehmen also an, dass die gesuchte Lösung eine Funktion der Form -%\begin{equation} -%x(t) = a\operatorname{zn}(bt,k) -%\label{buch:elliptisch:eqn:loesungsansatz} -%\end{equation} -%ist. -%Die erste Ableitung von $x(t)$ ist -%\[ -%\dot{x}(t) -%= -%a\operatorname{zn}'(bt,k). -%\] -% -%Indem wir diesen Lösungsansatz in die -%Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} -%einsetzen, erhalten wir -%\begin{equation} -%a^2b^2 \operatorname{zn}'(bt,k)^2 -%= -%a^4A\operatorname{zn}(bt,k)^4 -%+ -%a^2B\operatorname{zn}(bt,k)^2 -%+C -%\label{buch:elliptisch:eqn:dglx} -%\end{equation} -%Andererseits wissen wir, dass $\operatorname{zn}(u,k)$ einer -%Differentilgleichung der Form~\eqref{buch:elliptisch:eqn:1storderdglell} -%erfüllt. -%Wenn wir \eqref{buch:elliptisch:eqn:dglx} durch $a^2b^2$ teilen, können wir -%die rechte Seite von \eqref{buch:elliptisch:eqn:dglx} mit der rechten -%Seite von \eqref{buch:elliptisch:eqn:1storderdglell} vergleichen: -%\[ -%\frac{a^2A}{b^2}\operatorname{zn}(bt,k)^4 -%+ -%\frac{B}{b^2}\operatorname{zn}(bt,k)^2 -%+\frac{C}{a^2b^2} -%= -%\alpha\operatorname{zn}(bt,k)^4 -%+ -%\beta\operatorname{zn}(bt,k)^2 -%+ -%\gamma\operatorname{zn}(bt,k). -%\] -%Daraus ergeben sich die Gleichungen -%\begin{align} -%\alpha &= \frac{a^2A}{b^2}, -%& -%\beta &= \frac{B}{b^2} -%&&\text{und} -%& -%\gamma &= \frac{C}{a^2b^2} -%\label{buch:elliptisch:eqn:koeffvergl} -%\intertext{oder aufgelöst nach den Koeffizienten der ursprünglichen -%Differentialgleichung} -%A&=\frac{\alpha b^2}{a^2} -%& -%B&=\beta b^2 -%&&\text{und}& -%C &= \gamma a^2b^2 -%\label{buch:elliptisch:eqn:koeffABC} -%\end{align} -%für die Koeffizienten der Differentialgleichung der zu verwendenden -%Funktion. -% -%Man beachte, dass nach \eqref{buch:elliptisch:eqn:koeffvergl} die -%Koeffizienten $A$, $B$ und $C$ die gleichen Vorzeichen haben wie -%$\alpha$, $\beta$ und $\gamma$, da in -%\eqref{buch:elliptisch:eqn:koeffvergl} nur mit Quadraten multipliziert -%wird, die immer positiv sind. -%Diese Vorzeichen bestimmen, welche der Funktionen gewählt werden muss. -% -%In den Differentialgleichungen für die elliptischen Funktionen gibt -%es nur den Parameter $k$, der angepasst werden kann. -%Es folgt, dass die Gleichungen -%\eqref{buch:elliptisch:eqn:koeffvergl} -%auch $a$ und $b$ bestimmen. -%Zum Beispiel folgt aus der letzten Gleichung, dass -%\[ -%b = \pm\sqrt{\frac{B}{\beta}}. -%\] -%Damit folgt dann aus der zweiten -%\[ -%a=\pm\sqrt{\frac{\beta C}{\gamma B}}. -%\] -%Die verbleibende Gleichung legt $k$ fest. -%Das folgende Beispiel illustriert das Vorgehen am Beispiel einer -%Gleichung, die Lösungsfunktion $\operatorname{sn}(u,k)$ verlangt. -% -%\begin{beispiel} -%Wir nehmen an, dass die Vorzeichen von $A$, $B$ und $C$ gemäss -%Tabelle~\ref{buch:elliptische:tabelle:loesungsfunktionen} verlangen, -%dass die Funktion $\operatorname{sn}(u,k)$ für die Lösung verwendet -%werden muss. -%Die Tabelle sagt dann auch, dass -%$\alpha=k^2$, $\beta=1$ und $\gamma=1$ gewählt werden müssen. -%Aus dem Koeffizientenvergleich~\eqref{buch:elliptisch:eqn:koeffvergl} -%folgt dann der Reihe nach -%\begin{align*} -%b&=\pm \sqrt{B} -%\\ -%a&=\pm \sqrt{\frac{C}{B}} -%\\ -%k^2 -%&= -%\frac{AC}{B^2}. -%\end{align*} -%Man beachte, dass man $k^2$ durch Einsetzen von -%\eqref{buch:elliptisch:eqn:koeffABC} -%auch direkt aus den Koeffizienten $\alpha$, $\beta$ und $\gamma$ -%erhalten kann, nämlich -%\[ -%\frac{AC}{B^2} -%= -%\frac{\frac{\alpha b^2}{a^2} \gamma a^2b^2}{\beta^2 b^4} -%= -%\frac{\alpha\gamma}{\beta^2}. -%\qedhere -%\] -%\end{beispiel} -% -%Da alle Parameter im -%Lösungsansatz~\eqref{buch:elliptisch:eqn:loesungsansatz} bereits -%festgelegt sind stellt sich die Frage, woher man einen weiteren -%Parameter nehmen kann, mit dem Anfangsbedingungen erfüllen kann. -%Die Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} ist -%autonom, die Koeffizienten der rechten Seite der Differentialgleichung -%sind nicht von der Zeit abhängig. -%Damit ist eine zeitverschobene Funktion $x(t-t_0)$ ebenfalls eine -%Lösung der Differentialgleichung. -%Die allgmeine Lösung der -%Differentialgleichung~\eqref{buch:elliptisch:eqn:allgdgl} hat -%also die Form -%\[ -%x(t) = a\operatorname{zn}(b(t-t_0)), -%\] -%wobei die Funktion $\operatorname{zn}(u,k)$ auf Grund der Vorzeichen -%von $A$, $B$ und $C$ gewählt werden müssen. -%% -%% Das mathematische Pendel -%% -%\subsection{Das mathematische Pendel -%\label{buch:elliptisch:subsection:mathpendel}} -%\begin{figure} -%\centering -%\includegraphics{chapters/110-elliptisch/images/pendel.pdf} -%\caption{Mathematisches Pendel -%\label{buch:elliptisch:fig:mathpendel}} -%\end{figure} -%Das in Abbildung~\ref{buch:elliptisch:fig:mathpendel} dargestellte -%Mathematische Pendel besteht aus einem Massepunkt der Masse $m$ -%im Punkt $P$, -%der über eine masselose Stange der Länge $l$ mit dem Drehpunkt $O$ -%verbunden ist. -%Das Pendel bewegt sich unter dem Einfluss der Schwerebeschleunigung $g$. -% -%Das Trägheitsmoment des Massepunktes um den Drehpunkt $O$ ist -%\( -%I=ml^2 -%\). -%Das Drehmoment der Schwerkraft ist -%\(M=gl\sin\vartheta\). -%Die Bewegungsgleichung wird daher -%\[ -%\begin{aligned} -%\frac{d}{dt} I\dot{\vartheta} -%&= -%M -%= -%gl\sin\vartheta -%\\ -%ml^2\ddot{\vartheta} -%&= -%gl\sin\vartheta -%&&\Rightarrow& -%\ddot{\vartheta} -%&=\frac{g}{l}\sin\vartheta -%\end{aligned} -%\] -%Dies ist eine nichtlineare Differentialgleichung zweiter Ordnung, die -%wir nicht unmittelbar mit den Differentialgleichungen erster Ordnung -%der elliptischen Funktionen vergleichen können. -% -%Die Differentialgleichungen erster Ordnung der elliptischen Funktionen -%enthalten das Quadrat der ersten Ableitung. -%In unserem Fall entspricht das einer Gleichung, die $\dot{\vartheta}^2$ -%enthält. -%Der Energieerhaltungssatz kann uns eine solche Gleichung geben. -%Die Summe von kinetischer und potentieller Energie muss konstant sein. -%Dies führt auf -%\[ -%E_{\text{kinetisch}} -%+ -%E_{\text{potentiell}} -%= -%\frac12I\dot{\vartheta}^2 -%+ -%mgl(1-\cos\vartheta) -%= -%\frac12ml^2\dot{\vartheta}^2 -%+ -%mgl(1-\cos\vartheta) -%= -%E -%\] -%Durch Auflösen nach $\dot{\vartheta}$ kann man jetzt die -%Differentialgleichung -%\[ -%\dot{\vartheta}^2 -%= -%- -%\frac{2g}{l}(1-\cos\vartheta) -%+\frac{2E}{ml^2} -%\] -%finden. -%In erster Näherung, d.h. wenn man die rechte Seite bis zu vierten -%Potenzen in eine Taylor-Reihe in $\vartheta$ entwickelt, ist dies -%tatsächlich eine Differentialgleichung der Art, wie wir sie für -%elliptische Funktionen gefunden haben, wir möchten aber eine exakte -%Lösung konstruieren. -% -%Die maximale Energie für eine Bewegung, bei der sich das Pendel gerade -%über den höchsten Punkt hinweg zu bewegen vermag, ist -%$E=2lmg$. -%Falls $E<2mgl$ ist, erwarten wir Schwingungslösungen, bei denen -%der Winkel $\vartheta$ immer im offenen Interval $(-\pi,\pi)$ -%bleibt. -%Für $E>2mgl$ wird sich das Pendel im Kreis bewegen, für sehr grosse -%Energie ist die kinetische Energie dominant, die Verlangsamung im -%höchsten Punkt wird immer weniger ausgeprägt sein. -% -%% -%% Koordinatentransformation auf elliptische Funktionen -%% -%\subsubsection{Koordinatentransformation auf elliptische Funktionen} -%Wir verwenden als neue Variable -%\[ -%y = \sin\frac{\vartheta}2 -%\] -%mit der Ableitung -%\[ -%\dot{y}=\frac12\cos\frac{\vartheta}{2}\cdot \dot{\vartheta}. -%\] -%Man beachte, dass $y$ nicht eine Koordinate in -%Abbildung~\ref{buch:elliptisch:fig:mathpendel} ist. -% -%Aus den Halbwinkelformeln finden wir -%\[ -%\cos\vartheta -%= -%1-2\sin^2 \frac{\vartheta}2 -%= -%1-2y^2. -%\] -%Dies können wir zusammen mit der -%Identität $\cos^2\vartheta/2 = 1-\sin^2\vartheta/2 = 1-y^2$ -%in die Energiegleichung einsetzen und erhalten -%\[ -%\frac12ml^2\dot{\vartheta}^2 + mgly^2 = E -%\qquad\Rightarrow\qquad -%\frac14 \dot{\vartheta}^2 = \frac{E}{2ml^2} - \frac{g}{2l}y^2. -%\] -%Der konstante Term auf der rechten Seite ist grösser oder kleiner als -%$1$ je nachdem, ob das Pendel sich im Kreis bewegt oder nicht. -% -%Durch Multiplizieren mit $\cos^2\frac{\vartheta}{2}=1-y^2$ -%erhalten wir auf der linken Seite einen Ausdruck, den wir -%als Funktion von $\dot{y}$ ausdrücken können. -%Wir erhalten -%\begin{align*} -%\frac14 -%\cos^2\frac{\vartheta}2 -%\cdot -%\dot{\vartheta}^2 -%&= -%\frac14 -%(1-y^2) -%\biggl(\frac{E}{2ml^2} -\frac{g}{2l}y^2\biggr) -%\\ -%\dot{y}^2 -%&= -%\frac{1}{4} -%(1-y^2) -%\biggl(\frac{E}{2ml^2} -\frac{g}{2l}y^2\biggr) -%\end{align*} -%Die letzte Gleichung hat die Form einer Differentialgleichung -%für elliptische Funktionen. -%Welche Funktion verwendet werden muss, hängt von der Grösse der -%Koeffizienten in der zweiten Klammer ab. -%Die Tabelle~\ref{buch:elliptisch:tabelle:loesungsfunktionen} -%zeigt, dass in der zweiten Klammer jeweils einer der Terme -%$1$ sein muss. -% -%% -%% Der Fall E < 2mgl -%% -%\subsubsection{Der Fall $E<2mgl$} -%\begin{figure} -%\centering -%\includegraphics[width=\textwidth]{chapters/110-elliptisch/images/jacobiplots.pdf} -%\caption{% -%Abhängigkeit der elliptischen Funktionen von $u$ für -%verschiedene Werte von $k^2=m$. -%Für $m=0$ ist $\operatorname{sn}(u,0)=\sin u$, -%$\operatorname{cn}(u,0)=\cos u$ und $\operatorname{dn}(u,0)=1$, diese -%sind in allen Plots in einer helleren Farbe eingezeichnet. -%Für kleine Werte von $m$ weichen die elliptischen Funktionen nur wenig -%von den trigonometrischen Funktionen ab, -%es ist aber klar erkennbar, dass die anharmonischen Terme in der -%Differentialgleichung die Periode mit steigender Amplitude verlängern. -%Sehr grosse Werte von $m$ nahe bei $1$ entsprechen der Situation, dass -%die Energie des Pendels fast ausreicht, dass es den höchsten Punkt -%erreichen kann, was es für $m$ macht. -%\label{buch:elliptisch:fig:jacobiplots}} -%\end{figure} -% -% -%Wir verwenden als neue Variable -%\[ -%y = \sin\frac{\vartheta}2 -%\] -%mit der Ableitung -%\[ -%\dot{y}=\frac12\cos\frac{\vartheta}{2}\cdot \dot{\vartheta}. -%\] -%Man beachte, dass $y$ nicht eine Koordinate in -%Abbildung~\ref{buch:elliptisch:fig:mathpendel} ist. -% -%Aus den Halbwinkelformeln finden wir -%\[ -%\cos\vartheta -%= -%1-2\sin^2 \frac{\vartheta}2 -%= -%1-2y^2. -%\] -%Dies können wir zusammen mit der -%Identität $\cos^2\vartheta/2 = 1-\sin^2\vartheta/2 = 1-y^2$ -%in die Energiegleichung einsetzen und erhalten -%\[ -%\frac12ml^2\dot{\vartheta}^2 + mgly^2 = E. -%\] -%Durch Multiplizieren mit $\cos^2\frac{\vartheta}{2}=1-y^2$ -%erhalten wir auf der linken Seite einen Ausdruck, den wir -%als Funktion von $\dot{y}$ ausdrücken können. -%Wir erhalten -%\begin{align*} -%\frac12ml^2 -%\cos^2\frac{\vartheta}2 -%\dot{\vartheta}^2 -%&= -%(1-y^2) -%(E -mgly^2) -%\\ -%\frac{1}{4}\cos^2\frac{\vartheta}{2}\dot{\vartheta}^2 -%&= -%\frac{1}{2} -%(1-y^2) -%\biggl(\frac{E}{ml^2} -\frac{g}{l}y^2\biggr) -%\\ -%\dot{y}^2 -%&= -%\frac{E}{2ml^2} -%(1-y^2)\biggl( -%1-\frac{2gml}{E}y^2 -%\biggr). -%\end{align*} -%Dies ist genau die Form der Differentialgleichung für die elliptische -%Funktion $\operatorname{sn}(u,k)$ -%mit $k^2 = 2gml/E< 1$. -% -%%% -%%% Der Fall E > 2mgl -%%% -%%\subsection{Der Fall $E > 2mgl$} -%%In diesem Fall hat das Pendel im höchsten Punkte immer noch genügend -%%kinetische Energie, so dass es sich im Kreise dreht. -%%Indem wir die Gleichung -% -% -%%\subsection{Soliton-Lösungen der Sinus-Gordon-Gleichung} -% -%%\subsection{Nichtlineare Differentialgleichung vierter Ordnung} -%%XXX Möbius-Transformation \\ -%%XXX Reduktion auf die Differentialgleichung elliptischer Funktionen diff --git a/buch/chapters/110-elliptisch/lemniskate.tex b/buch/chapters/110-elliptisch/lemniskate.tex index e766779..0df27a7 100644 --- a/buch/chapters/110-elliptisch/lemniskate.tex +++ b/buch/chapters/110-elliptisch/lemniskate.tex @@ -387,7 +387,7 @@ r(s) \begin{figure} \centering -\includegraphics{chapters/110-elliptisch/images/slcl.pdf} +\includegraphics[width=\textwidth]{chapters/110-elliptisch/images/slcl.pdf} \caption{ Lemniskatischer Sinus und Kosinus sowie Sinus und Kosinus mit derart skaliertem Argument, dass die Funktionen die gleichen Nullstellen -- cgit v1.2.1 From f9842b34a2b78bc340b861cc57aa29ccfbb13fd1 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 24 Apr 2022 15:35:47 +0200 Subject: Makefile fixes, lecture notes week 8 --- buch/Makefile | 4 ++-- buch/aufgaben3.tex | 4 ++-- buch/chapters/000-einleitung/Makefile.inc | 2 +- buch/chapters/010-potenzen/Makefile.inc | 2 +- buch/chapters/020-exponential/Makefile.inc | 2 +- buch/chapters/030-geometrie/Makefile.inc | 2 +- buch/chapters/040-rekursion/Makefile.inc | 2 +- buch/chapters/050-differential/Makefile.inc | 2 +- buch/chapters/060-integral/Makefile.inc | 6 +----- buch/chapters/070-orthogonalitaet/Makefile.inc | 2 +- buch/chapters/075-fourier/Makefile.inc | 2 +- buch/chapters/080-funktionentheorie/Makefile.inc | 2 +- buch/chapters/090-pde/Makefile.inc | 2 +- buch/chapters/110-elliptisch/Makefile.inc | 6 +++--- buch/chapters/110-elliptisch/uebungsaufgaben/1.tex | 19 ++++++++++++------- buch/common/macros.tex | 4 +++- buch/common/test-common.tex | 1 + buch/common/test3.tex | 1 + 18 files changed, 35 insertions(+), 30 deletions(-) (limited to 'buch') diff --git a/buch/Makefile b/buch/Makefile index 00fcf42..af0e1e2 100755 --- a/buch/Makefile +++ b/buch/Makefile @@ -55,13 +55,13 @@ SeminarSpezielleFunktionen.ind: SeminarSpezielleFunktionen.idx # tests: test1.pdf test2.pdf test3.pdf -test1.pdf: common/test-common.tex common/test1.tex aufgaben1.tex +test1.pdf: common/test-common.tex common/test1.tex aufgaben1.tex $(TEXFILES) $(pdflatex) common/test1.tex test2.pdf: common/test-common.tex common/test1.tex aufgaben2.tex $(pdflatex) common/test2.tex -test3.pdf: common/test-common.tex common/test1.tex aufgaben3.tex +test3.pdf: common/test-common.tex common/test1.tex aufgaben3.tex $(CHAPTERFILES) $(pdflatex) common/test3.tex # diff --git a/buch/aufgaben3.tex b/buch/aufgaben3.tex index a39fc19..16288ec 100644 --- a/buch/aufgaben3.tex +++ b/buch/aufgaben3.tex @@ -4,6 +4,6 @@ % (c) 2022 Prof. Dr. Andreas Mueller, OST % -%\item -%\input chapters/60-gruppen/uebungsaufgaben/6001.tex +\item +\input chapters/110-elliptisch/uebungsaufgaben/1.tex diff --git a/buch/chapters/000-einleitung/Makefile.inc b/buch/chapters/000-einleitung/Makefile.inc index a870448..5840050 100644 --- a/buch/chapters/000-einleitung/Makefile.inc +++ b/buch/chapters/000-einleitung/Makefile.inc @@ -4,5 +4,5 @@ # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -CHAPTERFILES = $(CHAPTERFILES) \ +CHAPTERFILES += \ chapters/000-einleitung/chapter.tex diff --git a/buch/chapters/010-potenzen/Makefile.inc b/buch/chapters/010-potenzen/Makefile.inc index a4505cb..27ccdae 100644 --- a/buch/chapters/010-potenzen/Makefile.inc +++ b/buch/chapters/010-potenzen/Makefile.inc @@ -4,7 +4,7 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -CHAPTERFILES = $(CHAPTERFILES) \ +CHAPTERFILES += \ chapters/010-potenzen/loesbarkeit.tex \ chapters/010-potenzen/polynome.tex \ chapters/010-potenzen/tschebyscheff.tex \ diff --git a/buch/chapters/020-exponential/Makefile.inc b/buch/chapters/020-exponential/Makefile.inc index d6b3c7f..4d8f58b 100644 --- a/buch/chapters/020-exponential/Makefile.inc +++ b/buch/chapters/020-exponential/Makefile.inc @@ -4,7 +4,7 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -CHAPTERFILES = $(CHAPTERFILES) \ +CHAPTERFILES += \ chapters/020-exponential/zins.tex \ chapters/020-exponential/log.tex \ chapters/020-exponential/lambertw.tex \ diff --git a/buch/chapters/030-geometrie/Makefile.inc b/buch/chapters/030-geometrie/Makefile.inc index 0bf775f..d4940dc 100644 --- a/buch/chapters/030-geometrie/Makefile.inc +++ b/buch/chapters/030-geometrie/Makefile.inc @@ -4,7 +4,7 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -CHAPTERFILES = $(CHAPTERFILES) \ +CHAPTERFILES += \ chapters/030-geometrie/trigonometrisch.tex \ chapters/030-geometrie/sphaerisch.tex \ chapters/030-geometrie/hyperbolisch.tex \ diff --git a/buch/chapters/040-rekursion/Makefile.inc b/buch/chapters/040-rekursion/Makefile.inc index a222b1c..cd54c80 100644 --- a/buch/chapters/040-rekursion/Makefile.inc +++ b/buch/chapters/040-rekursion/Makefile.inc @@ -4,7 +4,7 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -CHAPTERFILES = $(CHAPTERFILES) \ +CHAPTERFILES += \ chapters/040-rekursion/gamma.tex \ chapters/040-rekursion/bohrmollerup.tex \ chapters/040-rekursion/integral.tex \ diff --git a/buch/chapters/050-differential/Makefile.inc b/buch/chapters/050-differential/Makefile.inc index b72a275..7151c07 100644 --- a/buch/chapters/050-differential/Makefile.inc +++ b/buch/chapters/050-differential/Makefile.inc @@ -4,7 +4,7 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -CHAPTERFILES = $(CHAPTERFILES) \ +CHAPTERFILES += \ chapters/050-differential/beispiele.tex \ chapters/050-differential/potenzreihenmethode.tex \ chapters/050-differential/bessel.tex \ diff --git a/buch/chapters/060-integral/Makefile.inc b/buch/chapters/060-integral/Makefile.inc index e19cb0c..d85caad 100644 --- a/buch/chapters/060-integral/Makefile.inc +++ b/buch/chapters/060-integral/Makefile.inc @@ -4,13 +4,9 @@ # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -CHAPTERFILES = $(CHAPTERFILES) \ +CHAPTERFILES += \ chapters/060-integral/fehlerfunktion.tex \ chapters/060-integral/eulertransformation.tex \ chapters/060-integral/differentialkoerper.tex \ chapters/060-integral/risch.tex \ - chapters/060-integral/orthogonal.tex \ - chapters/060-integral/legendredgl.tex \ - chapters/060-integral/sturm.tex \ - chapters/060-integral/gaussquadratur.tex \ chapters/060-integral/chapter.tex diff --git a/buch/chapters/070-orthogonalitaet/Makefile.inc b/buch/chapters/070-orthogonalitaet/Makefile.inc index 286ab2e..8f58489 100644 --- a/buch/chapters/070-orthogonalitaet/Makefile.inc +++ b/buch/chapters/070-orthogonalitaet/Makefile.inc @@ -4,7 +4,7 @@ # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -CHAPTERFILES = $(CHAPTERFILES) \ +CHAPTERFILES += \ chapters/070-orthogonalitaet/orthogonal.tex \ chapters/070-orthogonalitaet/rekursion.tex \ chapters/070-orthogonalitaet/rodrigues.tex \ diff --git a/buch/chapters/075-fourier/Makefile.inc b/buch/chapters/075-fourier/Makefile.inc index c153dc4..a762e63 100644 --- a/buch/chapters/075-fourier/Makefile.inc +++ b/buch/chapters/075-fourier/Makefile.inc @@ -4,7 +4,7 @@ # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -CHAPTERFILES = $(CHAPTERFILES) \ +CHAPTERFILES += \ chapters/075-fourier/bessel.tex \ chapters/075-fourier/2d.tex \ chapters/075-fourier/chapter.tex diff --git a/buch/chapters/080-funktionentheorie/Makefile.inc b/buch/chapters/080-funktionentheorie/Makefile.inc index affaa94..779cd80 100644 --- a/buch/chapters/080-funktionentheorie/Makefile.inc +++ b/buch/chapters/080-funktionentheorie/Makefile.inc @@ -4,7 +4,7 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -CHAPTERFILES = $(CHAPTERFILES) \ +CHAPTERFILES += \ chapters/080-funktionentheorie/holomorph.tex \ chapters/080-funktionentheorie/analytisch.tex \ chapters/080-funktionentheorie/cauchy.tex \ diff --git a/buch/chapters/090-pde/Makefile.inc b/buch/chapters/090-pde/Makefile.inc index c64af06..5b52d27 100644 --- a/buch/chapters/090-pde/Makefile.inc +++ b/buch/chapters/090-pde/Makefile.inc @@ -4,7 +4,7 @@ # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -CHAPTERFILES = $(CHAPTERFILES) \ +CHAPTERFILES += \ chapters/090-pde/gleichung.tex \ chapters/090-pde/separation.tex \ chapters/090-pde/rechteck.tex \ diff --git a/buch/chapters/110-elliptisch/Makefile.inc b/buch/chapters/110-elliptisch/Makefile.inc index b23df52..639cb8f 100644 --- a/buch/chapters/110-elliptisch/Makefile.inc +++ b/buch/chapters/110-elliptisch/Makefile.inc @@ -4,12 +4,12 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -CHAPTERFILES = $(CHAPTERFILES) \ +CHAPTERFILES += \ chapters/110-elliptisch/ellintegral.tex \ chapters/110-elliptisch/jacobi.tex \ chapters/110-elliptisch/elltrigo.tex \ chapters/110-elliptisch/dglsol.tex \ chapters/110-elliptisch/mathpendel.tex \ chapters/110-elliptisch/lemniskate.tex \ - chapters/110-elliptisch/uebungsaufgaben/001.tex \ - chapters/110-geometrie/chapter.tex + chapters/110-elliptisch/uebungsaufgaben/1.tex \ + chapters/110-elliptisch/chapter.tex diff --git a/buch/chapters/110-elliptisch/uebungsaufgaben/1.tex b/buch/chapters/110-elliptisch/uebungsaufgaben/1.tex index 8e4b39f..67d5148 100644 --- a/buch/chapters/110-elliptisch/uebungsaufgaben/1.tex +++ b/buch/chapters/110-elliptisch/uebungsaufgaben/1.tex @@ -28,9 +28,11 @@ for den anharmonischen Oszillator ab, die sie in der Form $\frac12m\dot{x}^2 = f(x)$ schreiben. \item Die Amplitude der Schwingung ist derjenige $x$-Wert, für den die -Geschwindigkeit verschwindet. +Geschwindigkeit $\dot{x}(t)$ verschwindet. Leiten Sie die Amplitude aus der Differentialgleichung von -\ref{buch:1101:basic-dgl} ab. +%\ref{buch:1101:basic-dgl} +Teilaufgabe c) +ab. Sie erhalten zwei Werte $x_{\pm}$, wobei der kleinere $x_-$ die Amplitude einer beschränkten Schwingung beschreibt, während die $x_+$ die minimale Ausgangsamplitude einer gegen @@ -66,13 +68,16 @@ wobei $k^2=x_-^2/x_+^2$ und $A$ geeignet gewählt werden müssen. \label{buch:1101:teilaufgabe:dgl3} Verwenden Sie $t(\tau) = \alpha\tau$ und -$Y(\tau)=X(t(\tau))$ um eine Differentialgleichung für die Funktion -$Y(\tau)$ zu gewinnen, die die Form der Differentialgleichung -von $\operatorname{sn}(u,k)$ hat, für die also $A=0$ in -\eqref{buch:1101:eqn:dgl3} ist. +$Y(\tau)=X(t(\tau))=X(\alpha\tau)$ um eine Differentialgleichung für +die Funktion $Y(\tau)$ zu gewinnen, die die Form der Differentialgleichung +von $\operatorname{sn}(u,k)$ hat (Abschnitt +\ref{buch:elliptisch:subsection:differentialgleichungen}), +für die also $A=0$ in \eqref{buch:1101:eqn:dgl3} ist. \item Verwenden Sie die Lösung $\operatorname{sn}(u,k)$ der in -\ref{buch:1101:teilaufgabe:dgl3} erhaltenen Differentialgleichung, +Teilaufgabe h) +%\ref{buch:1101:teilaufgabe:dgl3} +erhaltenen Differentialgleichung, um die Lösung $x(t)$ der ursprünglichen Gleichung aufzuschreiben. \end{teilaufgaben} diff --git a/buch/common/macros.tex b/buch/common/macros.tex index 7c82180..bb6e9b0 100644 --- a/buch/common/macros.tex +++ b/buch/common/macros.tex @@ -23,7 +23,9 @@ \vfill\pagebreak} \newenvironment{teilaufgaben}{ \begin{enumerate} -\renewcommand{\labelenumi}{\alph{enumi})} +\renewcommand{\theenumi}{\alph{enumi})} +%\renewcommand{\labelenumi}{\alph{enumi})} +\renewcommand{\labelenumi}{\theenumi} }{\end{enumerate}} % Aufgabe \newcounter{problemcounter}[chapter] diff --git a/buch/common/test-common.tex b/buch/common/test-common.tex index 289e59c..3f49701 100644 --- a/buch/common/test-common.tex +++ b/buch/common/test-common.tex @@ -30,6 +30,7 @@ \usepackage{standalone} \usepackage{environ} \usepackage{tikz} +\usepackage{xr} \input{../common/linsys.tex} \newcounter{beispiel} \newenvironment{beispiele}{ diff --git a/buch/common/test3.tex b/buch/common/test3.tex index 8b24262..22d6b63 100644 --- a/buch/common/test3.tex +++ b/buch/common/test3.tex @@ -4,6 +4,7 @@ % (c) 2021 Prof. Dr. Andreas Mueller, OST % \input{common/test-common.tex} +\externaldocument{buch} \begin{document} {\parindent0pt\hbox to\hsize{% -- cgit v1.2.1 From bc23a25ab1aaa67f78998d34d8bf75afbe70606d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 25 Apr 2022 21:54:35 +0200 Subject: fix typos --- buch/chapters/110-elliptisch/uebungsaufgaben/1.tex | 20 +++++++++++++------- 1 file changed, 13 insertions(+), 7 deletions(-) (limited to 'buch') diff --git a/buch/chapters/110-elliptisch/uebungsaufgaben/1.tex b/buch/chapters/110-elliptisch/uebungsaufgaben/1.tex index 67d5148..694f18a 100644 --- a/buch/chapters/110-elliptisch/uebungsaufgaben/1.tex +++ b/buch/chapters/110-elliptisch/uebungsaufgaben/1.tex @@ -267,15 +267,21 @@ Die Ableitung von $Y(\tau)=X(t(\tau))$ nach $\tau$ ist = \alpha \dot{X}(t(\tau)) -\qquad\Rightarrow\qquad -\frac{1}{\alpha^2}\frac{dY}{d\tau} +\quad\Rightarrow\quad +\frac{1}{\alpha}\frac{dY}{d\tau} = -\dot{X}(t(\tau)). +\dot{X}(t(\tau)) +\quad\Rightarrow\quad +\frac{1}{\alpha^2}\biggl(\frac{dY}{d\tau}\biggr)^2 += +\dot{X}(t(\tau))^2. \] Die Differentialgleichung für $Y(\tau)$ ist \[ -\frac{2mk^2}{\delta x_+^2\alpha^2} +\frac{2m}{\delta x_+^2\alpha^2} +\biggl( \frac{dY}{d\tau} +\biggr)^2 = (1-Y^2)(1-k^2Y^2). \] @@ -283,7 +289,7 @@ Der Koeffizient vor der Ableitung wird $1$, wenn man \[ \alpha^2 = -\frac{2mk^2}{\delta x_+^2} +\frac{2m}{\delta x_+^2} \] wählt. Diese Differentialgleichug hat die Lösung @@ -299,9 +305,9 @@ x(t) x_- X(t) = x_- \operatorname{sn}\biggl( -t\sqrt{\frac{\delta x_+^2}{2mk^2} } +t\sqrt{\frac{\delta x_+^2}{2m} } ,k -\biggr) +\biggr). \end{align*} Das Produkt $\delta x_+^2$ kann auch als \[ -- cgit v1.2.1 From c771727f3d404d7d79f36b3871e540a8539edfcf Mon Sep 17 00:00:00 2001 From: runterer Date: Sat, 30 Apr 2022 22:03:05 +0200 Subject: wip copying my handwritten stuff to LaTex --- buch/papers/zeta/Makefile.inc | 7 +- buch/papers/zeta/analytic_continuation.tex | 165 +++++++++++++++++++++++++++++ buch/papers/zeta/einleitung.tex | 11 ++ buch/papers/zeta/main.tex | 32 ++---- buch/papers/zeta/teil0.tex | 22 ---- buch/papers/zeta/teil1.tex | 55 ---------- buch/papers/zeta/teil2.tex | 40 ------- buch/papers/zeta/teil3.tex | 40 ------- buch/papers/zeta/zeta_gamma.tex | 53 +++++++++ 9 files changed, 239 insertions(+), 186 deletions(-) create mode 100644 buch/papers/zeta/analytic_continuation.tex create mode 100644 buch/papers/zeta/einleitung.tex delete mode 100644 buch/papers/zeta/teil0.tex delete mode 100644 buch/papers/zeta/teil1.tex delete mode 100644 buch/papers/zeta/teil2.tex delete mode 100644 buch/papers/zeta/teil3.tex create mode 100644 buch/papers/zeta/zeta_gamma.tex (limited to 'buch') diff --git a/buch/papers/zeta/Makefile.inc b/buch/papers/zeta/Makefile.inc index 11c7697..14babe2 100644 --- a/buch/papers/zeta/Makefile.inc +++ b/buch/papers/zeta/Makefile.inc @@ -7,8 +7,7 @@ dependencies-zeta = \ papers/zeta/packages.tex \ papers/zeta/main.tex \ papers/zeta/references.bib \ - papers/zeta/teil0.tex \ - papers/zeta/teil1.tex \ - papers/zeta/teil2.tex \ - papers/zeta/teil3.tex + papers/zeta/einleitung.tex \ + papers/zeta/analytic_continuation.tex \ + papers/zeta/zeta_gamma.tex \ diff --git a/buch/papers/zeta/analytic_continuation.tex b/buch/papers/zeta/analytic_continuation.tex new file mode 100644 index 0000000..943647a --- /dev/null +++ b/buch/papers/zeta/analytic_continuation.tex @@ -0,0 +1,165 @@ +\section{Analytische Fortsetzung} \label{zeta:section:analytische_fortsetzung} +\rhead{Analytische Fortsetzung} + +%TODO missing Text + +\subsection{Fortsetzung auf $\Re(s) > 0$} \label{zeta:subsection:auf_bereich_ge_0} +Zuerst definieren die Dirichletsche Etafunktion als +\begin{equation}\label{zeta:equation:eta} + \eta(s) + = + \sum_{n=1}^{\infty} + \frac{(-1)^{n-1}}{n^s}, +\end{equation} +wobei die Reihe bis auf die alternierenden Vorzeichen die selbe wie in der Zetafunktion ist. +Diese Etafunktion konvergiert gemäss dem Leibnitz-Kriterium im Bereich $\Re(s) > 0$, da dann die einzelnen Glieder monoton fallend sind. + +Wenn wir es nun schaffen, die sehr ähnliche Zetafunktion mit der Etafunktion auszudrücken, dann haben die gesuchte Fortsetzung. +Die folgenden Schritte zeigen, wie man dazu kommt: +\begin{align} + \zeta(s) + &= + \sum_{n=1}^{\infty} + \frac{1}{n^s} \label{zeta:align1} + \\ + \frac{1}{2^{s-1}} + \zeta(s) + &= + \sum_{n=1}^{\infty} + \frac{2}{(2n)^s} \label{zeta:align2} + \\ + \left(1 - \frac{1}{2^{s-1}} \right) + \zeta(s) + &= + \frac{1}{1^s} + \underbrace{-\frac{2}{2^s} + \frac{1}{2^s}}_{-\frac{1}{2^s}} + + \frac{1}{3^s} + \underbrace{-\frac{2}{4^s} + \frac{1}{4^s}}_{-\frac{1}{4^s}} + \ldots + && \text{\eqref{zeta:align1}} - \text{\eqref{zeta:align2}} + \\ + &= \eta(s) + \\ + \zeta(s) + &= + \left(1 - \frac{1}{2^{s-1}} \right)^{-1} \eta(s). +\end{align} + +\subsection{Fortsetzung auf ganz $\mathbb{C}$} \label{zeta:subsection:auf_ganz} +Für die Fortsetzung auf den Rest von $\mathbb{C}$, verwenden wir den Zusammenhang von Gamma- und Zetafunktion aus \ref{zeta:section:zusammenhang_mit_gammafunktion}. +Wir beginnen damit, die Gammafunktion für den halben Funktionswert zu berechnen als +\begin{equation} + \Gamma \left( \frac{s}{2} \right) + = + \int_0^{\infty} t^{\frac{s}{2}-1} e^{-t} dt. +\end{equation} +Nun substituieren wir $t$ mit $t = \pi n^2 x$ und $dt=\pi n^2 dx$ und erhalten +\begin{align} + \Gamma \left( \frac{s}{2} \right) + &= + \int_0^{\infty} + (\pi n^2)^{\frac{s}{2}} + x^{\frac{s}{2}-1} + e^{-\pi n^2 x} + dx + && \text{Division durch } (\pi n^2)^{\frac{s}{2}} + \\ + \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}} n^s} + &= + \int_0^{\infty} + x^{\frac{s}{2}-1} + e^{-\pi n^2 x} + dx + && \text{Zeta durch Summenbildung } \sum_{n=1}^{\infty} + \\ + \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} + \zeta(s) + &= + \int_0^{\infty} + x^{\frac{s}{2}-1} + \sum_{n=1}^{\infty} + e^{-\pi n^2 x} + dx. \label{zeta:equation:integral1} +\end{align} +Die Summe kürzen wir ab als $\psi(x) = \sum_{n=1}^{\infty} e^{-\pi n^2 x}$. +%TODO Wieso folgendes -> aus Fourier Signal +Es gilt +\begin{equation}\label{zeta:equation:psi} + \psi(x) + = + - \frac{1}{2} + + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}} + + \frac{1}{2 \sqrt{x}}. +\end{equation} + +Zunächst teilen wir nun das Integral aus \eqref{zeta:equation:integral1} auf als +\begin{equation}\label{zeta:equation:integral2} + \int_0^{\infty} + x^{\frac{s}{2}-1} + \psi(x) + dx + = + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + dx + + + \int_1^{\infty} + x^{\frac{s}{2}-1} + \psi(x) + dx, +\end{equation} +wobei wir uns nun auf den ersten Teil konzentrieren werden. +Dabei setzen wir das Wissen aus \eqref{zeta:equation:integral2} ein und erhalten +\begin{align} + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + dx + &= + \int_0^{1} + x^{\frac{s}{2}-1} + \left( + - \frac{1}{2} + + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}} + + \frac{1}{2 \sqrt{x}}. + \right) + dx + \\ + &= + \int_0^{1} + x^{\frac{s}{2}-\frac{3}{2}} + \psi \left( \frac{1}{x} \right) + + \frac{1}{2} + \left( + x^{\frac{s}{2}-\frac{3}{2}} + - + x^{\frac{s}{2}-1} + \right) + dx + \\ + &= + \int_0^{1} + x^{\frac{s}{2}-\frac{3}{2}} + \psi \left( \frac{1}{x} \right) + dx + + \frac{1}{2} + \int_0^1 + x^{\frac{s}{2}-\frac{3}{2}} + - + x^{\frac{s}{2}-1} + dx. +\end{align} +Dabei kann das zweite integral gelöst werden als +\begin{equation} + \frac{1}{2} + \int_0^1 + x^{\frac{s}{2}-\frac{3}{2}} + - + x^{\frac{s}{2}-1} + dx + = + \frac{1}{s(s-1)}. +\end{equation} + + diff --git a/buch/papers/zeta/einleitung.tex b/buch/papers/zeta/einleitung.tex new file mode 100644 index 0000000..3b70531 --- /dev/null +++ b/buch/papers/zeta/einleitung.tex @@ -0,0 +1,11 @@ +\section{Einleitung} \label{zeta:section:einleitung} +\rhead{Einleitung} + +Die Riemannsche Zetafunktion ist für alle komplexe $s$ mit $\Re(s) > 1$ definiert als +\begin{equation}\label{zeta:equation1} + \zeta(s) + = + \sum_{n=1}^{\infty} + \frac{1}{n^s}. +\end{equation} + diff --git a/buch/papers/zeta/main.tex b/buch/papers/zeta/main.tex index 1d9e059..e0ea8e1 100644 --- a/buch/papers/zeta/main.tex +++ b/buch/papers/zeta/main.tex @@ -3,34 +3,16 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:zeta}} -\lhead{Thema} +\chapter{Riemannsche Zetafunktion\label{chapter:zeta}} +\lhead{Riemannsche Zetafunktion} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Raphael Unterer} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} +%TODO Einleitung -\input{papers/zeta/teil0.tex} -\input{papers/zeta/teil1.tex} -\input{papers/zeta/teil2.tex} -\input{papers/zeta/teil3.tex} +\input{papers/zeta/einleitung.tex} +\input{papers/zeta/zeta_gamma.tex} +\input{papers/zeta/analytic_continuation.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/zeta/teil0.tex b/buch/papers/zeta/teil0.tex deleted file mode 100644 index 56c0b1b..0000000 --- a/buch/papers/zeta/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{zeta:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{zeta:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/zeta/teil1.tex b/buch/papers/zeta/teil1.tex deleted file mode 100644 index 4017ee8..0000000 --- a/buch/papers/zeta/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{zeta:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{zeta:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{zeta:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{zeta:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{zeta:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/zeta/teil2.tex b/buch/papers/zeta/teil2.tex deleted file mode 100644 index 9e8a96e..0000000 --- a/buch/papers/zeta/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{zeta:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{zeta:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/zeta/teil3.tex b/buch/papers/zeta/teil3.tex deleted file mode 100644 index 6610cc3..0000000 --- a/buch/papers/zeta/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{zeta:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{zeta:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/zeta/zeta_gamma.tex b/buch/papers/zeta/zeta_gamma.tex new file mode 100644 index 0000000..59c8744 --- /dev/null +++ b/buch/papers/zeta/zeta_gamma.tex @@ -0,0 +1,53 @@ +\section{Zusammenhang mit Gammafunktion} \label{zeta:section:zusammenhang_mit_gammafunktion} +\rhead{Zusammenhang mit Gammafunktion} + +Dieser Abschnitt stellt die Verbindung zwischen der Gamma- und der Zetafunktion her. + +%TODO ref Gamma +Wenn in der Gammafunkion die Integrationsvariable $t$ substituieren mit $t = nu$ und $dt = n du$, dann können wir die Gleichung umstellen und erhalten den Zusammenhang mit der Zetafunktion +\begin{align} + \Gamma(s) + &= + \int_0^{\infty} t^{s-1} e^{-t} dt + \\ + &= + \int_0^{\infty} n^{s\cancel{-1}}u^{s-1} e^{-nu} \cancel{n}du + && + \text{Division durch }n^s + \\ + \frac{\Gamma(s)}{n^s} + &= + \int_0^{\infty} u^{s-1} e^{-nu}du + && + \text{Zeta durch Summenbildung } \sum_{n=1}^{\infty} + \\ + \Gamma(s) \zeta(s) + &= + \int_0^{\infty} u^{s-1} + \sum_{n=1}^{\infty}e^{-nu} + du. + \label{zeta:equation:zeta_gamma1} +\end{align} +Die Summe über $e^{-nu}$ können wir als geometrische Reihe schreiben und erhalten +\begin{align} + \sum_{n=1}^{\infty}e^{-u^n} + &= + \sum_{n=0}^{\infty}e^{-u^n} + - + 1 + \\ + &= + \frac{1}{1 - e^{-u}} - 1 + \\ + &= + \frac{1}{e^u - 1}. +\end{align} +Wenn wir dieses Resultat einsetzen in \eqref{zeta:equation:zeta_gamma1} und durch $\Gamma(s)$ teilen, erhalten wir +\begin{equation}\label{zeta:equation:zeta_gamma_final} + \zeta(s) + = + \frac{1}{\Gamma(s)} + \int_0^{\infty} + \frac{u^{s-1}}{e^u -1} + du. +\end{equation} -- cgit v1.2.1 From e26f12668c78fab5f0d8c5c9625396fd34970c82 Mon Sep 17 00:00:00 2001 From: runterer Date: Sat, 30 Apr 2022 22:39:22 +0200 Subject: Erster Entwurf der analytischen Fortsetzung geschrieben --- buch/papers/zeta/analytic_continuation.tex | 107 +++++++++++++++++++++++++++-- 1 file changed, 103 insertions(+), 4 deletions(-) (limited to 'buch') diff --git a/buch/papers/zeta/analytic_continuation.tex b/buch/papers/zeta/analytic_continuation.tex index 943647a..f5de6e7 100644 --- a/buch/papers/zeta/analytic_continuation.tex +++ b/buch/papers/zeta/analytic_continuation.tex @@ -110,7 +110,7 @@ Zunächst teilen wir nun das Integral aus \eqref{zeta:equation:integral1} auf al dx, \end{equation} wobei wir uns nun auf den ersten Teil konzentrieren werden. -Dabei setzen wir das Wissen aus \eqref{zeta:equation:integral2} ein und erhalten +Dabei setzen wir das Wissen aus \eqref{zeta:equation:psi} ein und erhalten \begin{align} \int_0^{1} x^{\frac{s}{2}-1} @@ -148,9 +148,9 @@ Dabei setzen wir das Wissen aus \eqref{zeta:equation:integral2} ein und erhalten x^{\frac{s}{2}-\frac{3}{2}} - x^{\frac{s}{2}-1} - dx. + dx. \label{zeta:equation:integral3} \end{align} -Dabei kann das zweite integral gelöst werden als +Dabei kann das zweite Integral gelöst werden als \begin{equation} \frac{1}{2} \int_0^1 @@ -161,5 +161,104 @@ Dabei kann das zweite integral gelöst werden als = \frac{1}{s(s-1)}. \end{equation} - +Das erste Integral aus \eqref{zeta:equation:integral3} mit $\psi \left(\frac{1}{x} \right)$ ist nicht lösbar in dieser Form. +Deshalb substituieren wir $x = \frac{1}{u}$ und $dx = -\frac{1}{u^2}du$. +Die untere Integralgrenze wechselt ebenfalls zu $x_0 = 0 \rightarrow u_0 = \infty$. +Dies ergibt +\begin{align} + \int_{\infty}^{1} + {\frac{1}{u}}^{\frac{s}{2}-\frac{3}{2}} + \psi(u) + \frac{-du}{u^2} + &= + \int_{1}^{\infty} + {\frac{1}{u}}^{\frac{s}{2}-\frac{3}{2}} + \psi(u) + \frac{du}{u^2} + \\ + &= + \int_{1}^{\infty} + x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} + \psi(x) + dx, +\end{align} +wobei wir durch Multiplikation mit $(-1)$ die Integralgrenzen tauschen dürfen. +Es ist zu beachten das diese Grenzen nun identisch mit den Grenzen des zweiten Integrals von \eqref{zeta:equation:integral2} sind. +Wir setzen beide Lösungen ein in Gleichung \eqref{zeta:equation:integral3} und erhalten +\begin{equation} + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + dx + = + \int_{1}^{\infty} + x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} + \psi(x) + dx, + + + \frac{1}{s(s-1)}. +\end{equation} +Dieses Resultat setzen wir wiederum ein in \eqref{zeta:equation:integral2}, um schlussendlich +\begin{align} + \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} + \zeta(s) + &= + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + dx + + + \int_1^{\infty} + x^{\frac{s}{2}-1} + \psi(x) + dx + \nonumber + \\ + &= + \frac{1}{s(s-1)} + + + \int_{1}^{\infty} + x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} + \psi(x) + dx, + + + \int_1^{\infty} + x^{\frac{s}{2}-1} + \psi(x) + dx + \\ + &= + \frac{1}{s(s-1)} + + + \int_{1}^{\infty} + \left( + x^{-\frac{s}{2}-\frac{1}{2}} + + + x^{\frac{s}{2}-1} + \right) + \psi(x) + dx + \\ + &= + \frac{-1}{s(1-s)} + + + \int_{1}^{\infty} + \left( + x^{\frac{1-s}{2}} + + + x^{\frac{s}{2}} + \right) + \frac{\psi(x)}{x} + dx, +\end{align} +zu erhalten. +Wenn wir dieses Resultat genau anschauen, erkennen wir dass sich nichts verändert wenn $s$ mit $1-s$ ersetzt wird. +Somit haben wir die analytische Fortsetzung gefunden als +\begin{equation}\label{zeta:equation:functional} + \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} + \zeta(s) + = + \frac{\Gamma \left( \frac{1-s}{2} \right)}{\pi^{\frac{1-s}{2}}} + \zeta(1-s). +\end{equation} -- cgit v1.2.1 From 2041283fe8afc6c80451208e239913f52f767d93 Mon Sep 17 00:00:00 2001 From: runterer Date: Sat, 30 Apr 2022 22:46:13 +0200 Subject: minor fix --- buch/papers/zeta/analytic_continuation.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'buch') diff --git a/buch/papers/zeta/analytic_continuation.tex b/buch/papers/zeta/analytic_continuation.tex index f5de6e7..bb95b92 100644 --- a/buch/papers/zeta/analytic_continuation.tex +++ b/buch/papers/zeta/analytic_continuation.tex @@ -194,7 +194,7 @@ Wir setzen beide Lösungen ein in Gleichung \eqref{zeta:equation:integral3} und \int_{1}^{\infty} x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} \psi(x) - dx, + dx + \frac{1}{s(s-1)}. \end{equation} @@ -220,7 +220,7 @@ Dieses Resultat setzen wir wiederum ein in \eqref{zeta:equation:integral2}, um s \int_{1}^{\infty} x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} \psi(x) - dx, + dx + \int_1^{\infty} x^{\frac{s}{2}-1} -- cgit v1.2.1 From ce72c8b27b09ecbf98a454f3b37019aaa948a57e Mon Sep 17 00:00:00 2001 From: Andrea Mozzini Vellen Date: Mon, 2 May 2022 16:02:40 +0200 Subject: Intro chapters --- buch/papers/kreismembran/main.tex | 23 ++---- buch/papers/kreismembran/teil0.tex | 16 +---- buch/papers/kreismembran/teil1.tex | 142 +++++++++++++++++++++++++------------ buch/papers/kreismembran/teil2.tex | 79 ++++++++++++--------- 4 files changed, 148 insertions(+), 112 deletions(-) (limited to 'buch') diff --git a/buch/papers/kreismembran/main.tex b/buch/papers/kreismembran/main.tex index 67b436c..eafec18 100644 --- a/buch/papers/kreismembran/main.tex +++ b/buch/papers/kreismembran/main.tex @@ -3,28 +3,19 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:kreismembran}} -\lhead{Thema} +\chapter{Schwingungen einer kreisförmligen Membran\label{chapter:kreismembran}} +\lhead{Schwingungen einer kreisförmligen Membran} \begin{refsection} -\chapterauthor{Hans Muster} - -Ein paar Hinweise für die korrekte Formatierung des Textes +\chapterauthor{Andrea Mozzini Vellen und Tim Tönz} \begin{itemize} \item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. +Tim ist ein snitch \item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. +ich dachte wir sind gute Freunden \item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. +du schuldest mir ein bier \item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. +auch ein gin tonic \end{itemize} \input{papers/kreismembran/teil0.tex} diff --git a/buch/papers/kreismembran/teil0.tex b/buch/papers/kreismembran/teil0.tex index e4b1711..1552259 100644 --- a/buch/papers/kreismembran/teil0.tex +++ b/buch/papers/kreismembran/teil0.tex @@ -3,20 +3,8 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 0\label{kreismembran:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{kreismembran:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. +\section{Einleitung\label{kreismembran:section:teil0}} +\rhead{Einleitung} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. diff --git a/buch/papers/kreismembran/teil1.tex b/buch/papers/kreismembran/teil1.tex index b715075..29a47a6 100644 --- a/buch/papers/kreismembran/teil1.tex +++ b/buch/papers/kreismembran/teil1.tex @@ -2,54 +2,102 @@ % teil1.tex -- Beispiel-File für das Paper % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{kreismembran:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt + +\section{Die Hankel Transformation \label{kreismembran:section:teil1}} +\rhead{Die Hankel Transformation} + +Hermann Hankel (1839-1873) war ein deutscher Mathematiker, der für seinen Beitrag zur mathematischen Analyse und insbesondere für seine namensgebende Transformation bekannt ist. +Diese Transformation tritt bei der Untersuchung von funktionen auf, die nur von der Enternung des Ursprungs abhängen. +Er studierte auch funktionen, jetzt Hankel- oder Bessel- Funktionen genannt, der dritten Art. +Die Hankel Transformation mit Bessel Funktionen al Kern taucht natürlich bei achsensymmetrischen Problemen auf, die in Zylindrischen Polarkoordinaten formuliert sind. +In diesem Kapitel werden die Theorie der Transformation und einige Eigenschaften der Grundoperationen erläutert. + +Wir führen die Definition der Hankel Transformation aus der zweidimensionalen Fourier Transformation und ihrer Umkehrung ein, die durch: +\begin{align} + \mathscr{F}\{f(x,y)\} & = F(k,l)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-i( \bm{\kappa}\cdot \mathbf{r})}f(x,y) dx dy,\label{equation:fourier_transform}\\ + \mathscr{F}^{-1}\{F(x,y)\} & = f(x,y)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{i(\bm{\kappa}\cdot \mathbf{r}))}F(k,l) dx dy \label{equation:inv_fourier_transform} +\end{align} +wo $\mathbf{r}=(x,y)$ und $\bm{\kappa}=(k,l)$. Wie bereits erwähnt, sind Polarkoordinaten für diese Art von Problemen am besten geeignet, also mit, $(x,y)=r(\cos\theta,\sin\theta)$ und $(k,l)=\kappa(\cos\phi,\sin\phi)$, findet man $\bm{\kappa}\cdot\mathbf{r}=\kappa r(\cos(\theta-\phi))$ und danach: +\begin{align} + F(k,\phi)=\frac{1}{2\pi}\int_{0}^{\infty}r dr \int_{0}^{2\pi}e^{-ikr\cos(\theta-\phi)}f(r,\theta) d\phi. + \label{equation:F_ohne_variable_wechsel} +\end{align} +Dann wird angenommen dass, $f(r,\theta)=e^{in\theta}f(r)$, was keine strenge Einschränkung ist, und es wird eine Änderung der Variabeln vorgenommen $\theta-\phi=\alpha-\frac{\pi}{2}$, um \ref{equation:F_ohne_variable_wechsel} zu reduzieren: +\begin{align} + F(k,\phi)=\frac{1}{2\pi}\int_{0}^{\infty}rf(r) dr \int_{\phi_{0}}^{2\pi+\phi_{0}}e^{in(\phi-\frac{\pi}{2})+i(n\alpha-kr\sin\alpha)} d\alpha, + \label{equation:F_ohne_bessel} +\end{align} +wo $\phi_{0}=(\frac{\pi}{2}-\phi)$. + +Unter Verwendung der Integral Darstellung der Besselfunktion vom Ordnung n +\begin{align} + J_n(\kappa r)=\frac{1}{2\pi}\int_{\phi_{0}}^{2\pi + \phi_{0}}e^{i(n\alpha-\kappa r \sin \alpha)} d\alpha + \label{equation:bessel_n_ordnung} +\end{align} +\eqref{equation:F_ohne_bessel} wird sie zu: +\begin{align} + F(k,\phi)&=e^{in(\phi-\frac{\pi}{2})}\int_{0}^{\infty}rJ_n(\kappa r) f(r) dr \label{equation:F_mit_bessel_step_1} \\ + &=e^{in(\phi-\frac{\pi}{2})}\tilde{f}_n(\kappa), + \label{equation:F_mit_bessel_step_2} +\end{align} +wo $\tilde{f}_n(\kappa)$ ist die \textit{Hankel Transformation} von $f(r)$ und ist formell definiert durch: +\begin{align} + \mathscr{H}_n\{f(r)\}=\tilde{f}_n(\kappa)=\int_{0}^{\infty}rJ_n(\kappa r) f(r) dr. + \label{equation:hankel} +\end{align} + +Ähnlich verhält es sich mit der inversen Fourier Transformation in Form von polaren Koordinaten unter der Annahme $f(r,\theta)=e^{in\theta}f(r)$ mit \ref{equation:F_mit_bessel_step_2}, wird die inverse Fourier Transformation \ref{equation:inv_fourier_transform}: + +\begin{align*} + e^{in\theta}f(r)&=\frac{1}{2\pi}\int_{0}^{\infty}\kappa d\kappa \int_{0}^{2\pi}e^{i\kappa r \cos (\theta - \phi)}F(\kappa,\phi) d\phi\\ + &= \frac{1}{2\pi}\int_{0}^{\infty}\kappa \tilde{f}_n(\kappa) d\kappa \int_{0}^{2\pi}e^{in(\phi - \frac{\pi}{2})- i\kappa r \cos (\theta - \phi)} d\phi, +\end{align*} +was durch den Wechsel der Variablen $\theta-\phi=-(\alpha+\frac{\pi}{2})$ und $\theta_0=-(\theta+\frac{\pi}{2})$, + +\begin{align} + &= \frac{1}{2\pi}\int_{0}^{\infty}\kappa \tilde{f}_n(\kappa) d\kappa \int_{\theta_0}^{2\pi+\theta_0}e^{in(\theta + \alpha - i\kappa r \sin\alpha)} d\alpha \nonumber \\ + &= e^{in\theta}\int_{0}^{\infty}\kappa J_n(\kappa r) \tilde{f}_n(\kappa) d\kappa,\quad \text{von \eqref{equation:bessel_n_ordnung}} +\end{align} + +Also, die inverse \textit{Hankel Transformation} ist so definiert: +\begin{align} + \mathscr{H}^{-1}_n\{\tilde{f}_n(\kappa)\}=f(r)=\int_{0}^{\infty}\kappa J_n(\kappa r) \tilde{f}_n(\kappa) d\kappa. + \label{equation:inv_hankel} +\end{align} + +Anstelle von $\tilde{f}_n(\kappa)$, wird häufig für die Hankel Transformation verwendet, indem die Ordnung angegeben wird. +\eqref{equation:hankel} und \eqref{equation:inv_hankel} Integralen existieren für eine grosse Klasse von Funktionen, die normalerweise in physikalischen Anwendungen benötigt werden. +Alternativ kann auch die berühmte Hankel Transformationsformel verwendet werden, + +\begin{align} + f(r) = \int_{0}^{\infty}\kappa J_n(\kappa r) d\kappa \int_{0}^{\infty} p J_n(\kappa p)f(p) dp, + \label{equation:hankel_integral_formula} +\end{align} +um die Hankel Transformation \eqref{equation:hankel} und ihre Inverse \eqref{equation:inv_hankel} zu definieren. +Insbesondere die Hankel Transformation der nullten Ordnung ($n=0$) und der ersten Ordnung ($n=1$) sind häufig nützlich, um Lösungen für Probleme mit der Laplace Gleichung in einer achsensymmetrischen zylindrischen Geometrie zu finden. + +\subsection{Operative Eigenschaften der Hankel Transformation\label{sub:op_properties_hankel}} +In diesem Kapitel werden die operativen Eigenschaften der Hankel Transformation aufgeführt. Der Beweis für ihre Gültigkeit wird jedoch nicht analysiert. + +\subsubsection{Skalierung \label{subsub:skalierung}} +Wenn $\mathscr{H}_n\{f(r)\}=\tilde{f}_n(\kappa)$, dann: + +\begin{equation} + \mathscr{H}_n\{f(ar)\}=\frac{1}{a^{2}}\tilde{f}_n \left(\frac{\kappa}{a}\right), \quad a>0. +\end{equation} + +\subsubsection{Persevalsche Relation \label{subsub:perseval}} +Wenn $\tilde{f}(\kappa)=\mathscr{H}_n\{f(r)\}$ und $\tilde{g}(\kappa)=\mathscr{H}_n\{g(r)\}$, dann: + \begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{kreismembran:equation1} + \int_{0}^{\infty}rf(r) dr = \int_{0}^{\infty}\kappa\tilde{f}(\kappa)\tilde{g}(\kappa) d\kappa. \end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{kreismembran:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{kreismembran:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{kreismembran:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. +\subsubsection{Hankel Transformationen von Ableitungen \label{subsub:ableitungen}} +Wenn $\tilde{f}_n(\kappa)=\mathscr{H}_n\{f(r)\}$, dann: +\begin{align} + &\mathscr{H}_n\{f'(r)\}=\frac{\kappa}{2n}\left[(n-1)\tilde{f}_{n+1}(\kappa)-(n+1)\tilde{f}_{n-1}(\kappa)\right], \quad n\geq1, \\ + &\mathscr{H}_1\{f'(r)\}=-\kappa \tilde{f}_0(\kappa), +\end{align} +bereitgestellt dass $[rf(r)]$ verschwindet als $r\to0$ und $r\to\infty=0$. \ No newline at end of file diff --git a/buch/papers/kreismembran/teil2.tex b/buch/papers/kreismembran/teil2.tex index 7ed217f..45357f2 100644 --- a/buch/papers/kreismembran/teil2.tex +++ b/buch/papers/kreismembran/teil2.tex @@ -1,40 +1,49 @@ % -% teil2.tex -- Beispiel-File für teil2 -% % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 2 -\label{kreismembran:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{kreismembran:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +\section{Lösung der partiellen Differentialgleichung + \label{kreismembran:section:teil2}} +\rhead{Lösung der partiellen Differentialgleichung} + +Wie im vorherigen Kapitel gezeigt, lautet die partielle Differentialgleichung, die die Schwingungen einer Membran beschreibt: +\begin{equation*} + \frac{1}{c^2}\frac{\partial^2u}{\partial t^2} = \Delta u +\end{equation*} +Da es sich um eine Kreisscheibe handelt, werden Polarkoordinaten verwendet, so dass sich der Laplaceoperator ergibt: +\begin{equation*} + \Delta + = + \frac{\partial^2}{\partial r^2} + + + \frac1r + \frac{\partial}{\partial r} + + + \frac{1}{r 2} + \frac{\partial^2}{\partial\varphi^2}. + \label{buch:pde:kreis:laplace} +\end{equation*} + +Es wird eine runde elastische Membran berücksichtigt, die den Gebietbereich $\Omega$ abdeckt und am Rand $\Gamma$ befestigt ist. +Es wird daher davon ausgegangen, dass die Membran aus einem homogenen Material von vernachlässigbarer Dicke gefertigt ist. +Die Membran kann verformt werden, aber innere elastische Kräfte wirken den Verformungen entgegen. Es wirken keine äusseren Kräfte. Es handelt sich somit von einer kreisförmligen eigespannten homogenen schwingenden Membran. + +Daher ist die Membranabweichung im Punkt $(r,\theta)$ $\in$ $\overline{\rm \Omega}$ zum Zeitpunkt $t$: +\begin{align*} + u: \overline{\rm \Omega} \times \mathbb{R}_{\geq 0} &\longrightarrow \mathbb{R}\\ + (r,\theta,t) &\longmapsto u(r,\theta,t) +\end{align*} +Da die Membran am Rand befestigt ist, kann es keine Schwingungen geben, so dass die \textit{Dirichlet-Randbedingung} gilt: +\begin{equation*} + u\big|_{\Gamma} = 0 +\end{equation*} + + +Um eine eindeutige Lösung bestimmen zu können, werden die folgenden Anfangsbedingungen festgelegt: + +\begin{align*} + u(r,\theta, 0) &:= f(x,y)\\ + \frac{\partial}{\partial t} u(r,\theta, 0) &:= g(x,y) +\end{align*} +An dieser Stelle könnte man zum Beispiel die bereits in Kapitel (TODO:refKAPITEL) vorgestellte Methode der Separation anwenden. Da es sich in diesem Fall jedoch um einem achsensymmetrischen Problem handelt, das in Polarkoordinaten formuliert ist, wird man die Transformationsmethode verwenden, insbesondere die Hankel Transformation. -- cgit v1.2.1 From 32093bf360c25dded5b3b02f97c5fe8d93dfcd2a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 3 May 2022 21:38:35 +0200 Subject: some fixes --- buch/chapters/070-orthogonalitaet/gaussquadratur.tex | 8 +++++--- buch/chapters/070-orthogonalitaet/sturm.tex | 2 +- 2 files changed, 6 insertions(+), 4 deletions(-) (limited to 'buch') diff --git a/buch/chapters/070-orthogonalitaet/gaussquadratur.tex b/buch/chapters/070-orthogonalitaet/gaussquadratur.tex index acfdb1a..2e43cec 100644 --- a/buch/chapters/070-orthogonalitaet/gaussquadratur.tex +++ b/buch/chapters/070-orthogonalitaet/gaussquadratur.tex @@ -263,7 +263,7 @@ werden können, muss auch = \int_{-1}^1 q(x)p(x)\,dx = -\sum_{i=0}^n q(x_i)p(x_i) +\sum_{i=0}^n A_iq(x_i)p(x_i) \] für jedes beliebige Polynom $q\in R_{n-1}$ gelten. Da man für $q$ die Interpolationspolynome $l_j(x)$ verwenden @@ -272,9 +272,11 @@ kann, den Grad $n-1$ haben, folgt 0 = \sum_{i=0}^n -l_j(x_i)p(x_i) +A_il_j(x_i)p(x_i) = -\sum_{i=0}^n \delta_{ij}p(x_i), +\sum_{i=0}^n A_i\delta_{ij}p(x_i) += +A_jp(x_j), \] die Stützstellen $x_i$ müssen also die Nullstellen des Polynoms $p(x)$ sein. diff --git a/buch/chapters/070-orthogonalitaet/sturm.tex b/buch/chapters/070-orthogonalitaet/sturm.tex index c9c9cc6..35054ab 100644 --- a/buch/chapters/070-orthogonalitaet/sturm.tex +++ b/buch/chapters/070-orthogonalitaet/sturm.tex @@ -375,7 +375,7 @@ automatisch für diese Funktionenfamilien. \subsubsection{Trigonometrische Funktionen} Die trigonometrischen Funktionen sind Eigenfunktionen des Operators $d^2/dx^2$, also eines Sturm-Liouville-Operators mit $p(x)=1$, $q(x)=0$ -und $w(x)=0$. +und $w(x)=1$. Auf dem Intervall $(-\pi,\pi)$ können wir die Randbedingungen \bgroup \renewcommand{\arraycolsep}{2pt} -- cgit v1.2.1 From 3259e5f02e8a1b8e9fd22038bbbbec2929f8105c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 4 May 2022 08:18:47 +0200 Subject: remove template paper --- buch/papers/common/addpapers.tex | 1 - buch/papers/common/paperlist | 1 - 2 files changed, 2 deletions(-) (limited to 'buch') diff --git a/buch/papers/common/addpapers.tex b/buch/papers/common/addpapers.tex index dd2b07a..eb353d7 100644 --- a/buch/papers/common/addpapers.tex +++ b/buch/papers/common/addpapers.tex @@ -3,7 +3,6 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\input{papers/000template/main.tex} \input{papers/lambertw/main.tex} \input{papers/fm/main.tex} \input{papers/parzyl/main.tex} diff --git a/buch/papers/common/paperlist b/buch/papers/common/paperlist index d4e5c20..f607279 100644 --- a/buch/papers/common/paperlist +++ b/buch/papers/common/paperlist @@ -1,4 +1,3 @@ -000template lambertw fm parzyl -- cgit v1.2.1 From a7a12c313b1a4fb528337eb354668e69d6d20942 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 11 May 2022 22:13:59 +0200 Subject: dreiecksgraphik --- buch/papers/nav/images/Makefile | 11 ++++++ buch/papers/nav/images/dreieck.tex | 68 ++++++++++++++++++++++++++++++++++++++ buch/papers/nav/images/macros.tex | 54 ++++++++++++++++++++++++++++++ buch/papers/nav/images/pk.m | 55 ++++++++++++++++++++++++++++++ 4 files changed, 188 insertions(+) create mode 100644 buch/papers/nav/images/Makefile create mode 100644 buch/papers/nav/images/dreieck.tex create mode 100644 buch/papers/nav/images/macros.tex create mode 100644 buch/papers/nav/images/pk.m (limited to 'buch') diff --git a/buch/papers/nav/images/Makefile b/buch/papers/nav/images/Makefile new file mode 100644 index 0000000..a0d7b34 --- /dev/null +++ b/buch/papers/nav/images/Makefile @@ -0,0 +1,11 @@ +# +# Makefile to build images +# +# (c) 2022 +# + +dreieck.pdf: dreieck.tex dreieckdata.tex macros.tex + pdflatex dreieck.tex + +dreieckdata.tex: pk.m + octave pk.m diff --git a/buch/papers/nav/images/dreieck.tex b/buch/papers/nav/images/dreieck.tex new file mode 100644 index 0000000..55f6a81 --- /dev/null +++ b/buch/papers/nav/images/dreieck.tex @@ -0,0 +1,68 @@ +% +% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +\def\punktbeschriftung{ + \node at (A) [above] {$A$}; + \node at (B) [left] {$B$}; + \node at (C) [right] {$C$}; + \node at (P) [below] {$P$}; +} + +\winkelKappa{gray} + +\winkelAlpha{red} +\winkelGamma{blue} +\winkelBeta{darkgreen} + +\winkelOmega{gray} +\winkelBetaEins{brown} + +\seiteC{black} +\seiteB{black} +\seiteA{black} + +\seiteL{gray} +\seitePB{gray} +\seitePC{gray} + +\draw[line width=1.4pt] \kanteAB; +\draw[line width=1.4pt] \kanteAC; +\draw[color=gray] \kanteAP; +\draw[line width=1.4pt] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[color=gray] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{gray}; + +\punktbeschriftung + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/macros.tex b/buch/papers/nav/images/macros.tex new file mode 100644 index 0000000..69a620d --- /dev/null +++ b/buch/papers/nav/images/macros.tex @@ -0,0 +1,54 @@ +\def\winkelAlpha#1{ + \begin{scope} + \clip (A) circle[radius=1.1]; + \fill[color=#1!20] \kanteAB -- \kanteCA -- cycle; + \end{scope} + \node[color=#1] at ($(A)+(222:0.82)$) {$\alpha$}; +} + +\def\winkelOmega#1{ + \begin{scope} + \clip (A) circle[radius=0.7]; + \fill[color=#1!20] \kanteAP -- \kanteCA -- cycle; + \end{scope} + \node[color=#1] at ($(A)+(285:0.50)$) {$\omega$}; +} + +\def\winkelGamma#1{ + \begin{scope} + \clip (C) circle[radius=1.0]; + \fill[color=#1!20] \kanteCA -- \kanteBC -- cycle; + \end{scope} + \node[color=#1] at ($(C)+(155:0.60)$) {$\gamma$}; +} + +\def\winkelKappa#1{ + \begin{scope} + \clip (B) circle[radius=1.2]; + \fill[color=#1!20] \kanteBP -- \kanteAB -- cycle; + \end{scope} + \node[color=#1] at ($(B)+(15:1.00)$) {$\kappa$}; +} + +\def\winkelBeta#1{ + \begin{scope} + \clip (B) circle[radius=0.8]; + \fill[color=#1!20] \kanteBC -- \kanteAB -- cycle; + \end{scope} + \node[color=#1] at ($(B)+(35:0.40)$) {$\beta$}; +} + +\def\winkelBetaEins#1{ + \begin{scope} + \clip (B) circle[radius=0.8]; + \fill[color=#1!20] \kanteBP -- \kanteCB -- cycle; + \end{scope} + \node[color=#1] at ($(B)+(330:0.60)$) {$\beta_1$}; +} + +\def\seiteC#1{ \node[color=#1] at (-1.9,5.9) {$c$}; } +\def\seiteB#1{ \node[color=#1] at (3.2,6.5) {$b$}; } +\def\seiteL#1{ \node[color=#1] at (-0.2,4.5) {$l$}; } +\def\seiteA#1{ \node[color=#1] at (2,3) {$a$}; } +\def\seitePB#1{ \node[color=#1] at (-2.1,1) {$p_b$}; } +\def\seitePC#1{ \node[color=#1] at (2.5,1.5) {$p_c$}; } diff --git a/buch/papers/nav/images/pk.m b/buch/papers/nav/images/pk.m new file mode 100644 index 0000000..6e89e9a --- /dev/null +++ b/buch/papers/nav/images/pk.m @@ -0,0 +1,55 @@ +# +# pk.m -- Punkte und Kanten für sphärisches Dreieck +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +A = [ 1, 8 ]; +B = [ -3, 3 ]; +C = [ 4, 4 ]; +P = [ 0, 0 ]; + +global fn; +fn = fopen("dreieckdata.tex", "w"); + +fprintf(fn, "\\coordinate (P) at (%.4f,%.4f);\n", P(1,1), P(1,2)); +fprintf(fn, "\\coordinate (A) at (%.4f,%.4f);\n", A(1,1), A(1,2)); +fprintf(fn, "\\coordinate (B) at (%.4f,%.4f);\n", B(1,1), B(1,2)); +fprintf(fn, "\\coordinate (C) at (%.4f,%.4f);\n", C(1,1), C(1,2)); + +function retval = seite(A, B, l, nameA, nameB) + global fn; + d = fliplr(B - A); + d(1, 2) = -d(1, 2); + # Zentrum + C = 0.5 * (A + B) + l * d; + # Radius: + r = hypot(C(1,1)-A(1,1), C(1,2)-A(1,2)) + # Winkel von + winkelvon = atan2(A(1,2)-C(1,2),A(1,1)-C(1,1)); + # Winkel bis + winkelbis = atan2(B(1,2)-C(1,2),B(1,1)-C(1,1)); + if (abs(winkelvon - winkelbis) > pi) + if (winkelbis < winkelvon) + winkelbis = winkelbis + 2 * pi + else + winkelvon = winkelvon + 2 * pi + end + end + # Kurve + fprintf(fn, "\\def\\kante%s%s{(%.4f,%.4f) arc (%.5f:%.5f:%.4f)}\n", + nameA, nameB, + A(1,1), A(1,2), winkelvon * 180 / pi, winkelbis * 180 / pi, r); + fprintf(fn, "\\def\\kante%s%s{(%.4f,%.4f) arc (%.5f:%.5f:%.4f)}\n", + nameB, nameA, + B(1,1), B(1,2), winkelbis * 180 / pi, winkelvon * 180 / pi, r); +endfunction + +seite(A, B, -1, "A", "B"); +seite(A, C, 1, "A", "C"); +seite(A, P, -1, "A", "P"); +seite(B, C, -2, "B", "C"); +seite(B, P, -1, "B", "P"); +seite(C, P, 2, "C", "P"); + +fclose(fn); -- cgit v1.2.1 From 1067af52a6b066174b7306e96766e9e4e11cbac7 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 11 May 2022 22:14:11 +0200 Subject: dreiecksdaten --- buch/papers/nav/images/dreieckdata.tex | 16 ++++++++++++++++ 1 file changed, 16 insertions(+) create mode 100644 buch/papers/nav/images/dreieckdata.tex (limited to 'buch') diff --git a/buch/papers/nav/images/dreieckdata.tex b/buch/papers/nav/images/dreieckdata.tex new file mode 100644 index 0000000..c0fb720 --- /dev/null +++ b/buch/papers/nav/images/dreieckdata.tex @@ -0,0 +1,16 @@ +\coordinate (P) at (0.0000,0.0000); +\coordinate (A) at (1.0000,8.0000); +\coordinate (B) at (-3.0000,3.0000); +\coordinate (C) at (4.0000,4.0000); +\def\kanteAB{(1.0000,8.0000) arc (114.77514:167.90524:7.1589)} +\def\kanteBA{(-3.0000,3.0000) arc (167.90524:114.77514:7.1589)} +\def\kanteAC{(1.0000,8.0000) arc (63.43495:10.30485:5.5902)} +\def\kanteCA{(4.0000,4.0000) arc (10.30485:63.43495:5.5902)} +\def\kanteAP{(1.0000,8.0000) arc (146.30993:199.44003:9.0139)} +\def\kantePA{(0.0000,0.0000) arc (199.44003:146.30993:9.0139)} +\def\kanteBC{(-3.0000,3.0000) arc (-95.90614:-67.83365:14.5774)} +\def\kanteCB{(4.0000,4.0000) arc (-67.83365:-95.90614:14.5774)} +\def\kanteBP{(-3.0000,3.0000) arc (-161.56505:-108.43495:4.7434)} +\def\kantePB{(0.0000,0.0000) arc (-108.43495:-161.56505:4.7434)} +\def\kanteCP{(4.0000,4.0000) arc (-30.96376:-59.03624:11.6619)} +\def\kantePC{(0.0000,0.0000) arc (-59.03624:-30.96376:11.6619)} -- cgit v1.2.1 From b7ee1c1a6836f30d2267cfc9e6dbfa206b2cb737 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Patrik=20M=C3=BCller?= Date: Thu, 12 May 2022 18:19:49 +0200 Subject: Derive Laguerre-Polynomials from Laguerre-ODE, proof orthogonality with Sturm-Liouville --- buch/.gitignore | 37 ---- buch/papers/laguerre/Makefile | 6 +- buch/papers/laguerre/Makefile.inc | 4 +- buch/papers/laguerre/definition.tex | 160 ++++++++------- buch/papers/laguerre/eigenschaften.tex | 94 ++++++++- buch/papers/laguerre/images/laguerre_polynomes.pdf | Bin 0 -> 16239 bytes buch/papers/laguerre/scripts/gamma_approx.ipynb | 224 +++++++++------------ buch/papers/laguerre/scripts/laguerre_plot.py | 103 ++++++++-- .../laguerre/scripts/lanczos_approximation.py | 47 ----- buch/papers/laguerre/scripts/quadrature_gama.py | 178 ---------------- buch/standalone.tex | 35 ---- 11 files changed, 361 insertions(+), 527 deletions(-) delete mode 100644 buch/.gitignore create mode 100644 buch/papers/laguerre/images/laguerre_polynomes.pdf delete mode 100644 buch/papers/laguerre/scripts/lanczos_approximation.py delete mode 100644 buch/papers/laguerre/scripts/quadrature_gama.py delete mode 100644 buch/standalone.tex (limited to 'buch') diff --git a/buch/.gitignore b/buch/.gitignore deleted file mode 100644 index 4d056e5..0000000 --- a/buch/.gitignore +++ /dev/null @@ -1,37 +0,0 @@ -*.acn -*.acr -*.alg -*.aux -*.bbl -*.blg -*.dvi -*.fdb_latexmk -*.glg -*.glo -*.gls -*.idx -*.ilg -*.ind -*.ist -*.lof -*.log -*.lot -*.maf -*.mtc -*.mtc0 -*.nav -*.nlo -*.out -*.pdfsync -.vscode/* -*.fls -*.xdv -*.ps -*.snm -*.synctex.gz -*.toc -*.vrb -*.xdy -*.tdo -*-blx.bib -*.synctex \ No newline at end of file diff --git a/buch/papers/laguerre/Makefile b/buch/papers/laguerre/Makefile index 606d7e1..0f0985a 100644 --- a/buch/papers/laguerre/Makefile +++ b/buch/papers/laguerre/Makefile @@ -4,6 +4,8 @@ # (c) 2020 Prof Dr Andreas Mueller # -images: - @echo "no images to be created in laguerre" +images: images/laguerre_polynomes.pdf + +images/laguerre_polynomes.pdf: scripts/laguerre_plot.py + python3 scripts/laguerre_plot.py diff --git a/buch/papers/laguerre/Makefile.inc b/buch/papers/laguerre/Makefile.inc index 1eb5034..aae51f9 100644 --- a/buch/papers/laguerre/Makefile.inc +++ b/buch/papers/laguerre/Makefile.inc @@ -9,8 +9,6 @@ dependencies-laguerre = \ papers/laguerre/references.bib \ papers/laguerre/definition.tex \ papers/laguerre/eigenschaften.tex \ - papers/laguerre/quadratur.tex \ - papers/laguerre/transformation.tex \ - papers/laguerre/wasserstoff.tex + papers/laguerre/quadratur.tex diff --git a/buch/papers/laguerre/definition.tex b/buch/papers/laguerre/definition.tex index 84a26cf..edd2b7b 100644 --- a/buch/papers/laguerre/definition.tex +++ b/buch/papers/laguerre/definition.tex @@ -4,11 +4,11 @@ % (c) 2022 Patrik Müller, Ostschweizer Fachhochschule % \section{Definition -\label{laguerre:section:definition}} + \label{laguerre:section:definition}} \rhead{Definition} -Die Laguerre-Differentialgleichung ist gegeben durch +Die verallgemeinerte Laguerre-Differentialgleichung ist gegeben durch \begin{align} -x y''(x) + (1 - x) y'(x) + n y(x) +x y''(x) + (\nu + 1 - x) y'(x) + n y(x) = 0 , \quad @@ -18,22 +18,27 @@ x \in \mathbb{R} . \label{laguerre:dgl} \end{align} -Zur Lösung der Gleichung \eqref{laguerre:dgl} -verwenden wir einen Potenzreihenansatz. +Hier wird die verallgemeinerte Laguerre-Differentialgleichung verwendet, +weil die Lösung gleich berechnet werden kann, +aber man zusätzlich die Lösung für den allgmeinen Fall erhält. +Zur Lösung der Gleichung \eqref{laguerre:dgl} verwenden wir einen +Potenzreihenansatz. +Da wir bereits wissen, dass die Lösung orthogonale Polynome sind, +erscheint dieser Ansatz sinnvoll. Setzt man nun den Ansatz \begin{align*} -y(x) -&= +y(x) + & = \sum_{k=0}^\infty a_k x^k \\ y'(x) -& = + & = \sum_{k=1}^\infty k a_k x^{k-1} = \sum_{k=0}^\infty (k+1) a_{k+1} x^k \\ y''(x) -&= + & = \sum_{k=2}^\infty k (k-1) a_k x^{k-2} = \sum_{k=1}^\infty (k+1) k a_{k+1} x^{k-1} @@ -41,98 +46,109 @@ y''(x) in die Differentialgleichung ein, erhält man: \begin{align*} \sum_{k=1}^\infty (k+1) k a_{k+1} x^k -+ \sum_{k=0}^\infty (k+1) a_{k+1} x^k -- \sum_{k=0}^\infty k a_k x^k -+ n \sum_{k=0}^\infty a_k x^k -&= -0\\ -\sum_{k=0}^\infty -\left[ (k+1) k a_{k+1} + (k+1) a_{k+1} - k a_k + n a_k \right] x^k -&= ++ +(\nu + 1)\sum_{k=0}^\infty (k+1) a_{k+1} x^k +- +\sum_{k=0}^\infty k a_k x^k ++ +n \sum_{k=0}^\infty a_k x^k + & = +0 \\ +\sum_{k=1}^\infty +\left[ (k+1) k a_{k+1} + (\nu + 1)(k+1) a_{k+1} - k a_k + n a_k \right] x^k + & = 0. \end{align*} Daraus lässt sich die Rekursionsbeziehung \begin{align*} a_{k+1} -&= -\frac{k-n}{(k+1) ^ 2} a_k + & = +\frac{k-n}{(k+1) (k + \nu + 1)} a_k \end{align*} ableiten. -Für ein konstantes $n$ erhalten wir als Potenzreihenlösung ein Polynom vom Grad $n$, +Für ein konstantes $n$ erhalten wir als Potenzreihenlösung ein Polynom vom Grad +$n$, denn für $k=n$ wird $a_{n+1} = 0$ und damit auch $a_{n+2}=a_{n+3}=\ldots=0$. -Aus der Rekursionsbeziehung ist zudem ersichtlich, +Aus der Rekursionsbeziehung ist zudem ersichtlich, dass $a_0 \neq 0$ beliebig gewählt werden kann. -Wählen wir nun $c_0 = 1$, dann folgt für die Koeffizienten $a_1, a_2, a_3$ +Wählen wir nun $a_0 = 1$, dann folgt für die Koeffizienten $a_1, a_2, a_3$ \begin{align*} -a_1 -= --\frac{n}{1^2} -,&& -a_2 -= -\frac{(n-1)n}{1^2 2^2} -,&& +a_1 += +-\frac{n}{1 \cdot (\nu + 1)} +, & & +a_2 += +\frac{(n-1)n}{1 \cdot 2 \cdot (\nu + 1)(\nu + 2)} +, & & a_3 = --\frac{(n-2)(n-1)n}{1^2 2^2 3^2} +-\frac{(n-2)(n-1)n}{1 \cdot 2 \cdot 3 \cdot (\nu + 1)(\nu + 2)(\nu + 3)} \end{align*} und allgemein \begin{align*} -k&\leq n: -& -a_k -&= -(-1)^k \frac{n!}{(n-k)!} \frac{1}{(k!)^2} -= -\frac{(-1)^k}{k!} -\begin{pmatrix} -n -\\ k -\end{pmatrix} + & \leq +n: + & +a_k + & = +(-1)^k \frac{n!}{(n-k)!} \frac{1}{k!(\nu + 1)_k} += +\frac{(-1)^k}{(\nu + 1)_k} \binom{n}{k} \\ -k&>n: -& +k & >n: + & a_k -&= + & = 0. \end{align*} -Somit haben wir die Laguerre-Polynome $L_n(x)$ erhalten: +Somit erhalten wir für $\nu = 0$ die Laguerre-Polynome \begin{align} L_n(x) = -\sum_{k=0}^{n} -\frac{(-1)^k}{k!} -\begin{pmatrix} -n \\ -k -\end{pmatrix} -x^k +\sum_{k=0}^{n} \frac{(-1)^k}{k!} \binom{n}{k} x^k \label{laguerre:polynom} \end{align} - -\subsection{Assoziierte Laguerre-Polynome -\label{laguerre:subsection:assoz_laguerre} -} +und mit $\nu \in \mathbb{R}$ die verallgemeinerten Laguerre-Polynome \begin{align} -x y''(x) + (\alpha + 1 - x) y'(x) + n y(x) +L_n^\nu(x) = -0 -\label{laguerre:generell_dgl} +\sum_{k=0}^{n} \frac{(-1)^k}{(\nu + 1)_k} \binom{n}{k} x^k. +\label{laguerre:allg_polynom} \end{align} - -\begin{align} -L_n^\alpha (x) +Durch die analytische Fortsetzung erhalten wir zudem noch die zweite Lösung der +Differentialgleichung mit der Form +\begin{align*} +\Xi_n(x) = -\sum_{k=0}^{n} -\frac{(-1)^k}{k!} -\begin{pmatrix} -n + \alpha \\ -n - k -\end{pmatrix} -x^k -\label{laguerre:polynom} -\end{align} +L_n(x) \ln(x) + \sum_{k=1}^\infty d_k x^k +\end{align*} +Nach einigen mühsamen Rechnungen, +die den Rahmen dieses Kapitel sprengen würden, +erhalten wir +\begin{align*} +\Xi_n += +L_n(x) \ln(x) ++ +\sum_{k=1}^n \frac{(-1)^k}{k!} \binom{n}{k} +(\alpha_{n-k} - \alpha_n - 2 \alpha_k)x^k ++ +(-1)^n \sum_{k=1}^\infty \frac{(k-1)!n!}{((n+k)!)^2} x^{n+k}, +\end{align*} +wobei $\alpha_0 = 0$ und $\alpha_k =\sum_{i=1}^k i^{-1}$, +$\forall k \in \mathbb{N}$. +Die Laguerre-Polynome von Grad $0$ bis $7$ sind in +Abbildung~\ref{laguerre:fig:polyeval} dargestellt. +\begin{figure} +\centering +\includegraphics[width=0.7\textwidth]{% + papers/laguerre/images/laguerre_polynomes.pdf% +} +\caption{Laguerre-Polynome vom Grad $0$ bis $7$} +\label{laguerre:fig:polyeval} +\end{figure} % https://www.math.kit.edu/iana1/lehre/hm3phys2012w/media/laguerre.pdf % http://www.physics.okayama-u.ac.jp/jeschke_homepage/E4/kapitel4.pdf diff --git a/buch/papers/laguerre/eigenschaften.tex b/buch/papers/laguerre/eigenschaften.tex index b7597e5..c589c92 100644 --- a/buch/papers/laguerre/eigenschaften.tex +++ b/buch/papers/laguerre/eigenschaften.tex @@ -4,5 +4,95 @@ % (c) 2022 Patrik Müller, Ostschweizer Fachhochschule % \section{Eigenschaften -\label{laguerre:section:eigenschaften}} -\rhead{Eigenschaften} \ No newline at end of file + \label{laguerre:section:eigenschaften}} +\rhead{Eigenschaften} + +\subsection{Orthogonalität} +Wenn wir die Laguerre\--Differentialgleichung in ein +Sturm\--Liouville\--Problem umwandeln können, haben wir bewiesen, dass es sich +bei +den Laguerre\--Polynomen um orthogonale Polynome handelt (siehe +Abschnitt~\ref{buch:integrale:subsection:sturm-liouville-problem}). +Der Sturm-Liouville-Operator hat die Form +\begin{align} +S += +\frac{1}{w(x)} \left(-\frac{d}{dx}p(x) \frac{d}{dx} + q(x) \right). +\label{laguerre:slop} +\end{align} +Aus der Beziehung +\begin{align} +S + & = +\Lambda +\nonumber +\\ +\frac{1}{w(x)} \left(-\frac{d}{dx}p(x) \frac{d}{dx} + q(x) \right) + & = +x \frac{d^2}{dx^2} + (\nu + 1 - x) \frac{d}{dx} +\label{laguerre:sl-lag} +\end{align} +lässt sich sofort erkennen, dass $q(x) = 0$. +Ausserdem ist ersichtlich, dass $p(x)$ die Differentialgleichung +\begin{align*} +x \frac{dp}{dx} += +-(\nu + 1 - x) p, +\end{align*} +erfüllen muss. +Durch Separation erhalten wir dann +\begin{align*} +\int \frac{dp}{p} + & = +-\int \frac{\nu + 1 - x}{x}dx +\\ +\log p + & = +-\log \nu + 1 - x + C +\\ +p(x) + & = +-C x^{\nu + 1} e^{-x} +\end{align*} +Eingefügt in Gleichung~\eqref{laguerre:sl-lag} erhalten wir +\begin{align*} +\frac{C}{w(x)} +\left( +x^{\nu+1} e^{-x} \frac{d^2}{dx^2} + +(\nu + 1 - x) x^{\nu} e^{-x} \frac{d}{dx} +\right) += +x \frac{d^2}{dx^2} + (\nu + 1 - x) \frac{d}{dx}. +\end{align*} +Mittels Koeffizientenvergleich kann nun abgelesen werden, dass $w(x) = x^\nu +e^{-x}$ und $C=1$ mit $\nu > -1$. +Die Gewichtsfunktion $w(x)$ wächst für $x\rightarrow-\infty$ sehr schnell an, +deshalb ist die Laguerre-Gewichtsfunktion nur geeignet für den +Definitionsbereich $(0, \infty)$. +Bleibt nur noch sicherzustellen, dass die Randbedingungen, +\begin{align} +k_0 y(0) + h_0 p(0)y'(0) + & = +0 +\label{laguerre:sllag_randa} +\\ +k_\infty y(\infty) + h_\infty p(\infty) y'(\infty) + & = +0 +\label{laguerre:sllag_randb} +\end{align} +mit $|k_i|^2 + |h_i|^2 \neq 0,\,\forall i \in \{0, \infty\}$, erfüllt sind. +Am linken Rand (Gleichung~\eqref{laguerre:sllag_randa}) kann $y(0) = 1$, $k_0 = +0$ und $h_0 = 1$ verwendet werden, +was auch die Laguerre-Polynome ergeben haben. +Für den rechten Rand ist die Bedingung (Gleichung~\eqref{laguerre:sllag_randb}) +\begin{align*} +\lim_{x \rightarrow \infty} p(x) y'(x) + & = +\lim_{x \rightarrow \infty} -x^{\nu + 1} e^{-x} y'(x) += +0 +\end{align*} +für beliebige Polynomlösungen erfüllt für $k_\infty=0$ und $h_\infty=1$. +Damit können wir schlussfolgern, dass die Laguerre-Polynome orthogonal +bezüglich des Skalarproduktes mit der Laguerre\--Gewichtsfunktion sind. diff --git a/buch/papers/laguerre/images/laguerre_polynomes.pdf b/buch/papers/laguerre/images/laguerre_polynomes.pdf new file mode 100644 index 0000000..3976bc7 Binary files /dev/null and b/buch/papers/laguerre/images/laguerre_polynomes.pdf differ diff --git a/buch/papers/laguerre/scripts/gamma_approx.ipynb b/buch/papers/laguerre/scripts/gamma_approx.ipynb index 9a1fee6..44f3abd 100644 --- a/buch/papers/laguerre/scripts/gamma_approx.ipynb +++ b/buch/papers/laguerre/scripts/gamma_approx.ipynb @@ -28,13 +28,13 @@ " =\n", " \\frac{(n!)^2}{(2n)!} f^{(2n)}(\\xi) \n", " = \n", - " (-2n + z)_{2n} \\frac{(n!)^2}{(2n)!} \\xi^{z - 2n - 1}\n", + " \\frac{(-2n + z)_{2n}}{(z-m)_m} \\frac{(n!)^2}{(2n)!} \\xi^{z + m - 2n - 1}\n", "$$" ] }, { "cell_type": "code", - "execution_count": 11, + "execution_count": null, "metadata": {}, "outputs": [], "source": [ @@ -48,7 +48,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": null, "metadata": {}, "outputs": [], "source": [ @@ -86,35 +86,36 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": null, "metadata": {}, "outputs": [], "source": [ - "zeros = np.array(\n", - " [\n", - " 1.70279632305101000e-1,\n", - " 9.03701776799379912e-1,\n", - " 2.25108662986613069e0,\n", - " 4.26670017028765879e0,\n", - " 7.04590540239346570e0,\n", - " 1.07585160101809952e1,\n", - " 1.57406786412780046e1,\n", - " 2.28631317368892641e1,\n", - " ]\n", - ")\n", - "\n", - "weights = np.array(\n", - " [\n", - " 3.69188589341637530e-1,\n", - " 4.18786780814342956e-1,\n", - " 1.75794986637171806e-1,\n", - " 3.33434922612156515e-2,\n", - " 2.79453623522567252e-3,\n", - " 9.07650877335821310e-5,\n", - " 8.48574671627253154e-7,\n", - " 1.04800117487151038e-9,\n", - " ]\n", - ")\n", + "zeros, weights = np.polynomial.laguerre.laggauss(8)\n", + "# zeros = np.array(\n", + "# [\n", + "# 1.70279632305101000e-1,\n", + "# 9.03701776799379912e-1,\n", + "# 2.25108662986613069e0,\n", + "# 4.26670017028765879e0,\n", + "# 7.04590540239346570e0,\n", + "# 1.07585160101809952e1,\n", + "# 1.57406786412780046e1,\n", + "# 2.28631317368892641e1,\n", + "# ]\n", + "# )\n", + "\n", + "# weights = np.array(\n", + "# [\n", + "# 3.69188589341637530e-1,\n", + "# 4.18786780814342956e-1,\n", + "# 1.75794986637171806e-1,\n", + "# 3.33434922612156515e-2,\n", + "# 2.79453623522567252e-3,\n", + "# 9.07650877335821310e-5,\n", + "# 8.48574671627253154e-7,\n", + "# 1.04800117487151038e-9,\n", + "# ]\n", + "# )\n", "\n", "\n", "def pochhammer(z, n):\n", @@ -149,7 +150,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": null, "metadata": {}, "outputs": [], "source": [ @@ -183,27 +184,32 @@ "### Test with real values" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Empirische Tests zeigen:\n", + "- $n=4 \\Rightarrow m=6$\n", + "- $n=5 \\Rightarrow m=7$ oder $m=8$\n", + "- $n=6 \\Rightarrow m=9$\n", + "- $n=7 \\Rightarrow m=10$\n", + "- $n=8 \\Rightarrow m=11$ oder $m=12$\n", + "- $n=9 \\Rightarrow m=13$\n", + "- $n=10 \\Rightarrow m=14$\n", + "- $n=11 \\Rightarrow m=15$ oder $m=16$\n", + "- $n=12 \\Rightarrow m=17$\n", + "- $n=13 \\Rightarrow m=18 \\Rightarrow $ Beginnt numerisch instabil zu werden \n" + ] + }, { "cell_type": "code", - "execution_count": 60, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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", - "text/plain": [ - "
" - ] - }, - "metadata": { - "needs_background": "light" - }, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ - "targets = np.arange(8, 15)\n", - "mean_targets = ((11, 12),)\n", + "zeros, weights = np.polynomial.laguerre.laggauss(12)\n", + "targets = np.arange(16, 21)\n", + "mean_targets = ((16, 17),)\n", "x = np.linspace(EPSILON, 1 - EPSILON, 101)\n", "_, axs = plt.subplots(\n", " 2, sharex=True, clear=True, constrained_layout=True, figsize=(12, 12)\n", @@ -216,14 +222,18 @@ " axs[0].plot(x, rel_error_mean, label=mean_target)\n", " axs[1].semilogy(x, np.abs(rel_error_mean), label=mean_target)\n", "\n", + "mins = []\n", + "maxs = []\n", "for target in targets:\n", " rel_error = evaluate(x, target)\n", + " mins.append(np.min(np.abs(rel_error[(0.1 <= x) & (x <= 0.9)])))\n", + " maxs.append(np.max(np.abs(rel_error)))\n", " axs[0].plot(x, rel_error, label=target)\n", " axs[1].semilogy(x, np.abs(rel_error), label=target)\n", "# axs[0].set_ylim(*(np.array([-1, 1]) * 3.5e-8))\n", "\n", "axs[0].set_xlim(x[0], x[-1])\n", - "axs[1].set_ylim(1e-10, 2e-7)\n", + "axs[1].set_ylim(np.min(mins), 1.04*np.max(maxs))\n", "for ax in axs:\n", " ax.legend()\n", " ax.grid(which=\"both\")\n" @@ -231,46 +241,11 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "(-7.5, 25.0)" - ] - }, - "execution_count": 7, - "metadata": {}, - "output_type": "execute_result" - }, - { - "data": { - "image/png": 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", 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", - "text/plain": [ - "
" - ] - }, - "metadata": { - "needs_background": "light" - }, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ - "targets = (11, 12)\n", + "targets = (16, 17)\n", "xmax = 15\n", "x = np.linspace(-xmax + EPSILON, xmax - EPSILON, 1000)\n", "\n", @@ -289,7 +264,7 @@ "axs[1].semilogy(x, np.abs(rel_error_simple), label=targets[-1])\n", "axs[0].set_xlim(x[0], x[-1])\n", "# axs[0].set_ylim(*(np.array([-1, 1]) * 4.2e-8))\n", - "axs[1].set_ylim(1e-10, 5e-8)\n", + "# axs[1].set_ylim(1e-10, 5e-8)\n", "for ax in axs:\n", " ax.legend()\n", "\n", @@ -309,24 +284,11 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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", - "text/plain": [ - "
" - ] - }, - "metadata": { - "needs_background": "light" - }, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ - "targets = (11, 12)\n", + "targets = (16, 17)\n", "vals = np.linspace(-5 + EPSILON, 5, 100)\n", "x, y = np.meshgrid(vals, vals)\n", "mesh = x + 1j * y\n", @@ -361,45 +323,47 @@ }, { "cell_type": "code", - "execution_count": 67, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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", - "text/plain": [ - "
" - ] - }, - "metadata": { - "needs_background": "light" - }, - "output_type": "display_data" - } - ], + "outputs": [], "source": [ - "# TODO: Macht kei Sinn!\n", - "n = 8\n", - "ms = np.arange(4, 5)\n", - "xi = np.linspace(EPSILON, 20, 201)[:, None]\n", - "z = np.arange(6, 16)[None]+0.1\n", - "c = scipy.special.factorial(n) ** 2 / scipy.special.factorial(2 * n)\n", - "\n", + "z = 0.5\n", + "ns = [4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12] # np.arange(4, 13)\n", + "ms = np.arange(6, 18)\n", + "xi = np.logspace(0, 2, 201)[:, None]\n", + "lanczos = eval_lanczos([z])[0]\n", "\n", "_, ax = plt.subplots(clear=True, constrained_layout=True, figsize=(12, 8))\n", "ax.grid(1)\n", - "for m, color in zip(ms, ['r', 'b', 'g', 'c', 'm', 'y']):\n", + "for n, m in zip(ns, ms):\n", + " zeros, weights = np.polynomial.laguerre.laggauss(n)\n", + " c = scipy.special.factorial(n) ** 2 / scipy.special.factorial(2 * n)\n", " e = np.abs(\n", " scipy.special.poch(z - 2 * n, 2 * n)\n", " / scipy.special.poch(z - m, m)\n", " * c\n", " * xi ** (z - 2 * n + m - 1)\n", " )\n", + " ez = np.sum(\n", + " scipy.special.poch(z - 2 * n, 2 * n)\n", + " / scipy.special.poch(z - m, m)\n", + " * c\n", + " * zeros[:, None] ** (z - 2 * n + m - 1),\n", + " 0,\n", + " )\n", + " lag = eval_laguerre([z], m)[0]\n", + " err = np.abs(lanczos - lag)\n", + " # print(m+z,ez)\n", + " # for zi,ezi in zip(z[0], ez):\n", + " # print(f\"{m+zi}: {ezi}\")\n", " # ax.semilogy(xi, e, color=color)\n", - " ax.semilogy(xi, e)\n", - " ax.set_xticks(np.arange(xi[-1] +1))\n", - " ax.set_ylim(1e-8, 1e5)\n", - " _ = ax.legend([f'z={zi}' for zi in z[0]])\n" + " lines = ax.loglog(xi, e, label=str(n))\n", + " ax.axhline(err, color=lines[0].get_color())\n", + " # ax.set_xticks(np.arange(xi[-1] + 1))\n", + " # ax.set_ylim(1e-8, 1e5)\n", + "_ = ax.legend()\n", + "# _ = ax.legend([f\"z={zi}\" for zi in z[0]])\n", + "# _ = [ax.axvline(x) for x in zeros]\n" ] } ], diff --git a/buch/papers/laguerre/scripts/laguerre_plot.py b/buch/papers/laguerre/scripts/laguerre_plot.py index cd90df1..b9088d0 100644 --- a/buch/papers/laguerre/scripts/laguerre_plot.py +++ b/buch/papers/laguerre/scripts/laguerre_plot.py @@ -2,38 +2,99 @@ # -*- coding:utf-8 -*- """Some plots for Laguerre Polynomials.""" +import os from pathlib import Path import matplotlib.pyplot as plt import numpy as np import scipy.special as ss + +def get_ticks(start, end, step=1): + ticks = np.arange(start, end, step) + return ticks[ticks != 0] + + N = 1000 -t = np.linspace(0, 12.5, N)[:, None] +step = 5 +t = np.linspace(-1.05, 10.5, N)[:, None] root = str(Path(__file__).parent) +img_path = f"{root}/../images" +os.makedirs(img_path, exist_ok=True) + +# fig = plt.figure(num=1, clear=True, tight_layout=True, figsize=(5.5, 3.7)) +# ax = fig.add_subplot(axes_class=AxesZero) fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(6, 4)) -for n in np.arange(0, 10): +for n in np.arange(0, 8): k = np.arange(0, n + 1)[None] L = np.sum((-1) ** k * ss.binom(n, k) / ss.factorial(k) * t ** k, -1) ax.plot(t, L, label=f"n={n}") -ax.set_xticks(np.arange(1, t[-1])) -ax.set_xlim(t[0], t[-1] + 0.1*(t[1] - t[0])) -ax.set_ylim(-20, 20) -ax.legend(ncol=2) + +ax.set_xticks(get_ticks(int(t[0]), t[-1]), minor=True) +ax.set_xticks(get_ticks(0, t[-1], step)) +ax.set_xlim(t[0], t[-1] + 0.1 * (t[1] - t[0])) +ax.set_xlabel(r"$x$", x=1.0, labelpad=-10, rotation=0, fontsize="large") + +ylim = 13 +ax.set_yticks(np.arange(-ylim, ylim), minor=True) +ax.set_yticks(np.arange(-step * (ylim // step), ylim, step)) +ax.set_ylim(-ylim, ylim) +ax.set_ylabel(r"$y$", y=0.95, labelpad=-18, rotation=0, fontsize="large") + +ax.legend(ncol=2, loc=(0.125, 0.01), fontsize="large") + # set the x-spine -ax.spines['left'].set_position('zero') -ax.spines['right'].set_visible(False) -ax.spines['bottom'].set_position('zero') -ax.spines['top'].set_visible(False) -ax.xaxis.set_ticks_position('bottom') -ax.yaxis.set_ticks_position('left') - -# make arrows -# ax.plot((1), (0), ls="", marker=">", ms=10, color="k", -# transform=ax.get_yaxis_transform(), clip_on=False) -# ax.plot((0), (1), ls="", marker="^", ms=10, color="k", -# transform=ax.get_xaxis_transform(), clip_on=False) -# ax.grid(1) -fig.savefig(f'{root}/laguerre_polynomes.pdf') -# plt.show() +ax.spines[["left", "bottom"]].set_position("zero") +ax.spines[["right", "top"]].set_visible(False) +ax.xaxis.set_ticks_position("bottom") +hlx = 0.4 +dx = t[-1, 0] - t[0, 0] +dy = 2 * ylim +hly = dy / dx * hlx +dps = fig.dpi_scale_trans.inverted() +bbox = ax.get_window_extent().transformed(dps) +width, height = bbox.width, bbox.height + +# manual arrowhead width and length +hw = 1.0 / 60.0 * dy +hl = 1.0 / 30.0 * dx +lw = 0.5 # axis line width +ohg = 0.0 # arrow overhang + +# compute matching arrowhead length and width +yhw = hw / dy * dx * height / width +yhl = hl / dx * dy * width / height + +# draw x and y axis +ax.arrow( + t[-1, 0] - hl, + 0, + hl, + 0.0, + fc="k", + ec="k", + lw=lw, + head_width=hw, + head_length=hl, + overhang=ohg, + length_includes_head=True, + clip_on=False, +) + +ax.arrow( + 0, + ylim - yhl, + 0.0, + yhl, + fc="k", + ec="k", + lw=lw, + head_width=yhw, + head_length=yhl, + overhang=ohg, + length_includes_head=True, + clip_on=False, +) + +fig.savefig(f"{img_path}/laguerre_polynomes.pdf") diff --git a/buch/papers/laguerre/scripts/lanczos_approximation.py b/buch/papers/laguerre/scripts/lanczos_approximation.py deleted file mode 100644 index 3c48266..0000000 --- a/buch/papers/laguerre/scripts/lanczos_approximation.py +++ /dev/null @@ -1,47 +0,0 @@ -from cmath import exp, pi, sin, sqrt - -p = [ - 676.5203681218851, - -1259.1392167224028, - 771.32342877765313, - -176.61502916214059, - 12.507343278686905, - -0.13857109526572012, - 9.9843695780195716e-6, - 1.5056327351493116e-7, -] - -EPSILON = 1e-07 - - -def drop_imag(z): - if abs(z.imag) <= EPSILON: - z = z.real - return z - - -def gamma(z): - z = complex(z) - if z.real < 0.5: - y = pi / (sin(pi * z) * gamma(1 - z)) # Reflection formula - else: - z -= 1 - x = 0.99999999999980993 - for (i, pval) in enumerate(p): - x += pval / (z + i + 1) - t = z + len(p) - 0.5 - y = sqrt(2 * pi) * t ** (z + 0.5) * exp(-t) * x - return drop_imag(y) - - -""" -The above use of the reflection (thus the if-else structure) is necessary, even though -it may look strange, as it allows to extend the approximation to values of z where -Re(z) < 0.5, where the Lanczos method is not valid. -""" - -print(gamma(1)) -print(gamma(5)) -print(gamma(0.5)) -print(gamma(0.5* (1 + 1j))) -print(gamma(-0.5)) diff --git a/buch/papers/laguerre/scripts/quadrature_gama.py b/buch/papers/laguerre/scripts/quadrature_gama.py deleted file mode 100644 index 37a9cd8..0000000 --- a/buch/papers/laguerre/scripts/quadrature_gama.py +++ /dev/null @@ -1,178 +0,0 @@ -#!/usr/bin/env python3 -# -*- coding:utf-8 -*- -"""Use Gauss-Laguerre quadrature to calculate gamma function.""" -# import sympy -from cmath import exp, pi, sin, sqrt - -import matplotlib.pyplot as plt -import numpy as np -import scipy.special as ss - -p = [ - 676.5203681218851, - -1259.1392167224028, - 771.32342877765313, - -176.61502916214059, - 12.507343278686905, - -0.13857109526572012, - 9.9843695780195716e-6, - 1.5056327351493116e-7, -] - -EPSILON = 1e-07 - - -def drop_imag(z): - if abs(z.imag) <= EPSILON: - z = z.real - return z - - -def gamma(z): - z = complex(z) - if z.real < 0.5: - y = pi / (sin(pi * z) * gamma(1 - z)) # Reflection formula - else: - z -= 1 - x = 0.99999999999980993 - for (i, pval) in enumerate(p): - x += pval / (z + i + 1) - t = z + len(p) - 0.5 - y = sqrt(2 * pi) * t ** (z + 0.5) * exp(-t) * x - return drop_imag(y) - - -zeros = np.array( - [ - 3.22547689619392312e-1, - 1.74576110115834658e0, - 4.53662029692112798e0, - 9.39507091230113313e0, - ], - np.longdouble, -) -weights = np.array( - [ - 6.03154104341633602e-1, - 3.57418692437799687e-1, - 3.88879085150053843e-2, - 5.39294705561327450e-4, - ], - np.longdouble, -) - -zeros = np.array( - [ - 1.70279632305101000e-1, - 9.03701776799379912e-1, - 2.25108662986613069e0, - 4.26670017028765879e0, - 7.04590540239346570e0, - 1.07585160101809952e1, - 1.57406786412780046e1, - 2.28631317368892641e1, - ], - np.longdouble, -) - -weights = np.array( - [ - 3.69188589341637530e-1, - 4.18786780814342956e-1, - 1.75794986637171806e-1, - 3.33434922612156515e-2, - 2.79453623522567252e-3, - 9.07650877335821310e-5, - 8.48574671627253154e-7, - 1.04800117487151038e-9, - ], - np.longdouble, -) - - -def calc_gamma(z, n, x, w): - res = 0.0 - z = complex(z) - for xi, wi in zip(x, w): - res += xi ** (z + n - 1) * wi - for i in range(int(n)): - res /= z + i - res = drop_imag(res) - return res - -small = 1e-3 -Z = np.linspace(small, 1-small, 101) - -# Z = [-3/2, -1/2, 1/2, 3/2] -# target = -# targets = np.array([gamma(z) for z in Z]) -targets1 = ss.gamma(Z) -targets2 = np.array([gamma(z) for z in Z]) -approxs = np.array([calc_gamma(z, 11, zeros, weights) for z in Z]) -rel_error1 = np.abs(targets1 - approxs) / targets1 -rel_error2 = np.abs(targets2 - approxs) / targets2 - -_, axs = plt.subplots(2, num=1, clear=True, constrained_layout=True) -axs[0].plot(Z, rel_error1) -axs[1].semilogy(Z, rel_error1) -axs[0].plot(Z, rel_error2) -axs[1].semilogy(Z, rel_error2) -axs[1].semilogy(Z, np.abs(targets1-targets2)/targets1) -plt.show() -# values = np.array([calc_gamma]) -# _ = [ -# print( -# n, -# [ -# float( -# f"{np.abs((calc_gamma(z, n, zeros, weights) - gamma(z)) / gamma(z)):.3g}" -# ) -# for z in Z -# ], -# ) -# for n in range(21) -# ] - - -# target = ss.gamma(z) -# target = np.sqrt(np.pi) - -# _, ax = plt.subplots(num=1, clear=True, constrained_layout=True) -# for i, degree in enumerate(degrees): -# samples_points, weights = np.polynomial.laguerre.laggauss(degree) -# values = np.sum( -# samples_points[:, None] ** (z + shifts[None] - 1) * weights[:, None], 0 -# ) / ss.poch(z, shifts) -# # print(np.abs(target - values)) -# print(values) -# ax.plot(shifts, values, label=f"N={degree}") -# ax.legend() -# plt.show() - - -# def count_equal_digits(x, y): -# for i in range(1, 13): -# try: -# np.testing.assert_almost_equal(x, y, i) -# except AssertionError: -# break -# return i - - -# Z = np.linspace(1.0, 11.0, 11) -# # degrees = [2, 4, 8, 16, 32, 64, 100] -# d = 100 -# X = np.zeros(len(Z)) -# for i, z in enumerate(Z): -# samples_points, weights = np.polynomial.laguerre.laggauss(d) -# X[i] = np.sum(samples_points ** (z - 1) * weights) -# # X[i] = np.sum(np.sin(z * samples_points) * weights) -# Y = ss.gamma(Z) -# # Y = Z / (Z ** 2 + 1) -# ed = [count_equal_digits(x, y) for x, y in zip(X, Y)] -# for x,y in zip(X,Y): -# print(x,y) - -# _, ax = plt.subplots(num=1, clear=True, constrained_layout=True) -# ax.plot(Z, ed) -# plt.show() diff --git a/buch/standalone.tex b/buch/standalone.tex deleted file mode 100644 index e681460..0000000 --- a/buch/standalone.tex +++ /dev/null @@ -1,35 +0,0 @@ -% -% packages.tex -- common packages -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\documentclass{book} -\def\IncludeBookCover{0} -\input{common/packages.tex} -% additional packages used by the individual papers, add a line for -% each paper -\input{papers/common/addpackages.tex} - -% workaround for biblatex bug -\makeatletter -\def\blx@maxline{77} -\makeatother -\addbibresource{chapters/references.bib} - -% Bibresources for each article -\input{papers/common/addbibresources.tex} - -% make sure the last index starts on an odd page -\AtEndDocument{\clearpage\ifodd\value{page}\else\null\clearpage\fi} -\makeindex - -%\pgfplotsset{compat=1.12} -\setlength{\headheight}{15pt} % fix headheight warning -\DeclareGraphicsRule{*}{mps}{*}{} - -\begin{document} - \input{common/macros.tex} - \def\chapterauthor#1{{\large #1}\bigskip\bigskip} - - \input{papers/laguerre/main} -\end{document} \ No newline at end of file -- cgit v1.2.1 From 7397861ade0537bf8e501fa87bd57653d932d459 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Patrik=20M=C3=BCller?= Date: Thu, 12 May 2022 18:21:25 +0200 Subject: Remove deprecated files --- buch/papers/laguerre/images/wasserstoff_model.tex | 58 --------- buch/papers/laguerre/transformation.tex | 31 ----- buch/papers/laguerre/wasserstoff.tex | 142 ---------------------- 3 files changed, 231 deletions(-) delete mode 100644 buch/papers/laguerre/images/wasserstoff_model.tex delete mode 100644 buch/papers/laguerre/transformation.tex delete mode 100644 buch/papers/laguerre/wasserstoff.tex (limited to 'buch') diff --git a/buch/papers/laguerre/images/wasserstoff_model.tex b/buch/papers/laguerre/images/wasserstoff_model.tex deleted file mode 100644 index fe838c3..0000000 --- a/buch/papers/laguerre/images/wasserstoff_model.tex +++ /dev/null @@ -1,58 +0,0 @@ -\documentclass{standalone} - -\usepackage{pgfplots} -\usepackage{tikz-3dplot} - -\tdplotsetmaincoords{60}{115} -\pgfplotsset{compat=newest} - -\begin{document} - -\newcommand{\drawcircle}[4]{ -\shade[ball color=#3, opacity=#4] (#1) circle (#2 cm); -\tdplotsetrotatedcoords{0}{0}{0}; -\draw[dashed, tdplot_rotated_coords, #3!40!black] (#1) circle (#2); -} - -\begin{tikzpicture}[tdplot_main_coords, scale = 2] -\def\r{1.0} -\def\rp{0.2} -\def\rn{0.05} -\def\rvec{1.0} -\def\thetavec{45} -\def\phivec{60} - -\coordinate (O) at (0, 0, 0); -\tdplotsetcoord{P}{\rvec}{\thetavec}{\phivec} - -% Labels -\node[inner sep=1pt] at (0, -4.0*\rp, 1.0*\r) (plabel){Proton}; -\draw (plabel) -- (O); -\node[inner sep=1pt] at (-0.*\r, 1.0*\r, 1.3*\r) (elabel){Elektron}; -\draw (elabel) -- (P); -% Draw proton -\drawcircle{O}{\rp}{red}{1.0} - -% Draw spherical coordinates of electron -\draw (O) -- node[anchor=north west, yshift=4pt]{$r$} (P); -\draw[dashed] (O) -- (Pxy); -\draw[dashed] (P) -- (Pxy); -\tdplotdrawarc{(O)}{0.6}{0}{\phivec}{anchor=north}{$\varphi$} -\tdplotsetthetaplanecoords{\phivec} -\tdplotdrawarc[tdplot_rotated_coords]{(0,0,0)}{0.5}{0}% -{\thetavec}{anchor=south west, xshift=-2pt, yshift=-2pt}{$\vartheta$} - -% Draw electron -\drawcircle{P}{\rn}{blue}{1.0} - -% Draw surrounding sphere -\drawcircle{O}{\r}{gray}{0.3} - -% Draw cartesian coordinate system -\draw[-stealth, thick] (O) -- (1.8*\r,0,0) node[below left] {$x$}; -\draw[-stealth, thick] (O) -- (0,1.3*\r,0) node[below right] {$y$}; -\draw[-stealth, thick] (O) -- (0,0,1.3*\r) node[above] {$z$}; - -\end{tikzpicture} - -\end{document} \ No newline at end of file diff --git a/buch/papers/laguerre/transformation.tex b/buch/papers/laguerre/transformation.tex deleted file mode 100644 index 4de86b6..0000000 --- a/buch/papers/laguerre/transformation.tex +++ /dev/null @@ -1,31 +0,0 @@ -% -% transformation.tex -% -% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule -% -\section{Laguerre Transformation -\label{laguerre:section:transformation}} -\begin{align} - L \left\{ f(x) \right\} - = - \tilde{f}_\alpha(n) - = - \int_0^\infty e^{-x} x^\alpha L_n^\alpha(x) f(x) dx - \label{laguerre:transformation} -\end{align} - -\begin{align} - L^{-1} \left\{ \tilde{f}_\alpha(n) \right\} - = - f(x) - = - \sum_{n=0}^{\infty} - \begin{pmatrix} - n + \alpha \\ - n - \end{pmatrix}^{-1} - \frac{1}{\Gamma(\alpha + 1)} - \tilde{f}_\alpha(n) - L_n^\alpha(x) - \label{laguerre:inverse_transformation} -\end{align} \ No newline at end of file diff --git a/buch/papers/laguerre/wasserstoff.tex b/buch/papers/laguerre/wasserstoff.tex deleted file mode 100644 index 0da8be3..0000000 --- a/buch/papers/laguerre/wasserstoff.tex +++ /dev/null @@ -1,142 +0,0 @@ -% -% wasserstoff.tex -% -% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule -% -\section{Radialer Schwingungsanteil eines Wasserstoffatoms -\label{laguerre:section:radial_h_atom}} - -Das Wasserstoffatom besteht aus einem Proton im Kern -mit Masse $M$ und Ladung $+e$. -Ein Elektron mit Masse $m$ und Ladung $-e$ umkreist das Proton -(vgl. Abbildung~\ref{laguerre:fig:wasserstoff_model}). -Für das folgende Model werden folgende Annahmen getroffen: - -\begin{figure} -\centering -\includegraphics{papers/laguerre/images/wasserstoff_model.pdf} -\caption{Skizze eines Wasserstoffatoms. -Kartesische, wie auch Kugelkoordinaten sind eingezeichnet. -} -\label{laguerre:fig:wasserstoff_model} -\end{figure} - -\begin{enumerate} -\item -Das Elektron wird als nicht-relativistisches Teilchen betrachtet, -das heisst, -relativistische Effekte sind vernachlässigbar. -\item -Der Spin des Elektrons und des Protons -und das damit verbundene magnetische Moment -wird vernachlässigt. -\item -Fluktuationen des Vakuums werden nicht berücksichtigt. -\item -Wechselwirkung zwischen Elektron und Proton -ist durch die Coulombwechselwirkung gegeben. -Somit entspricht die potentielle Energie der Coulombenergie $V_C(r)$ -und nimmt damit die folgende Form an -\begin{align} - V_C(r) - = - -\frac{e^2}{4 \pi \epsilon_0 r} - \text{ mit } - r - = - \lvert\vec{r}\rvert - = - \sqrt{x^2 + y^2 + z^2} - . - \label{laguerre:coulombenergie} -\end{align} -Im Falle das der Kern einen endlichen Radius $r_0$ besitzt, -ist die $1/r$-Abhängigkeit in Gleichung \eqref{laguerre:coulombenergie} -als Näherung zu betrachten. -Diese Näherung darf nur angewendet werden, wenn die -Aufenthaltswahrscheinlicheit des Elektrons -innerhalb $r_0$ vernachlässigbar ist. -Für das Wasserstoffatom ist diese Näherung für alle Zustände gerechtfertigt. -\item -Da $M \gg m$, kann das Proton als in Ruhe angenommen werden. -\end{enumerate} - -\subsection{Herleitung zeitunabhängige Schrödinger-Gleichung} -\label{laguerre:subsection:herleitung_schroedinger} -Das Problem ist kugelsymmetrisch, -darum transformieren wir das Problem in Kugelkoordinaten. -Somit gilt: - -\begin{align*} - r - & = - \sqrt{x^2 + y^2 + z^2}\\ - \vartheta - & = - \arccos\left(\frac{z}{r}\right)\\ - \varphi - & = - \arctan\left(\frac{y}{x}\right) -\end{align*} - -Die potentielle Energie $V_C(r)$ hat keine direkte Zeitabhängigkeit. -Daraus folgt, dass die konstant ist Gesamtenergie $E$ -und es existieren stationäre Zustände - -\begin{align} - \psi(r, \vartheta, \varphi, t) - = - u(r, \vartheta, \varphi) e^{-i E t / h}, -\end{align} -wobei $u(r, \vartheta, \varphi)$ -die zeitunabhängige Schrödinger-Gleichung erfüllt. - -\begin{align} - -\frac{\hbar^2}{2m} \Delta u(r, \vartheta, \varphi) - + V_C(r) u(r, \vartheta, \varphi) - = - E u(r, \vartheta, \varphi) - \label{laguerre:schroedinger} -\end{align} - -Für Kugelkoordinaten hat der Laplace-Operator $\Delta$ die Form - -\begin{align} - \Delta - = - \frac{1}{r^2} \pdv{}{r} \left( r^2 \pdv{}{r} \right) - + \frac{1}{r^2 \sin\vartheta} \pdv{}{\vartheta} - \left(\sin\vartheta \pdv{}{\vartheta}\right) - + \frac{1}{r^2 \sin^2\vartheta} \pdv[2]{}{\varphi} - \label{laguerre:laplace_kugel} -\end{align} - -Setzt man nun -\eqref{laguerre:coulombenergie} und \eqref{laguerre:laplace_kugel} -in \eqref{laguerre:schroedinger} ein, -erhält man die zeitunabhängige Schrödinger-Gleichung für Kugelkoordinaten - -\begin{align} -\nonumber -- \frac{\hbar^2}{2m} -& -\left( -\frac{1}{r^2} \pdv{}{r} -\left( r^2 \pdv{}{r} \right) -+ -\frac{1}{r^2 \sin \vartheta} \pdv{}{\vartheta} -\left( \sin \vartheta \pdv{}{\vartheta} \right) -+ -\frac{1}{r^2 \sin^2 \vartheta} \pdv[2]{}{\varphi} -\right) -u(r, \vartheta, \varphi) -\\ -& - -\frac{e^2}{4 \pi \epsilon_0 r} u(r, \vartheta, \varphi) -= -E u(r, \vartheta, \varphi). -\label{laguerre:pdg_h_atom} -\end{align} - -\subsection{Separation der Schrödinger-Gleichung} -\label{laguerre:subsection:seperation_schroedinger} -- cgit v1.2.1 From 155989e49b70a4598dbf3ff3277d9e320f226a83 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Patrik=20M=C3=BCller?= Date: Fri, 13 May 2022 12:38:18 +0200 Subject: Add some information about Gauss Quadrature and application to Gamma integral --- buch/papers/laguerre/Makefile.inc | 3 +- buch/papers/laguerre/definition.tex | 5 ++- buch/papers/laguerre/eigenschaften.tex | 25 ++++++++++- buch/papers/laguerre/gamma.tex | 76 +++++++++++++++++++++++++++++++++ buch/papers/laguerre/main.tex | 3 +- buch/papers/laguerre/quadratur.tex | 78 ++++++++++++++++++++++++++++------ buch/papers/laguerre/references.bib | 45 +++++++------------- 7 files changed, 187 insertions(+), 48 deletions(-) create mode 100644 buch/papers/laguerre/gamma.tex (limited to 'buch') diff --git a/buch/papers/laguerre/Makefile.inc b/buch/papers/laguerre/Makefile.inc index aae51f9..12b0935 100644 --- a/buch/papers/laguerre/Makefile.inc +++ b/buch/papers/laguerre/Makefile.inc @@ -9,6 +9,7 @@ dependencies-laguerre = \ papers/laguerre/references.bib \ papers/laguerre/definition.tex \ papers/laguerre/eigenschaften.tex \ - papers/laguerre/quadratur.tex + papers/laguerre/quadratur.tex \ + papers/laguerre/gamma.tex diff --git a/buch/papers/laguerre/definition.tex b/buch/papers/laguerre/definition.tex index edd2b7b..d111f6f 100644 --- a/buch/papers/laguerre/definition.tex +++ b/buch/papers/laguerre/definition.tex @@ -18,8 +18,9 @@ x \in \mathbb{R} . \label{laguerre:dgl} \end{align} +Die klassische Laguerre-Diffentialgleichung erhält man, wenn $\nu = 0$. Hier wird die verallgemeinerte Laguerre-Differentialgleichung verwendet, -weil die Lösung gleich berechnet werden kann, +weil die Lösung mit der selben Methode berechnet werden kann, aber man zusätzlich die Lösung für den allgmeinen Fall erhält. Zur Lösung der Gleichung \eqref{laguerre:dgl} verwenden wir einen Potenzreihenansatz. @@ -117,6 +118,8 @@ L_n^\nu(x) \sum_{k=0}^{n} \frac{(-1)^k}{(\nu + 1)_k} \binom{n}{k} x^k. \label{laguerre:allg_polynom} \end{align} + +\subsection{Analytische Fortsetzung} Durch die analytische Fortsetzung erhalten wir zudem noch die zweite Lösung der Differentialgleichung mit der Form \begin{align*} diff --git a/buch/papers/laguerre/eigenschaften.tex b/buch/papers/laguerre/eigenschaften.tex index c589c92..b0cc3a3 100644 --- a/buch/papers/laguerre/eigenschaften.tex +++ b/buch/papers/laguerre/eigenschaften.tex @@ -5,9 +5,21 @@ % \section{Eigenschaften \label{laguerre:section:eigenschaften}} +{ +\large \color{red} +TODO: +Evtl. nur Orthogonalität hier behandeln, da nur diese für die Gauss-Quadratur +benötigt wird. +} + +Die Laguerre-Polynome besitzen einige interessante Eigenschaften \rhead{Eigenschaften} -\subsection{Orthogonalität} +\subsection{Orthogonalität + \label{laguerre:subsection:orthogonal}} +Im Abschnitt~\ref{laguerre:section:definition} haben wir behauptet, +dass die Laguerre-Polynome orthogonale Polynome sind. +Zu dieser Behauptung möchten wir nun einen Beweis liefern. Wenn wir die Laguerre\--Differentialgleichung in ein Sturm\--Liouville\--Problem umwandeln können, haben wir bewiesen, dass es sich bei @@ -95,4 +107,13 @@ Für den rechten Rand ist die Bedingung (Gleichung~\eqref{laguerre:sllag_randb}) \end{align*} für beliebige Polynomlösungen erfüllt für $k_\infty=0$ und $h_\infty=1$. Damit können wir schlussfolgern, dass die Laguerre-Polynome orthogonal -bezüglich des Skalarproduktes mit der Laguerre\--Gewichtsfunktion sind. +bezüglich des Skalarproduktes auf dem Intervall $(0, \infty)$ mit der Laguerre\--Gewichtsfunktion +$w(x)=x^\nu e^{-x}$ sind. + + +\subsection{Rodrigues-Formel} + +\subsection{Drei-Terme Rekursion} + +\subsection{Beziehung mit der Hypergeometrischen Funktion} + diff --git a/buch/papers/laguerre/gamma.tex b/buch/papers/laguerre/gamma.tex new file mode 100644 index 0000000..e3838b0 --- /dev/null +++ b/buch/papers/laguerre/gamma.tex @@ -0,0 +1,76 @@ +% +% gamma.tex +% +% (c) 2022 Patrik Müller, Ostschweizer Fachhochschule +% +\section{Anwendung: Berechnung der Gamma-Funktion + \label{laguerre:section:quad-gamma}} +Die Gauss-Laguerre-Quadratur kann nun verwendet werden, +um exponentiell abfallende Funktionen im Definitionsbereich $(0, \infty)$ zu +berechnen. +Dabei bietet sich z.B. die Gamma-Funkion bestens an, wie wir in den folgenden +Abschnitten sehen werden. + +\subsection{Gamma-Funktion} +Die Gamma-Funktion ist eine Erweiterung der Fakultät auf die reale und komplexe +Zahlenmenge. +Die Definition~\ref{buch:rekursion:def:gamma} beschreibt die Gamma-Funktion als +Integral der Form +\begin{align} +\Gamma(z) + & = +\int_0^\infty t^{z-1} e^{-t} dt +, +\quad +\text{wobei Realteil von $z$ grösser als $0$} +, +\label{laguerre:gamma} +\end{align} +welches alle Eigenschaften erfüllt, um mit der Gauss-Laguerre-Quadratur +berechnet zu werden. + +\subsubsection{Funktionalgleichung} +Die Funktionalgleichung besagt +\begin{align} +z \Gamma(z) = \Gamma(z+1). +\label{laguerre:gamma_funktional} +\end{align} +Mittels dieser Gleichung kann der Wert an einer bestimmten, +geeigneten Stelle evaluiert werden und dann zurückverschoben werden, +um das gewünschte Resultat zu erhalten. + +\subsection{Berechnung mittels Gauss-Laguerre-Quadratur} + +Fehlerterm: +\begin{align*} +R_n += +(z - 2n)_{2n} \frac{(n!)^2}{(2n)!} \xi^{z-2n-1} +\end{align*} + +\subsubsection{Finden der optimalen Berechnungsstelle} +Nun stellt sich die Frage, +ob die Approximation mittels Gauss-Laguerre-Quadratur verbessert werden kann, +wenn man das Problem an einer geeigneten Stelle evaluiert und +dann zurückverschiebt mit der Funktionalgleichung. +Dazu wollen wir den Fehlerterm in +Gleichung~\eqref{laguerre:lagurre:lag_error} anpassen und dann minimieren. +Zunächst wollen wir dies nur für $z\in \mathbb{R}$ und $0 Date: Fri, 13 May 2022 12:47:50 +0200 Subject: fix some bugs --- buch/chapters/070-orthogonalitaet/orthogonal.tex | 4 +- buch/chapters/070-orthogonalitaet/rekursion.tex | 10 ++-- buch/papers/nav/images/Makefile | 33 ++++++++++++ buch/papers/nav/images/dreieck1.pdf | Bin 0 -> 11578 bytes buch/papers/nav/images/dreieck1.tex | 59 +++++++++++++++++++++ buch/papers/nav/images/dreieck2.pdf | Bin 0 -> 8812 bytes buch/papers/nav/images/dreieck2.tex | 59 +++++++++++++++++++++ buch/papers/nav/images/dreieck3.pdf | Bin 0 -> 10636 bytes buch/papers/nav/images/dreieck3.tex | 59 +++++++++++++++++++++ buch/papers/nav/images/dreieck4.pdf | Bin 0 -> 13231 bytes buch/papers/nav/images/dreieck4.tex | 64 +++++++++++++++++++++++ buch/papers/nav/images/dreieck5.pdf | Bin 0 -> 8721 bytes buch/papers/nav/images/dreieck5.tex | 64 +++++++++++++++++++++++ buch/papers/nav/images/dreieck6.pdf | Bin 0 -> 10699 bytes buch/papers/nav/images/dreieck6.tex | 64 +++++++++++++++++++++++ buch/papers/nav/images/dreieck7.pdf | Bin 0 -> 11079 bytes buch/papers/nav/images/dreieck7.tex | 64 +++++++++++++++++++++++ 17 files changed, 473 insertions(+), 7 deletions(-) create mode 100644 buch/papers/nav/images/dreieck1.pdf create mode 100644 buch/papers/nav/images/dreieck1.tex create mode 100644 buch/papers/nav/images/dreieck2.pdf create mode 100644 buch/papers/nav/images/dreieck2.tex create mode 100644 buch/papers/nav/images/dreieck3.pdf create mode 100644 buch/papers/nav/images/dreieck3.tex create mode 100644 buch/papers/nav/images/dreieck4.pdf create mode 100644 buch/papers/nav/images/dreieck4.tex create mode 100644 buch/papers/nav/images/dreieck5.pdf create mode 100644 buch/papers/nav/images/dreieck5.tex create mode 100644 buch/papers/nav/images/dreieck6.pdf create mode 100644 buch/papers/nav/images/dreieck6.tex create mode 100644 buch/papers/nav/images/dreieck7.pdf create mode 100644 buch/papers/nav/images/dreieck7.tex (limited to 'buch') diff --git a/buch/chapters/070-orthogonalitaet/orthogonal.tex b/buch/chapters/070-orthogonalitaet/orthogonal.tex index a84248a..677e865 100644 --- a/buch/chapters/070-orthogonalitaet/orthogonal.tex +++ b/buch/chapters/070-orthogonalitaet/orthogonal.tex @@ -842,14 +842,14 @@ bei geeigneter Normierung die {\em Hermite-Polynome}. % % Laguerre-Gewichtsfunktion % -\subsection{Laguerre-Gewichtsfunktion} +\subsubsection{Laguerre-Gewichtsfunktion} Ähnlich wie die Hermite-Gewichtsfunktion ist die {\em Laguerre-Gewichtsfunktion} \index{Laguerre-Gewichtsfunktion}% \[ w_{\text{Laguerre}}(x) = -w^{-x} +e^{-x} \] auf ganz $\mathbb{R}$ definiert, und sie geht für $x\to\infty$ wieder sehr rasch gegen $0$. diff --git a/buch/chapters/070-orthogonalitaet/rekursion.tex b/buch/chapters/070-orthogonalitaet/rekursion.tex index 5ec7fed..dc5531b 100644 --- a/buch/chapters/070-orthogonalitaet/rekursion.tex +++ b/buch/chapters/070-orthogonalitaet/rekursion.tex @@ -30,7 +30,7 @@ Skalarproduktes $\langle\,\;,\;\rangle_w$, wenn für alle $n$, $m$. \end{definition} -\subsection{Allgemeine Drei-Term-Rekursion für orthogonale Polynome} +\subsubsection{Allgemeine Drei-Term-Rekursion für orthogonale Polynome} Der folgende Satz besagt, dass $p_n$ eine Rekursionsbeziehung erfüllt. \begin{satz} @@ -55,7 +55,7 @@ C_{n+1} = \frac{A_{n+1}}{A_n}\frac{h_{n+1}}{h_n}. \end{equation} \end{satz} -\subsection{Multiplikationsoperator mit $x$} +\subsubsection{Multiplikationsoperator mit $x$} Man kann die Relation auch nach dem Produkt $xp_n(x)$ auflösen, dann wird sie \begin{equation} @@ -72,7 +72,7 @@ Die Multiplikation mit $x$ ist eine lineare Abbildung im Raum der Funktionen. Die Relation~\eqref{buch:orthogonal:eqn:multixrelation} besagt, dass diese Abbildung in der Basis der Polynome $p_k$ tridiagonale Form hat. -\subsection{Drei-Term-Rekursion für die Tschebyscheff-Polynome} +\subsubsection{Drei-Term-Rekursion für die Tschebyscheff-Polynome} Eine Relation der Form~\eqref{buch:orthogonal:eqn:multixrelation} wurde bereits in Abschnitt~\ref{buch:potenzen:tschebyscheff:rekursionsbeziehungen} @@ -80,12 +80,12 @@ hergeleitet. In der Form~\eqref{buch:orthogonal:eqn:rekursion} geschrieben lautet sie \[ -T_{n+1}(x) = 2x\,T_n(x)-T_{n-1}(x). +T_{n+1}(x) = 2x\,T_n(x)-T_{n-1}(x), \] also $A_n=2$, $B_n=0$ und $C_n=1$. -\subsection{Beweis von Satz~\ref{buch:orthogonal:satz:drei-term-rekursion}} +\subsubsection{Beweis von Satz~\ref{buch:orthogonal:satz:drei-term-rekursion}} Die Relation~\eqref{buch:orthogonal:eqn:multixrelation} zeigt auch, dass der Beweis die Koeffizienten $\langle xp_k,p_j\rangle_w$ berechnen muss. diff --git a/buch/papers/nav/images/Makefile b/buch/papers/nav/images/Makefile index a0d7b34..0c1cbc3 100644 --- a/buch/papers/nav/images/Makefile +++ b/buch/papers/nav/images/Makefile @@ -3,9 +3,42 @@ # # (c) 2022 # +all: dreiecke dreieck.pdf: dreieck.tex dreieckdata.tex macros.tex pdflatex dreieck.tex dreieckdata.tex: pk.m octave pk.m + +DREIECKE = \ + dreieck1.pdf \ + dreieck2.pdf \ + dreieck3.pdf \ + dreieck4.pdf \ + dreieck5.pdf \ + dreieck6.pdf \ + dreieck7.pdf + +dreiecke: $(DREIECKE) + +dreieck1.pdf: dreieck1.tex dreieckdata.tex macros.tex + pdflatex dreieck1.tex + +dreieck2.pdf: dreieck2.tex dreieckdata.tex macros.tex + pdflatex dreieck2.tex + +dreieck3.pdf: dreieck3.tex dreieckdata.tex macros.tex + pdflatex dreieck3.tex + +dreieck4.pdf: dreieck4.tex dreieckdata.tex macros.tex + pdflatex dreieck4.tex + +dreieck5.pdf: dreieck5.tex dreieckdata.tex macros.tex + pdflatex dreieck5.tex + +dreieck6.pdf: dreieck6.tex dreieckdata.tex macros.tex + pdflatex dreieck6.tex + +dreieck7.pdf: dreieck7.tex dreieckdata.tex macros.tex + pdflatex dreieck7.tex diff --git a/buch/papers/nav/images/dreieck1.pdf b/buch/papers/nav/images/dreieck1.pdf new file mode 100644 index 0000000..5bdf23d Binary files /dev/null and b/buch/papers/nav/images/dreieck1.pdf differ diff --git a/buch/papers/nav/images/dreieck1.tex b/buch/papers/nav/images/dreieck1.tex new file mode 100644 index 0000000..436314c --- /dev/null +++ b/buch/papers/nav/images/dreieck1.tex @@ -0,0 +1,59 @@ +% +% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +\winkelAlpha{red} +\winkelGamma{blue} +\winkelBeta{darkgreen} + +\seiteC{black} +\seiteB{black} +\seiteA{black} + +%\seiteL{gray} +\seitePB{gray} +\seitePC{gray} + +\draw[line width=1.4pt] \kanteAB; +\draw[line width=1.4pt] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[line width=1.4pt] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[color=gray] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{gray}; + +\node at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node[color=gray] at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck2.pdf b/buch/papers/nav/images/dreieck2.pdf new file mode 100644 index 0000000..a872b25 Binary files /dev/null and b/buch/papers/nav/images/dreieck2.pdf differ diff --git a/buch/papers/nav/images/dreieck2.tex b/buch/papers/nav/images/dreieck2.tex new file mode 100644 index 0000000..99aabb7 --- /dev/null +++ b/buch/papers/nav/images/dreieck2.tex @@ -0,0 +1,59 @@ +% +% dreieck2.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +\seiteC{black} +\seiteB{black} +%\seiteA{black} + +%\seiteL{gray} +\seitePB{gray} +\seitePC{gray} + +\draw[line width=1.4pt] \kanteAB; +\draw[line width=1.4pt] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[line width=1.4pt] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[color=gray] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{gray}; + +\node at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node[color=gray] at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck3.pdf b/buch/papers/nav/images/dreieck3.pdf new file mode 100644 index 0000000..65070c6 Binary files /dev/null and b/buch/papers/nav/images/dreieck3.pdf differ diff --git a/buch/papers/nav/images/dreieck3.tex b/buch/papers/nav/images/dreieck3.tex new file mode 100644 index 0000000..0cf5363 --- /dev/null +++ b/buch/papers/nav/images/dreieck3.tex @@ -0,0 +1,59 @@ +% +% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +\seiteC{black} +\seiteB{black} +%\seiteA{black} + +%\seiteL{gray} +\seitePB{gray} +\seitePC{gray} + +\draw[line width=1.4pt] \kanteAB; +\draw[line width=1.4pt] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[line width=1.4pt] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[color=gray] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{gray}; + +\node at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node[color=gray] at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck4.pdf b/buch/papers/nav/images/dreieck4.pdf new file mode 100644 index 0000000..4871a1e Binary files /dev/null and b/buch/papers/nav/images/dreieck4.pdf differ diff --git a/buch/papers/nav/images/dreieck4.tex b/buch/papers/nav/images/dreieck4.tex new file mode 100644 index 0000000..19a7d12 --- /dev/null +++ b/buch/papers/nav/images/dreieck4.tex @@ -0,0 +1,64 @@ +% +% dreieck4.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +%\winkelKappa{gray} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +%\winkelOmega{gray} +\winkelBetaEins{brown} + +%\seiteC{gray} +%\seiteB{gray} +%\seiteL{gray} + +\seiteA{black} +\seitePB{black} +\seitePC{black} + +\draw[color=gray] \kanteAB; +\draw[color=gray] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[color=black,line width=1.4pt] \kanteBC; +\draw[color=black,line width=1.4pt] \kanteBP; +\draw[color=black,line width=1.4pt] \kanteCP; + +\punkt{(A)}{gray}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{black}; + +\node[color=gray] at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck5.pdf b/buch/papers/nav/images/dreieck5.pdf new file mode 100644 index 0000000..cf686e0 Binary files /dev/null and b/buch/papers/nav/images/dreieck5.pdf differ diff --git a/buch/papers/nav/images/dreieck5.tex b/buch/papers/nav/images/dreieck5.tex new file mode 100644 index 0000000..d1117d1 --- /dev/null +++ b/buch/papers/nav/images/dreieck5.tex @@ -0,0 +1,64 @@ +% +% dreieck4.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +%\winkelKappa{gray} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +%\winkelOmega{gray} +%\winkelBetaEins{brown} + +%\seiteC{gray} +%\seiteB{gray} +%\seiteL{gray} + +%\seiteA{black} +\seitePB{black} +\seitePC{black} + +\draw[color=gray] \kanteAB; +\draw[color=gray] \kanteAC; +%\draw[color=gray] \kanteAP; +\draw[color=black,line width=1.4pt] \kanteBC; +\draw[color=black,line width=1.4pt] \kanteBP; +\draw[color=black,line width=1.4pt] \kanteCP; + +\punkt{(A)}{gray}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{black}; + +\node[color=gray] at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck6.pdf b/buch/papers/nav/images/dreieck6.pdf new file mode 100644 index 0000000..7efd673 Binary files /dev/null and b/buch/papers/nav/images/dreieck6.pdf differ diff --git a/buch/papers/nav/images/dreieck6.tex b/buch/papers/nav/images/dreieck6.tex new file mode 100644 index 0000000..87db1c2 --- /dev/null +++ b/buch/papers/nav/images/dreieck6.tex @@ -0,0 +1,64 @@ +% +% dreieck6.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +\winkelKappa{gray} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +%\winkelOmega{gray} +%\winkelBetaEins{brown} + +\seiteC{black} +\seiteB{black} +%\seiteA{gray} + +\seiteL{black} +\seitePB{black} +\seitePC{black} + +\draw[color=black,line width=1.4pt] \kanteAB; +\draw[color=black,line width=1.4pt] \kanteAC; +\draw[color=black,line width=1.4pt] \kanteAP; +%\draw[color=gray] \kanteBC; +\draw[color=black,line width=1.4pt] \kanteBP; +\draw[color=black,line width=1.4pt] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{black}; +\punkt{(C)}{black}; +\punkt{(P)}{black}; + +\node at (A) [above] {$A$}; +\node at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/nav/images/dreieck7.pdf b/buch/papers/nav/images/dreieck7.pdf new file mode 100644 index 0000000..aa83e28 Binary files /dev/null and b/buch/papers/nav/images/dreieck7.pdf differ diff --git a/buch/papers/nav/images/dreieck7.tex b/buch/papers/nav/images/dreieck7.tex new file mode 100644 index 0000000..f084708 --- /dev/null +++ b/buch/papers/nav/images/dreieck7.tex @@ -0,0 +1,64 @@ +% +% dreieck.tex -- sphärische Dreiecke für Positionsbestimmung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\skala{1} + +\def\punkt#1#2{ + \fill[color=#2] #1 circle[radius=0.08]; +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{dreieckdata.tex} +\input{macros.tex} + +%\winkelKappa{gray} + +%\winkelAlpha{red} +%\winkelGamma{blue} +%\winkelBeta{darkgreen} + +\winkelOmega{gray} +%\winkelBetaEins{brown} + +\seiteC{black} +\seiteB{black} +\seiteA{gray} + +\seiteL{black} +\seitePB{gray} +\seitePC{black} + +\draw[color=gray] \kanteAB; +\draw[color=black,line width=1.4pt] \kanteAC; +\draw[color=black,line width=1.4pt] \kanteAP; +\draw[color=gray] \kanteBC; +\draw[color=gray] \kanteBP; +\draw[line width=1.4pt] \kanteCP; + +\punkt{(A)}{black}; +\punkt{(B)}{gray}; +\punkt{(C)}{black}; +\punkt{(P)}{black}; + +\node at (A) [above] {$A$}; +\node[color=gray] at (B) [left] {$B$}; +\node at (C) [right] {$C$}; +\node at (P) [below] {$P$}; + +\end{tikzpicture} +\end{document} + -- cgit v1.2.1 From 7b3657a77eeec57f2dd21de6fdc36e5240560c8e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 13 May 2022 15:13:30 +0200 Subject: improvements --- buch/papers/fm/anim/Makefile | 12 +++++ buch/papers/fm/anim/animation.tex | 85 +++++++++++++++++++++++++++++++++ buch/papers/fm/anim/fm.m | 98 +++++++++++++++++++++++++++++++++++++++ 3 files changed, 195 insertions(+) create mode 100644 buch/papers/fm/anim/Makefile create mode 100644 buch/papers/fm/anim/animation.tex create mode 100644 buch/papers/fm/anim/fm.m (limited to 'buch') diff --git a/buch/papers/fm/anim/Makefile b/buch/papers/fm/anim/Makefile new file mode 100644 index 0000000..f4c7850 --- /dev/null +++ b/buch/papers/fm/anim/Makefile @@ -0,0 +1,12 @@ +# +# Makefile +# +# (c) 2022 Prof Dr Andreas Müller +# +all: animation.pdf + +parts.tex: fm.m + octave fm.m + +animation.pdf: animation.tex parts.tex + pdflatex animation.tex diff --git a/buch/papers/fm/anim/animation.tex b/buch/papers/fm/anim/animation.tex new file mode 100644 index 0000000..4a6f428 --- /dev/null +++ b/buch/papers/fm/anim/animation.tex @@ -0,0 +1,85 @@ +% +% animation.tex +% +% (c) 2022 Prof Dr Andreas Müller, +% +\documentclass[aspectratio=169]{beamer} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{epic} +\usepackage{color} +\usepackage{array} +\usepackage{ifthen} +\usepackage{lmodern} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{nccmath} +\usepackage{mathtools} +\usepackage{adjustbox} +\usepackage{multimedia} +\usepackage{verbatim} +\usepackage{wasysym} +\usepackage{stmaryrd} +\usepackage{tikz} +\usetikzlibrary{shapes.geometric} +\usetikzlibrary{decorations.pathreplacing} +\usetikzlibrary{calc} +\usetikzlibrary{arrows} +\usetikzlibrary{3d} +\usetikzlibrary{arrows,shapes,math,decorations.text,automata} +\usepackage{pifont} +\usepackage[all]{xy} +\usepackage[many]{tcolorbox} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\begin{document} + +\def\spektrum#1#2{ +\only<#1>{ + \begin{scope} + \color{red} + \input{#2} + \end{scope} +} +} + +\begin{frame} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\df{0.37} +\def\da{1} + +\draw[->,color=gray] (0,-0.1) -- (0,6.3) [right] coordinate[label={right:$a$}]; + +\foreach \a in {1,...,5}{ + \draw[color=gray!50] (-6,{(6-\a)*\da}) -- (6,{(6-\a)*\da}); +} +\draw[color=gray!50] (-6,{6*\da}) -- (6,{6*\da}); +\foreach \f in {-15,-10,-5,5,10,15}{ + \draw[color=gray!50] ({\f*\df},0) -- ({\f*\df},{6*\da}); +} + +\input{parts.tex} + +\draw[->] (-6.1,0) -- (6.9,0) coordinate[label={$f$}]; +\foreach \f in {-16,...,16}{ + \draw ({\f*\df},-0.05) -- ({\f*\df},0.05); +} +\foreach \f in {-15,-10,-5,5,10,15}{ + \node at ({\f*\df},-0.1) [below] {$\f f_m$}; + \draw ({\f*\df},-0.1) -- ({\f*\df},0.1); +} +\node at (0,-0.1) [below] {$0$}; + +\foreach \a in {1,...,5}{ + \node at (6,{(6-\a)*\da}) [right] {$-\a$}; +} +\node at (6,{6*\da}) [right] {$\phantom{-}0$}; + +\end{tikzpicture} +\end{center} +\end{frame} + +\end{document} diff --git a/buch/papers/fm/anim/fm.m b/buch/papers/fm/anim/fm.m new file mode 100644 index 0000000..9062818 --- /dev/null +++ b/buch/papers/fm/anim/fm.m @@ -0,0 +1,98 @@ +# +# fm.m -- animation frequenzspektrum +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global fc; +fc = 1e6; +global width; +width = 16; +global fm; +fm = 1000; +global gamma; +gamma = 2; +global resolution; +resolution = 300; + +function retval = spektrum(beta, fm) + global width; + global fc; + retval = zeros(2 * width + 1, 2); + center = width + 1; + for k = (0:width) + retval(center - k, 1) = fc - k * fm; + retval(center + k, 1) = fc + k * fm; + a = besselj(k, beta); + retval(center - k, 2) = a; + retval(center + k, 2) = a; + endfor +endfunction + +function drawspectrum(fn, spectrum, foffset, fscale, beta) + n = size(spectrum)(1,1); + for i = (1:n) + f = (spectrum(i, 1) - foffset)/fscale; + a = log10(spectrum(i, 2)) + 6; + if (a < 0) + a = 0; + end + fprintf(fn, "\\draw[line width=3.5pt] "); + fprintf(fn, "({%.2f*\\df},0) -- ({%.2f*\\df},{%.5f*\\da});\n", + f, f, abs(a)); + fprintf(fn, "\\node at ({-15*\\df},5.5) [right] {$\\beta = %.3f$};", beta); + endfor +endfunction + +function drawhull(fn, beta) + global resolution; + fprintf(fn, "\\begin{scope}\n"); + fprintf(fn, "\\clip ({-16.5*\\df},0) rectangle ({16.5*\\df},{6*\\da});\n"); + p = zeros(resolution, 2); + for k = (1:resolution) + nu = 16.5 * (k - 1) / resolution; + p(k,1) = nu; + y = log10(abs(besselj(nu, beta))) + 6; + p(k,2) = y; + end + fprintf(fn, "\\draw[color=blue] ({%.4f*\\df},{%.5f*\\da})", + p(1,1), p(1,2)); + for k = (2:resolution) + fprintf(fn, "\n -- ({%.4f*\\df},{%.5f*\\da})", + p(k,1), p(k,2)); + endfor + fprintf(fn, ";\n\n"); + fprintf(fn, "\\draw[color=blue] ({%.4f*\\df},{%.5f*\\da})", + p(1,1), p(1,2)); + for k = (2:resolution) + fprintf(fn, "\n -- ({%.4f*\\df},{%.5f*\\da})", + -p(k,1), p(k,2)); + endfor + fprintf(fn, ";\n\n"); + fprintf(fn, "\\end{scope}\n"); +endfunction + +function animation(betamin, betamax, steps) + global fm; + global fc; + global gamma; + fa = fopen("parts.tex", "w"); + for k = (1:steps) + % add entry to parts.tex + fprintf(fa, "\\spektrum{%d}{texfiles/a%04d.tex}\n", k, k); + % compute beta + x = (k - 1) / (steps - 1); + beta = betamin + (betamax - betamin) * (x ^ gamma); + % create a new file + name = sprintf("texfiles/a%04d.tex", k); + fn = fopen(name, "w"); + % write the hull + drawhull(fn, beta); + % compute and write the spectrum + spectrum = spektrum(beta, fm); + drawspectrum(fn, spectrum, fc, fm, beta); + fclose(fn); + endfor + fclose(fa); +endfunction + +animation(0.001,10.1,200) -- cgit v1.2.1 From d223b0ff1fb5364b2b243b8fd4fd7a0e9ffba285 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 13 May 2022 20:12:50 +0200 Subject: 3dimages --- buch/papers/nav/images/Makefile | 66 ++++++++++++++- buch/papers/nav/images/common.inc | 149 ++++++++++++++++++++++++++++++++++ buch/papers/nav/images/dreieck3d1.pov | 58 +++++++++++++ buch/papers/nav/images/dreieck3d1.tex | 53 ++++++++++++ buch/papers/nav/images/dreieck3d2.pov | 26 ++++++ buch/papers/nav/images/dreieck3d2.tex | 53 ++++++++++++ buch/papers/nav/images/dreieck3d3.pov | 37 +++++++++ buch/papers/nav/images/dreieck3d3.tex | 53 ++++++++++++ buch/papers/nav/images/dreieck3d4.pov | 37 +++++++++ buch/papers/nav/images/dreieck3d4.tex | 54 ++++++++++++ buch/papers/nav/images/dreieck3d5.pov | 26 ++++++ buch/papers/nav/images/dreieck3d5.tex | 53 ++++++++++++ buch/papers/nav/images/dreieck3d6.pov | 37 +++++++++ buch/papers/nav/images/dreieck3d6.tex | 55 +++++++++++++ buch/papers/nav/images/dreieck3d7.pov | 39 +++++++++ buch/papers/nav/images/dreieck3d7.tex | 55 +++++++++++++ 16 files changed, 850 insertions(+), 1 deletion(-) create mode 100644 buch/papers/nav/images/common.inc create mode 100644 buch/papers/nav/images/dreieck3d1.pov create mode 100644 buch/papers/nav/images/dreieck3d1.tex create mode 100644 buch/papers/nav/images/dreieck3d2.pov create mode 100644 buch/papers/nav/images/dreieck3d2.tex create mode 100644 buch/papers/nav/images/dreieck3d3.pov create mode 100644 buch/papers/nav/images/dreieck3d3.tex create mode 100644 buch/papers/nav/images/dreieck3d4.pov create mode 100644 buch/papers/nav/images/dreieck3d4.tex create mode 100644 buch/papers/nav/images/dreieck3d5.pov create mode 100644 buch/papers/nav/images/dreieck3d5.tex create mode 100644 buch/papers/nav/images/dreieck3d6.pov create mode 100644 buch/papers/nav/images/dreieck3d6.tex create mode 100644 buch/papers/nav/images/dreieck3d7.pov create mode 100644 buch/papers/nav/images/dreieck3d7.tex (limited to 'buch') diff --git a/buch/papers/nav/images/Makefile b/buch/papers/nav/images/Makefile index 0c1cbc3..c9dcacc 100644 --- a/buch/papers/nav/images/Makefile +++ b/buch/papers/nav/images/Makefile @@ -3,7 +3,7 @@ # # (c) 2022 # -all: dreiecke +all: dreiecke3d dreieck.pdf: dreieck.tex dreieckdata.tex macros.tex pdflatex dreieck.tex @@ -42,3 +42,67 @@ dreieck6.pdf: dreieck6.tex dreieckdata.tex macros.tex dreieck7.pdf: dreieck7.tex dreieckdata.tex macros.tex pdflatex dreieck7.tex + +DREIECKE3D = \ + dreieck3d1.pdf \ + dreieck3d2.pdf \ + dreieck3d3.pdf \ + dreieck3d4.pdf \ + dreieck3d5.pdf \ + dreieck3d6.pdf \ + dreieck3d7.pdf + +dreiecke3d: $(DREIECKE3D) + +POVRAYOPTIONS = -W1080 -H1080 +#POVRAYOPTIONS = -W480 -H480 + +dreieck3d1.png: dreieck3d1.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d1.png dreieck3d1.pov +dreieck3d1.jpg: dreieck3d1.png + convert dreieck3d1.png -density 300 -units PixelsPerInch dreieck3d1.jpg +dreieck3d1.pdf: dreieck3d1.tex dreieck3d1.jpg + pdflatex dreieck3d1.tex + +dreieck3d2.png: dreieck3d2.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d2.png dreieck3d2.pov +dreieck3d2.jpg: dreieck3d2.png + convert dreieck3d2.png -density 300 -units PixelsPerInch dreieck3d2.jpg +dreieck3d2.pdf: dreieck3d2.tex dreieck3d2.jpg + pdflatex dreieck3d2.tex + +dreieck3d3.png: dreieck3d3.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d3.png dreieck3d3.pov +dreieck3d3.jpg: dreieck3d3.png + convert dreieck3d3.png -density 300 -units PixelsPerInch dreieck3d3.jpg +dreieck3d3.pdf: dreieck3d3.tex dreieck3d3.jpg + pdflatex dreieck3d3.tex + +dreieck3d4.png: dreieck3d4.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d4.png dreieck3d4.pov +dreieck3d4.jpg: dreieck3d4.png + convert dreieck3d4.png -density 300 -units PixelsPerInch dreieck3d4.jpg +dreieck3d4.pdf: dreieck3d4.tex dreieck3d4.jpg + pdflatex dreieck3d4.tex + +dreieck3d5.png: dreieck3d5.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d5.png dreieck3d5.pov +dreieck3d5.jpg: dreieck3d5.png + convert dreieck3d5.png -density 300 -units PixelsPerInch dreieck3d5.jpg +dreieck3d5.pdf: dreieck3d5.tex dreieck3d5.jpg + pdflatex dreieck3d5.tex + +dreieck3d6.png: dreieck3d6.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d6.png dreieck3d6.pov +dreieck3d6.jpg: dreieck3d6.png + convert dreieck3d6.png -density 300 -units PixelsPerInch dreieck3d6.jpg +dreieck3d6.pdf: dreieck3d6.tex dreieck3d6.jpg + pdflatex dreieck3d6.tex + +dreieck3d7.png: dreieck3d7.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d7.png dreieck3d7.pov +dreieck3d7.jpg: dreieck3d7.png + convert dreieck3d7.png -density 300 -units PixelsPerInch dreieck3d7.jpg +dreieck3d7.pdf: dreieck3d7.tex dreieck3d7.jpg + pdflatex dreieck3d7.tex + diff --git a/buch/papers/nav/images/common.inc b/buch/papers/nav/images/common.inc new file mode 100644 index 0000000..33d9384 --- /dev/null +++ b/buch/papers/nav/images/common.inc @@ -0,0 +1,149 @@ +// +// common.inc -- 3d Darstellung +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.034; + +#declare A = vnormalize(< 0, 1, 0>); +#declare B = vnormalize(< 1, 2, -8>); +#declare C = vnormalize(< 5, 1, 0>); +#declare P = vnormalize(< 5, -1, -7>); + +camera { + location <40, 20, -20> + look_at <0, 0.24, -0.20> + right x * imagescale + up y * imagescale +} + +light_source { + <10, 10, -40> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +// +// draw an arrow from to with thickness with +// color +// +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +#macro grosskreis(normale, staerke) +union { + #declare v1 = vcross(normale, ); + #declare v1 = vnormalize(v1); + #declare v2 = vnormalize(vcross(v1, normale)); + #declare phisteps = 100; + #declare phistep = pi / phisteps; + #declare phi = 0; + #declare p1 = v1; + #while (phi < 2 * pi - phistep/2) + sphere { p1, staerke } + #declare phi = phi + phistep; + #declare p2 = v1 * cos(phi) + v2 * sin(phi); + cylinder { p1, p2, staerke } + #declare p1 = p2; + #end +} +#end + +#macro seite(p, q, staerke) + #declare n = vcross(p, q); + intersection { + grosskreis(n, staerke) + plane { -vcross(n, q) * vdot(vcross(n, q), p), 0 } + plane { -vcross(n, p) * vdot(vcross(n, p), q), 0 } + } +#end + +#macro winkel(w, p, q, staerke) + #declare n = vnormalize(w); + #declare pp = vnormalize(p - vdot(n, p) * n); + #declare qq = vnormalize(q - vdot(n, q) * n); + intersection { + sphere { <0, 0, 0>, 1 + staerke } + cone { <0, 0, 0>, 0, 1.2 * vnormalize(w), 0.4 } + plane { -vcross(n, qq) * vdot(vcross(n, qq), pp), 0 } + plane { -vcross(n, pp) * vdot(vcross(n, pp), qq), 0 } + } +#end + +#macro punkt(p, staerke) + sphere { p, 1.5 * staerke } +#end + +#declare fett = 0.015; +#declare fine = 0.010; + +#declare dreieckfarbe = rgb<0.6,0.6,0.6>; +#declare rot = rgb<0.8,0.2,0.2>; +#declare gruen = rgb<0,0.6,0>; +#declare blau = rgb<0.2,0.2,0.8>; + +sphere { + <0, 0, 0>, 1 + pigment { + color rgb<0.8,0.8,0.8> + } +} + +//union { +// sphere { A, 0.02 } +// sphere { B, 0.02 } +// sphere { C, 0.02 } +// sphere { P, 0.02 } +// pigment { +// color Red +// } +//} + +//union { +// winkel(A, B, C) +// winkel(B, P, C) +// seite(B, C, 0.01) +// seite(B, P, 0.01) +// pigment { +// color rgb<0,0.6,0> +// } +//} diff --git a/buch/papers/nav/images/dreieck3d1.pov b/buch/papers/nav/images/dreieck3d1.pov new file mode 100644 index 0000000..8afe60e --- /dev/null +++ b/buch/papers/nav/images/dreieck3d1.pov @@ -0,0 +1,58 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fett) + seite(B, C, fett) + seite(A, C, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fine) + seite(B, P, fine) + seite(C, P, fine) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, B, C, fine) + pigment { + color rot + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, C, A, fine) + pigment { + color gruen + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(C, A, B, fine) + pigment { + color blau + } + finish { + specular 0.95 + metallic + } +} diff --git a/buch/papers/nav/images/dreieck3d1.tex b/buch/papers/nav/images/dreieck3d1.tex new file mode 100644 index 0000000..799b21a --- /dev/null +++ b/buch/papers/nav/images/dreieck3d1.tex @@ -0,0 +1,53 @@ +% +% dreieck3d1.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d1.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +\node at (0.7,3) {$\alpha$}; +\node at (-2.5,-0.5) {$\beta$}; +\node at (2.3,-1.2) {$\gamma$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d2.pov b/buch/papers/nav/images/dreieck3d2.pov new file mode 100644 index 0000000..c23a54c --- /dev/null +++ b/buch/papers/nav/images/dreieck3d2.pov @@ -0,0 +1,26 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fett) + seite(B, C, fett) + seite(A, C, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fine) + seite(B, P, fine) + seite(C, P, fine) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d2.tex b/buch/papers/nav/images/dreieck3d2.tex new file mode 100644 index 0000000..0f6e10c --- /dev/null +++ b/buch/papers/nav/images/dreieck3d2.tex @@ -0,0 +1,53 @@ +% +% dreieck3d2.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d2.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +%\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d3.pov b/buch/papers/nav/images/dreieck3d3.pov new file mode 100644 index 0000000..f2496b5 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d3.pov @@ -0,0 +1,37 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fett) + seite(B, C, fett) + seite(A, C, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fine) + seite(B, P, fine) + seite(C, P, fine) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, B, C, fine) + pigment { + color rot + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d3.tex b/buch/papers/nav/images/dreieck3d3.tex new file mode 100644 index 0000000..a047b1b --- /dev/null +++ b/buch/papers/nav/images/dreieck3d3.tex @@ -0,0 +1,53 @@ +% +% dreieck3d3.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d3.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +%\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d4.pov b/buch/papers/nav/images/dreieck3d4.pov new file mode 100644 index 0000000..bddcf7c --- /dev/null +++ b/buch/papers/nav/images/dreieck3d4.pov @@ -0,0 +1,37 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fine) + seite(A, C, fine) + punkt(A, fine) + punkt(B, fett) + punkt(C, fett) + punkt(P, fett) + seite(B, C, fett) + seite(B, P, fett) + seite(C, P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, C, P, fine) + pigment { + color rgb<0.6,0.4,0.2> + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d4.tex b/buch/papers/nav/images/dreieck3d4.tex new file mode 100644 index 0000000..d49fb66 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d4.tex @@ -0,0 +1,54 @@ +% +% dreieck3d4.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d4.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +%\node at (-1.9,2.1) {$c$}; +\node at (-0.2,-1.2) {$a$}; +%\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; +\node at (-2.3,-1.5) {$\beta_1$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d5.pov b/buch/papers/nav/images/dreieck3d5.pov new file mode 100644 index 0000000..32fc9e6 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d5.pov @@ -0,0 +1,26 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fine) + seite(A, C, fine) + punkt(A, fine) + punkt(B, fett) + punkt(C, fett) + punkt(P, fett) + seite(B, C, fett) + seite(B, P, fett) + seite(C, P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d5.tex b/buch/papers/nav/images/dreieck3d5.tex new file mode 100644 index 0000000..8011b37 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d5.tex @@ -0,0 +1,53 @@ +% +% dreieck3d5.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d5.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +%\node at (-1.9,2.1) {$c$}; +%\node at (-0.2,-1.2) {$a$}; +%\node at (2.6,1.5) {$b$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d6.pov b/buch/papers/nav/images/dreieck3d6.pov new file mode 100644 index 0000000..7611950 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d6.pov @@ -0,0 +1,37 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fett) + seite(A, C, fett) + seite(B, P, fett) + seite(C, P, fett) + seite(A, P, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, A, P, fine) + pigment { + color rgb<0.6,0.2,0.6> + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d6.tex b/buch/papers/nav/images/dreieck3d6.tex new file mode 100644 index 0000000..bbca2ca --- /dev/null +++ b/buch/papers/nav/images/dreieck3d6.tex @@ -0,0 +1,55 @@ +% +% dreieck3d6.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d6.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +%\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; +\node at (-0.7,0.3) {$l$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; +\node at (-2.4,-0.6) {$\kappa$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/nav/images/dreieck3d7.pov b/buch/papers/nav/images/dreieck3d7.pov new file mode 100644 index 0000000..fa48f5b --- /dev/null +++ b/buch/papers/nav/images/dreieck3d7.pov @@ -0,0 +1,39 @@ +// +// dreiecke3d.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, C, fett) + seite(A, P, fett) + seite(C, P, fett) + + seite(A, B, fine) + seite(B, C, fine) + seite(B, P, fine) + punkt(A, fett) + punkt(C, fett) + punkt(P, fett) + punkt(B, fine) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, P, C, fine) + pigment { + color rgb<0.4,0.4,1> + } + finish { + specular 0.95 + metallic + } +} + diff --git a/buch/papers/nav/images/dreieck3d7.tex b/buch/papers/nav/images/dreieck3d7.tex new file mode 100644 index 0000000..4027a8b --- /dev/null +++ b/buch/papers/nav/images/dreieck3d7.tex @@ -0,0 +1,55 @@ +% +% dreieck3d7.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d7.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; +\node at (-0.7,0.3) {$l$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +%\node at (0.7,3) {$\alpha$}; +%\node at (-2.5,-0.5) {$\beta$}; +%\node at (2.3,-1.2) {$\gamma$}; +\node at (0.8,3.1) {$\omega$}; + +\end{tikzpicture} + +\end{document} + -- cgit v1.2.1 From fc8bf49548f168fe0a77e1446c73ff7be5d980cf Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 13 May 2022 23:11:38 +0200 Subject: fresnel paper erste Fassung --- buch/papers/fresnel/Makefile | 15 ++- buch/papers/fresnel/eulerspirale.m | 61 +++++++++ buch/papers/fresnel/eulerspirale.pdf | Bin 0 -> 22592 bytes buch/papers/fresnel/eulerspirale.tex | 41 ++++++ buch/papers/fresnel/fresnelgraph.pdf | Bin 0 -> 30018 bytes buch/papers/fresnel/fresnelgraph.tex | 46 +++++++ buch/papers/fresnel/main.tex | 24 +--- buch/papers/fresnel/pfad.pdf | Bin 0 -> 19126 bytes buch/papers/fresnel/pfad.tex | 34 +++++ buch/papers/fresnel/references.bib | 11 ++ buch/papers/fresnel/teil0.tex | 109 +++++++++++++--- buch/papers/fresnel/teil1.tex | 239 ++++++++++++++++++++++++++++------- buch/papers/fresnel/teil2.tex | 48 +++---- buch/papers/fresnel/teil3.tex | 136 +++++++++++++++----- 14 files changed, 617 insertions(+), 147 deletions(-) create mode 100644 buch/papers/fresnel/eulerspirale.m create mode 100644 buch/papers/fresnel/eulerspirale.pdf create mode 100644 buch/papers/fresnel/eulerspirale.tex create mode 100644 buch/papers/fresnel/fresnelgraph.pdf create mode 100644 buch/papers/fresnel/fresnelgraph.tex create mode 100644 buch/papers/fresnel/pfad.pdf create mode 100644 buch/papers/fresnel/pfad.tex (limited to 'buch') diff --git a/buch/papers/fresnel/Makefile b/buch/papers/fresnel/Makefile index c8aa073..11af3a7 100644 --- a/buch/papers/fresnel/Makefile +++ b/buch/papers/fresnel/Makefile @@ -1,9 +1,22 @@ # # Makefile -- make file for the paper fresnel # -# (c) 2020 Prof Dr Andreas Mueller +# (c) 2022 Prof Dr Andreas Mueller # +all: fresnelgraph.pdf eulerspirale.pdf pfad.pdf images: @echo "no images to be created in fresnel" +eulerpath.tex: eulerspirale.m + octave eulerspirale.m + +fresnelgraph.pdf: fresnelgraph.tex eulerpath.tex + pdflatex fresnelgraph.tex + +eulerspirale.pdf: eulerspirale.tex eulerpath.tex + pdflatex eulerspirale.tex + +pfad.pdf: pfad.tex + pdflatex pfad.tex + diff --git a/buch/papers/fresnel/eulerspirale.m b/buch/papers/fresnel/eulerspirale.m new file mode 100644 index 0000000..84e3696 --- /dev/null +++ b/buch/papers/fresnel/eulerspirale.m @@ -0,0 +1,61 @@ +# +# eulerspirale.m +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue +# +global n; +n = 1000; +global tmax; +tmax = 10; +global N; +N = round(n*5/tmax); + +function retval = f(x, t) + x = pi * t^2 / 2; + retval = [ cos(x); sin(x) ]; +endfunction + +x0 = [ 0; 0 ]; +t = tmax * (0:n) / n; + +c = lsode(@f, x0, t); + +fn = fopen("eulerpath.tex", "w"); + +fprintf(fn, "\\def\\fresnela{ (0,0)"); +for i = (2:n) + fprintf(fn, "\n\t-- (%.4f,%.4f)", c(i,1), c(i,2)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\fresnelb{ (0,0)"); +for i = (2:n) + fprintf(fn, "\n\t-- (%.4f,%.4f)", -c(i,1), -c(i,2)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\Cplotright{ (0,0)"); +for i = (2:N) + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", t(i), c(i,1)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\Cplotleft{ (0,0)"); +for i = (2:N) + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", -t(i), -c(i,1)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\Splotright{ (0,0)"); +for i = (2:N) + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", t(i), c(i,2)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\Splotleft{ (0,0)"); +for i = (2:N) + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", -t(i), -c(i,2)); +end +fprintf(fn, "\n}\n\n"); + +fclose(fn); diff --git a/buch/papers/fresnel/eulerspirale.pdf b/buch/papers/fresnel/eulerspirale.pdf new file mode 100644 index 0000000..4a85a50 Binary files /dev/null and b/buch/papers/fresnel/eulerspirale.pdf differ diff --git a/buch/papers/fresnel/eulerspirale.tex b/buch/papers/fresnel/eulerspirale.tex new file mode 100644 index 0000000..38ef756 --- /dev/null +++ b/buch/papers/fresnel/eulerspirale.tex @@ -0,0 +1,41 @@ +% +% eulerspirale.tex -- Darstellung der Eulerspirale +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{eulerpath.tex} + +\def\s{8} + +\begin{scope}[scale=\s] +\draw[color=blue] (-0.5,-0.5) rectangle (0.5,0.5); +\draw[color=darkgreen,line width=1.4pt] \fresnela; +\draw[color=darkgreen,line width=1.4pt] \fresnelb; +\fill[color=blue] (0.5,0.5) circle[radius={0.1/\s}]; +\fill[color=blue] (-0.5,-0.5) circle[radius={0.1/\s}]; +\draw (-0.5,{-0.05/\s}) -- (-0.5,{0.05/\s}); +\draw (0.5,{-0.05/\s}) -- (0.5,{-0.05/\s}); +\node at (-0.5,0) [above left] {$\frac12$}; +\node at (0.5,0) [below right] {$\frac12$}; +\node at (0,-0.5) [below right] {$\frac12$}; +\node at (0,0.5) [above left] {$\frac12$}; +\end{scope} + +\draw[->] (-6.7,0) -- (6.9,0) coordinate[label={$C(x)$}];; +\draw[->] (0,-5.8) -- (0,6.1) coordinate[label={left:$S(x)$}];; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/fresnelgraph.pdf b/buch/papers/fresnel/fresnelgraph.pdf new file mode 100644 index 0000000..9ccad56 Binary files /dev/null and b/buch/papers/fresnel/fresnelgraph.pdf differ diff --git a/buch/papers/fresnel/fresnelgraph.tex b/buch/papers/fresnel/fresnelgraph.tex new file mode 100644 index 0000000..20df951 --- /dev/null +++ b/buch/papers/fresnel/fresnelgraph.tex @@ -0,0 +1,46 @@ +% +% fresnelgraph.tex -- Graphs of the fresnel functions +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{eulerpath.tex} +\def\dx{1.3} +\def\dy{2.6} + +\draw[color=gray] (0,{0.5*\dy}) -- ({5*\dx},{0.5*\dy}); +\draw[color=gray] (0,{-0.5*\dy}) -- ({-5*\dx},{-0.5*\dy}); + +\draw[color=blue,line width=1.4pt] \Splotright; +\draw[color=blue,line width=1.4pt] \Splotleft; + +\draw[color=red,line width=1.4pt] \Cplotright; +\draw[color=red,line width=1.4pt] \Cplotleft; + +\draw[->] (-6.7,0) -- (6.9,0) coordinate[label={$x$}]; +\draw[->] (0,-2.3) -- (0,2.3) coordinate[label={$y$}]; + +\foreach \x in {1,2,3,4,5}{ + \draw ({\x*\dx},-0.05) -- ({\x*\dx},0.05); + \draw ({-\x*\dx},-0.05) -- ({-\x*\dx},0.05); + \node at ({\x*\dx},-0.05) [below] {$\x$}; + \node at ({-\x*\dx},0.05) [above] {$-\x$}; +} +\draw (-0.05,{0.5*\dy}) -- (0.05,{0.5*\dy}); +\node at (-0.05,{0.5*\dy}) [left] {$\frac12$}; +\draw (-0.05,{-0.5*\dy}) -- (0.05,{-0.5*\dy}); +\node at (0.05,{-0.5*\dy}) [right] {$-\frac12$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/main.tex b/buch/papers/fresnel/main.tex index bbaf7e6..e6ee3b5 100644 --- a/buch/papers/fresnel/main.tex +++ b/buch/papers/fresnel/main.tex @@ -3,29 +3,11 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:fresnel}} -\lhead{Thema} +\chapter{Fresnel-Integrale\label{chapter:fresnel}} +\lhead{Fresnel-Integrale} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Andreas Müller} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} \input{papers/fresnel/teil0.tex} \input{papers/fresnel/teil1.tex} diff --git a/buch/papers/fresnel/pfad.pdf b/buch/papers/fresnel/pfad.pdf new file mode 100644 index 0000000..ff514cc Binary files /dev/null and b/buch/papers/fresnel/pfad.pdf differ diff --git a/buch/papers/fresnel/pfad.tex b/buch/papers/fresnel/pfad.tex new file mode 100644 index 0000000..5439a71 --- /dev/null +++ b/buch/papers/fresnel/pfad.tex @@ -0,0 +1,34 @@ +% +% pfad.tex -- template for standalon tikz images +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\draw[->] (-1,0) -- (9,0) coordinate[label={$\operatorname{Re}$}]; +\draw[->] (0,-1) -- (0,6) coordinate[label={left:$\operatorname{Im}$}]; + +\draw[->,color=red,line width=1.4pt] (0,0) -- (7,0); +\draw[->,color=blue,line width=1.4pt] (7,0) arc (0:45:7); +\draw[->,color=darkgreen,line width=1.4pt] (45:7) -- (0,0); + +\node[color=red] at (3.5,0) [below] {$\gamma_1(t) = tR$}; +\node[color=blue] at (25:7) [right] {$\gamma_2(t) = Re^{it}$}; +\node[color=darkgreen] at (45:3.5) [above left] {$\gamma_3(t) = te^{i\pi/4}$}; + +\node at (7,0) [below] {$R$}; +\node at (45:7) [above] {$Re^{i\pi/4}$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/references.bib b/buch/papers/fresnel/references.bib index 84cd3bc..58e9242 100644 --- a/buch/papers/fresnel/references.bib +++ b/buch/papers/fresnel/references.bib @@ -33,3 +33,14 @@ url = {https://doi.org/10.1016/j.acha.2017.11.004} } +@online{fresnel:fresnelC, + url = { https://functions.wolfram.com/GammaBetaErf/FresnelC/introductions/FresnelIntegrals/ShowAll.html }, + title = { FresnelC }, + date = { 2022-05-13 } +} + +@online{fresnel:wikipedia, + url = { https://en.wikipedia.org/wiki/Fresnel_integral }, + title = { Fresnel Integral }, + date = { 2022-05-13 } +} diff --git a/buch/papers/fresnel/teil0.tex b/buch/papers/fresnel/teil0.tex index 5e9fdaf..253e2f3 100644 --- a/buch/papers/fresnel/teil0.tex +++ b/buch/papers/fresnel/teil0.tex @@ -1,22 +1,101 @@ % -% einleitung.tex -- Beispiel-File für die Einleitung +% teil0.tex -- Definition % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 0\label{fresnel:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{fresnel:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. +\section{Definition\label{fresnel:section:teil0}} +\rhead{Definition} +Die Funktion $e^{x^2}$ hat bekanntermassen keine elementare Stammfunktion, +weshalb die Fehlerfunktion als Stammfunktion definiert wurde. +Die Funktionen $\cos x^2$ und $\sin x^2$ sind eng mit $e^{x^2}$ +verwandt, es ist daher nicht überraschend, dass sie ebenfalls +keine elementare Stammfunktionen haben. +Dies rechtfertigt die Definition der Fresnel-Integrale als neue spezielle +Funktionen. -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. +\begin{definition} +Die Funktionen +\begin{align*} +C(x) &= \int_0^x \cos\biggl(\frac{\pi}2 t^2\biggr)\,dt +\\ +S(x) &= \int_0^x \sin\biggl(\frac{\pi}2 t^2\biggr)\,dt +\end{align*} +heissen die Fesnel-Integrale. +\end{definition} +Der Faktor $\frac{\pi}2$ ist einigermassen willkürlich, man könnte +daher noch allgemeiner die Funktionen +\begin{align*} +C_a(x) &= \int_0^x \cos(at^2)\,dt +\\ +S_a(x) &= \int_0^x \sin(at^2)\,dt +\end{align*} +definieren, so dass die Funktionen $C(x)$ und $S(x)$ der Fall +$a=\frac{\pi}2$ werden, also +\[ +\begin{aligned} +C(x) &= C_{\frac{\pi}2}(x), +& +S(x) &= S_{\frac{\pi}2}(x). +\end{aligned} +\] +Durch eine Substution $t=bs$ erhält man +\begin{align*} +C_a(x) +&= +\int_0^x \cos(at^2)\,dt += +b +\int_0^{\frac{x}b} \cos(ab^2s^2)\,ds += +b +C_{ab^2}\biggl(\frac{x}b\biggr) +\\ +S_a(x) +&= +\int_0^x \sin(at^2)\,dt += +b +\int_0^{\frac{x}b} \sin(ab^2s^2)\,ds += +b +S_{ab^2}\biggl(\frac{x}b\biggr). +\end{align*} +Indem man $ab^2=\frac{\pi}2$ setzt, also +\[ +b += +\sqrt{\frac{\pi}{2a}} +, +\] +kann man die Funktionen $C_a(x)$ und $S_a(x)$ durch $C(x)$ und $S(x)$ +ausdrücken: +\begin{align} +C_a(x) +&= +\sqrt{\frac{\pi}{2a}} +C\biggl(x +\sqrt{\frac{2a}{\pi}} +\biggr) +&&\text{und}& +S_a(x) +&= +\sqrt{\frac{\pi}{2a}} +S\biggl(x +\sqrt{\frac{2a}{\pi}} +\biggr). +\label{fresnel:equation:arg} +\end{align} +Im Folgenden werden wir meistens nur den Fall $a=1$, also die Funktionen +$C_1(x)$ und $S_1(x)$ betrachten, da in diesem Fall die Formeln einfacher +werden. +\begin{figure} +\centering +\includegraphics{papers/fresnel/fresnelgraph.pdf} +\caption{Graph der Funktionen $C(x)$ ({\color{red}rot}) +und $S(x)$ ({\color{blue}blau}) +\label{fresnel:figure:plot}} +\end{figure} +Die Abbildung~\ref{fresnel:figure:plot} zeigt die Graphen der +Funktion $C(x)$ und $S(x)$. diff --git a/buch/papers/fresnel/teil1.tex b/buch/papers/fresnel/teil1.tex index a2df138..df84797 100644 --- a/buch/papers/fresnel/teil1.tex +++ b/buch/papers/fresnel/teil1.tex @@ -1,55 +1,202 @@ % -% teil1.tex -- Beispiel-File für das Paper +% teil1.tex -- Euler-Spirale % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 1 -\label{fresnel:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{fresnel:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. +\section{Euler-Spirale +\label{fresnel:section:eulerspirale}} +\rhead{Euler-Spirale} +\begin{figure} +\centering +\includegraphics{papers/fresnel/eulerspirale.pdf} +\caption{Die Eulerspirale ist die Kurve mit der Parameterdarstellung +$x\mapsto (C(x),S(x))$, sie ist rot dargestellt. +Sie windet sich unendlich oft um die beiden Punkte $(\pm\frac12,\pm\frac12)$. +\label{fresnel:figure:eulerspirale}} +\end{figure} +Ein besseres Verständnis für die beiden Funktionen $C(x)$ und $S(x)$ +als die Darstellung~\ref{fresnel:figure:plot} ermöglicht die +Abbildung~\ref{fresnel:figure:eulerspirale}, die die beiden Funktionen +als die $x$- und $y$-Koordinaten der Parameterdarstellung einer Kurve +zeigt. +Sie heisst die {\em Euler-Spirale}. +Die Spirale scheint sich für $x\to\pm\infty$ um die Punkte +$(\pm\frac12,\pm\frac12)$ zu winden. -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? +\begin{figure} +\centering +\includegraphics{papers/fresnel/pfad.pdf} +\caption{Pfad zur Berechnung der Grenzwerte $C_1(\infty)$ und +$S_1(\infty)$ mit Hilfe des Cauchy-Integralsatzes +\label{fresnel:figure:pfad}} +\end{figure} -\subsection{De finibus bonorum et malorum -\label{fresnel:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. -Et harum quidem rerum facilis est et expedita distinctio -\ref{fresnel:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{fresnel:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. +\begin{satz} +Die Grenzwerte der Fresnel-Integrale für $x\to\pm\infty$ sind +\[ +\lim_{x\to\pm\infty} C(x) += +\lim_{x\to\pm\infty} S(x) += +\frac12. +\] +\end{satz} +\begin{proof}[Beweis] +Die komplexe Funktion +\[ +f(z) = e^{-z^2} +\] +ist eine ganze Funktion, das Integral über einen geschlossenen +Pfad in der komplexen Ebene verschwindet daher. +Wir verwenden den Pfad in Abbildung~\ref{fresnel:figure:pfad} +bestehend aus den drei Segmenten $\gamma_1$ entlang der reellen +Achse von $0$ bis $R$, dem Kreisbogen $\gamma_2$ um $0$ mit Radius $R$ +und $\gamma_3$ mit der Parametrisierung $t\mapsto te^{i\pi/4}$. + +Das Teilintegral über $\gamma_1$ ist +\[ +\lim_{R\to\infty} +\int_{\gamma_1} e^{-z^2}\,dz += +\int_0^\infty e^{-t^2}\,dt += +\frac{\sqrt{\pi}}2. +\] +Das Integral über $\gamma_3$ ist +\begin{align*} +\lim_{R\to\infty} +\int_{\gamma_3} +e^{-z^2}\,dz +&= +-\int_0^\infty \exp(-t^2 e^{i\pi/2}) e^{i\pi/4}\,dt += +- +\int_0^\infty e^{-it^2}\,dt\, +e^{i\pi/4} +\\ +&= +-e^{i\pi/4}\int_0^\infty \cos t^2 - i \sin t^2\,dt +\\ +&= +-\frac{1}{\sqrt{2}}(1+i) +\bigl( +C_1(\infty) +-i +S_1(\infty) +\bigr) +\\ +&= +-\frac{1}{\sqrt{2}} +\bigl( +C_1(\infty)+S_1(\infty) ++ +i(C_1(\infty)-S_1(\infty)) +\bigr), +\end{align*} +wobei wir +\[ +C_1(\infty) = \lim_{R\to\infty} C_1(R) +\qquad\text{und}\qquad +S_1(\infty) = \lim_{R\to\infty} S_1(R) +\] +abgekürzt haben. +Das Integral über das Segment $\gamma_2$ lässt sich +mit der Parametrisierung +\( +\gamma_2(t) += +Re^{it} += +R(\cos t + i\sin t) +\) +wie folgt +abschätzen: +\begin{align*} +\biggl|\int_{\gamma_2} e^{-z^2} \,dz\biggr| +&= +\biggl| +\int_0^{\frac{\pi}4} +\exp(-R^2(\cos 2t + i\sin 2t)) iR e^{it}\,dt +\biggr| +\\ +&\le +R +\int_0^{\frac{\pi}4} +e^{-R^2\cos 2t} +\,dt +\le +R +\int_0^{\frac{\pi}4} +e^{-R^2(1-\frac{4}{\pi}t)} +\,dt. +\intertext{Dabei haben wir $\cos 2t\ge 1-\frac{4}\pi t$ verwendet. +Mit dieser Vereinfachung kann das Integral ausgewertet werden und +ergibt} +&= +Re^{-R^2} +\int_0^{\frac{\pi}4} +e^{R^2\frac{\pi}4t} +\,dt += +Re^{-R^2} +\biggl[ +\frac{4}{\pi R^2} +e^{R^2\frac{\pi}4t} +\biggr]_0^{\frac{\pi}4} += +\frac{4}{\pi R} +e^{-R^2}(e^{R^2}-1) += +\frac{4}{\pi R} +(1-e^{-R^2}) +\to 0 +\end{align*} +für $R\to \infty$. +Im Grenzwert $R\to \infty$ kann der Teil $\gamma_2$ des Pfades +vernachlässigt werden. + +Das Integral über den geschlossenen Pfad $\gamma$ verschwindet. +Da der Teil $\gamma_2$ keine Rolle spielt, müssen sich die +Integrale über $\gamma_1$ und $\gamma_3$ wegheben, also +\begin{align*} +0 += +\int_\gamma e^{-z^2}\,dz +&= +\int_{\gamma_1} e^{-z^2}\,dz ++ +\int_{\gamma_2} e^{-z^2}\,dz ++ +\int_{\gamma_3} e^{-z^2}\,dz +\\ +&\to +\frac{\sqrt{\pi}}2 +-\frac{1}{\sqrt{2}}(C_1(\infty)+S_1(\infty)) +-\frac{i}{\sqrt{2}}(C_1(\infty)-S_1(\infty)). +\end{align*} +Der Imaginärteil ist $C_1(\infty)-S_1(\infty)$, da er verschwinden +muss, folgt $C_1(\infty)=S_1(\infty)$. +Nach Multlikation mit $\sqrt{2}$ folgt aus der Tatsache, dass auch +der Realteil verschwinden muss +\[ +\frac{\sqrt{\pi}}{\sqrt{2}} = C_1(\infty)+S_1(\infty) +\qquad +\Rightarrow +\qquad +C_1(\infty) += +S_1(\infty) += +\frac{\sqrt{\pi}}{2\sqrt{2}} +\] +Aus +\eqref{fresnel:equation:arg} +erhält man dann auch die Grenzwerte +\[ +C(\infty)=S(\infty)=\frac12. +\qedhere +\] +\end{proof} diff --git a/buch/papers/fresnel/teil2.tex b/buch/papers/fresnel/teil2.tex index 701c3ee..22d2a89 100644 --- a/buch/papers/fresnel/teil2.tex +++ b/buch/papers/fresnel/teil2.tex @@ -3,38 +3,22 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 2 -\label{fresnel:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? +\section{Klothoide +\label{fresnel:section:klothoide}} +\rhead{Klothoide} +In diesem Abschnitt soll gezeigt werden, dass die Krümmung der +Euler-Spirale proportional zur vom Nullpunkt aus gemessenen Bogenlänge +ist. -\subsection{De finibus bonorum et malorum -\label{fresnel:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +\begin{definition} +Eine ebene Kurve, deren Krümmung proportionale zur Kurvenlänge ist, +heisst {\em Klothoide}. +\end{definition} +Die Klothoide wird zum Beispiel im Strassenbau bei Autobahnkurven +angewendet. +Fährt man mit konstanter Geschwindigkeit mit entlang einer Klothoide, +muss man die Krümmung mit konstaner Geschwindigkeit ändern, +also das Lenkrad mit konstanter Geschwindigkeit drehen. +Dies ermöglicht eine ruhige Fahrweise. diff --git a/buch/papers/fresnel/teil3.tex b/buch/papers/fresnel/teil3.tex index d4f15f6..a5b5878 100644 --- a/buch/papers/fresnel/teil3.tex +++ b/buch/papers/fresnel/teil3.tex @@ -3,38 +3,110 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 3 -\label{fresnel:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? +\section{Numerische Berechnung der Fresnel-Integrale +\label{fresnel:section:numerik}} +\rhead{Numerische Berechnung} +Die Fresnel-Integrale können mit verschiedenen Methoden effizient berechnet +werden. -\subsection{De finibus bonorum et malorum -\label{fresnel:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +\subsection{Komplexe Fehlerfunktionen} +Es wurde schon darauf hingewiesen, dass der Integrand der Fresnel-Integrale +mit $e^{t^2}$ verwandt ist. +Tatsächlich kann gezeigt werden dass sich die Fresnel-Integrale mit +Hilfe der komplexen Fehlerfunktion als +\[ +\left. +\begin{matrix} +S_1(z) +\\ +C_1(z) +\end{matrix} +\; +\right\} += +\frac{1\pm i}4\biggl( +\operatorname{erf}\biggl(\frac{1+i}2\sqrt{\pi}z\biggr) +\mp +\operatorname{erf}\biggl(\frac{1-i}2\sqrt{\pi}z\biggr) +\biggr) +\] +ausdrücken lassen. +Diese Darstellung ist jedoch für die numerische Berechnung nur +beschränkt nützlich, weil die meisten Bibliotheken für die Fehlerfunktion +diese nur für reelle Argument auszuwerten gestatten. + +\subsection{Als Lösung einer Differentialgleichung} +Da die Fresnel-Integrale die sehr einfachen Differentialgleichungen +\[ +C'(x) = \cos \biggl(\frac{\pi}2 x^2\biggr) +\qquad\text{und}\qquad +S'(x) = \sin \biggl(\frac{\pi}2 x^2\biggr) +\] +erfüllen, kann man eine Methode zur Lösung von Differentialgleichung +verwenden. +Die Abbildungen~\ref{fresnel:figure:plot} und \ref{fresnel:figure:eulerspirale} +wurden auf diese Weise erzeugt. + +\subsection{Taylor-Reihe integrieren} +Die Taylorreihen +\begin{align*} +\cos x +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k)!} x^{2k} +&&\text{und}& +\sin x +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} x^{2k+1} +\intertext{% +der trigonometrischen Funktionen werden durch Einsetzen von $x=t^2$ +zu} +\cos t^2 +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k)!} t^{4k} +&&\text{und}& +\sin t^2 +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} t^{4k+2}. +\intertext{% +Die Fresnel-Integrale $C_1(x)$ und $S_1(x)$ können daher durch +termweise Integration mit Hilfe der Reihen} +C_1(x) +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k)!} \frac{x^{4k+1}}{4k+1} +&&\text{und}& +S_1(x) +&= +\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} \frac{x^{4k+3}}{4k+3} +\end{align*} +berechnet werden. +Diese Reihen sind insbesondere für kleine Werte von $x$ sehr +schnell konvergent. + +\subsection{Hypergeometrische Reihen} +Aus der Reihenentwicklung kann jetzt auch eine Darstellung der +Fresnel-Integrale durch hypergeometrische Reihen gefunden werden +\cite{fresnel:fresnelC}. +Es ergibt sich +\begin{align*} +S(z) +&= +\frac{\pi z^3}{6} +\cdot +\mathstrut_1F_2\biggl( +\begin{matrix}\frac34\\\frac32,\frac74\end{matrix} +; +-\frac{\pi^2z^4}{16} +\biggr) +\\ +C(z) +&= +z +\cdot +\mathstrut_1F_2\biggl( +\begin{matrix}\frac14\\\frac12,\frac54\end{matrix} +; +-\frac{\pi^2z^4}{16} +\biggr). +\end{align*} -- cgit v1.2.1 From 2b1142edd31d88f9c4b050abf4aeb1e885925ad5 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 13 May 2022 23:15:21 +0200 Subject: typos --- buch/papers/fresnel/teil1.tex | 6 +++--- buch/papers/fresnel/teil3.tex | 2 +- 2 files changed, 4 insertions(+), 4 deletions(-) (limited to 'buch') diff --git a/buch/papers/fresnel/teil1.tex b/buch/papers/fresnel/teil1.tex index df84797..a41ddb7 100644 --- a/buch/papers/fresnel/teil1.tex +++ b/buch/papers/fresnel/teil1.tex @@ -45,9 +45,9 @@ Die Grenzwerte der Fresnel-Integrale für $x\to\pm\infty$ sind \begin{proof}[Beweis] Die komplexe Funktion -\[ +\( f(z) = e^{-z^2} -\] +\) ist eine ganze Funktion, das Integral über einen geschlossenen Pfad in der komplexen Ebene verschwindet daher. Wir verwenden den Pfad in Abbildung~\ref{fresnel:figure:pfad} @@ -190,7 +190,7 @@ C_1(\infty) = S_1(\infty) = -\frac{\sqrt{\pi}}{2\sqrt{2}} +\frac{\sqrt{\pi}}{2\sqrt{2}}. \] Aus \eqref{fresnel:equation:arg} diff --git a/buch/papers/fresnel/teil3.tex b/buch/papers/fresnel/teil3.tex index a5b5878..37e6bee 100644 --- a/buch/papers/fresnel/teil3.tex +++ b/buch/papers/fresnel/teil3.tex @@ -30,7 +30,7 @@ C_1(z) \operatorname{erf}\biggl(\frac{1-i}2\sqrt{\pi}z\biggr) \biggr) \] -ausdrücken lassen. +ausdrücken lassen \cite{fresnel:fresnelC}. Diese Darstellung ist jedoch für die numerische Berechnung nur beschränkt nützlich, weil die meisten Bibliotheken für die Fehlerfunktion diese nur für reelle Argument auszuwerten gestatten. -- cgit v1.2.1 From a28b0e8a16564e78aaecc299526fa8bb96964e0e Mon Sep 17 00:00:00 2001 From: runterer Date: Sat, 14 May 2022 18:21:13 +0200 Subject: corrections to zeta_gamma --- buch/papers/zeta/zeta_gamma.tex | 53 ++++++++++++++++++++++++----------------- 1 file changed, 31 insertions(+), 22 deletions(-) (limited to 'buch') diff --git a/buch/papers/zeta/zeta_gamma.tex b/buch/papers/zeta/zeta_gamma.tex index 59c8744..bed4262 100644 --- a/buch/papers/zeta/zeta_gamma.tex +++ b/buch/papers/zeta/zeta_gamma.tex @@ -1,38 +1,47 @@ -\section{Zusammenhang mit Gammafunktion} \label{zeta:section:zusammenhang_mit_gammafunktion} -\rhead{Zusammenhang mit Gammafunktion} +\section{Zusammenhang mit der Gammafunktion} \label{zeta:section:zusammenhang_mit_gammafunktion} +\rhead{Zusammenhang mit der Gammafunktion} -Dieser Abschnitt stellt die Verbindung zwischen der Gamma- und der Zetafunktion her. +In diesem Abschnitt wird gezeigt, wie sich die Zetafunktion durch die Gammafunktion $\Gamma(s)$ ausdrücken lässt. +Dieser Zusammenhang der Art $\zeta(s) = f(\Gamma(s))$ wird später für die Herleitung der analytischen Fortsetzung gebraucht. %TODO ref Gamma -Wenn in der Gammafunkion die Integrationsvariable $t$ substituieren mit $t = nu$ und $dt = n du$, dann können wir die Gleichung umstellen und erhalten den Zusammenhang mit der Zetafunktion -\begin{align} +Wir erinnern uns an die Definition der Gammafunktion in \ref{buch:rekursion:gamma:integralbeweis} +\begin{equation*} + \Gamma(s) + = + \int_0^{\infty} t^{s-1} e^{-t} \,dt, +\end{equation*} +wobei die Notation an die Zetafunktion angepasst ist. +Durch die Substitution von $t$ mit $t = nu$ und $dt = n\,du$ wird daraus +\begin{align*} \Gamma(s) &= - \int_0^{\infty} t^{s-1} e^{-t} dt - \\ + \int_0^{\infty} n^{s-1}u^{s-1} e^{-nu} n \,du \\ &= - \int_0^{\infty} n^{s\cancel{-1}}u^{s-1} e^{-nu} \cancel{n}du - && - \text{Division durch }n^s - \\ + \int_0^{\infty} n^s u^{s-1} e^{-nu} \,du. +\end{align*} +Durch Division mit durch $n^s$ ergibt sich die Quotienten +\begin{equation*} \frac{\Gamma(s)}{n^s} - &= - \int_0^{\infty} u^{s-1} e^{-nu}du - && - \text{Zeta durch Summenbildung } \sum_{n=1}^{\infty} - \\ + = + \int_0^{\infty} u^{s-1} e^{-nu} \,du, +\end{equation*} +welche sich zur Zetafunktion summieren +\begin{equation} + \sum_{n=1}^{\infty} \frac{\Gamma(s)}{n^s} + = \Gamma(s) \zeta(s) - &= + = \int_0^{\infty} u^{s-1} \sum_{n=1}^{\infty}e^{-nu} - du. + \,du. \label{zeta:equation:zeta_gamma1} -\end{align} +\end{equation} Die Summe über $e^{-nu}$ können wir als geometrische Reihe schreiben und erhalten \begin{align} - \sum_{n=1}^{\infty}e^{-u^n} + \sum_{n=1}^{\infty}\left(e^{-u}\right)^n &= - \sum_{n=0}^{\infty}e^{-u^n} + \sum_{n=0}^{\infty}\left(e^{-u}\right)^n - 1 \\ @@ -42,7 +51,7 @@ Die Summe über $e^{-nu}$ können wir als geometrische Reihe schreiben und erhal &= \frac{1}{e^u - 1}. \end{align} -Wenn wir dieses Resultat einsetzen in \eqref{zeta:equation:zeta_gamma1} und durch $\Gamma(s)$ teilen, erhalten wir +Wenn wir dieses Resultat einsetzen in \eqref{zeta:equation:zeta_gamma1} und durch $\Gamma(s)$ teilen, erhalten wir %TODO formulieren als Satz \begin{equation}\label{zeta:equation:zeta_gamma_final} \zeta(s) = -- cgit v1.2.1 From 8f643765aa134d48da27f161890f07038d2223f3 Mon Sep 17 00:00:00 2001 From: runterer Date: Sat, 14 May 2022 22:17:18 +0200 Subject: Alle einfachen Korrekturen umgesetzt --- buch/papers/zeta/analytic_continuation.tex | 108 ++++++++++++++++++----------- buch/papers/zeta/zeta_gamma.tex | 2 +- 2 files changed, 68 insertions(+), 42 deletions(-) (limited to 'buch') diff --git a/buch/papers/zeta/analytic_continuation.tex b/buch/papers/zeta/analytic_continuation.tex index bb95b92..5e09e42 100644 --- a/buch/papers/zeta/analytic_continuation.tex +++ b/buch/papers/zeta/analytic_continuation.tex @@ -14,8 +14,8 @@ Zuerst definieren die Dirichletsche Etafunktion als wobei die Reihe bis auf die alternierenden Vorzeichen die selbe wie in der Zetafunktion ist. Diese Etafunktion konvergiert gemäss dem Leibnitz-Kriterium im Bereich $\Re(s) > 0$, da dann die einzelnen Glieder monoton fallend sind. -Wenn wir es nun schaffen, die sehr ähnliche Zetafunktion mit der Etafunktion auszudrücken, dann haben die gesuchte Fortsetzung. -Die folgenden Schritte zeigen, wie man dazu kommt: +Wenn wir es nun schaffen, die sehr ähnliche Zetafunktion durch die Etafunktion auszudrücken, dann haben die gesuchte Fortsetzung. +Zuerst wiederholen wir zweimal die Definition der Zetafunktion \eqref{zeta:equation1}, wobei wir sie einmal durch $2^{s-1}$ teilen \begin{align} \zeta(s) &= @@ -26,8 +26,10 @@ Die folgenden Schritte zeigen, wie man dazu kommt: \zeta(s) &= \sum_{n=1}^{\infty} - \frac{2}{(2n)^s} \label{zeta:align2} - \\ + \frac{2}{(2n)^s}. \label{zeta:align2} +\end{align} +Durch Subtraktion der beiden Gleichungen \eqref{zeta:align1} minus \eqref{zeta:align2}, ergibt sich +\begin{align} \left(1 - \frac{1}{2^{s-1}} \right) \zeta(s) &= @@ -36,14 +38,15 @@ Die folgenden Schritte zeigen, wie man dazu kommt: + \frac{1}{3^s} \underbrace{-\frac{2}{4^s} + \frac{1}{4^s}}_{-\frac{1}{4^s}} \ldots - && \text{\eqref{zeta:align1}} - \text{\eqref{zeta:align2}} - \\ - &= \eta(s) \\ + &= \eta(s). +\end{align} +Dies ist die Fortsetzung auf den noch unbekannten Bereich $0 < \Re(s) < 1$ +\begin{equation} \zeta(s) - &= + := \left(1 - \frac{1}{2^{s-1}} \right)^{-1} \eta(s). -\end{align} +\end{equation} \subsection{Fortsetzung auf ganz $\mathbb{C}$} \label{zeta:subsection:auf_ganz} Für die Fortsetzung auf den Rest von $\mathbb{C}$, verwenden wir den Zusammenhang von Gamma- und Zetafunktion aus \ref{zeta:section:zusammenhang_mit_gammafunktion}. @@ -61,7 +64,7 @@ Nun substituieren wir $t$ mit $t = \pi n^2 x$ und $dt=\pi n^2 dx$ und erhalten (\pi n^2)^{\frac{s}{2}} x^{\frac{s}{2}-1} e^{-\pi n^2 x} - dx + \,dx && \text{Division durch } (\pi n^2)^{\frac{s}{2}} \\ \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}} n^s} @@ -69,7 +72,7 @@ Nun substituieren wir $t$ mit $t = \pi n^2 x$ und $dt=\pi n^2 dx$ und erhalten \int_0^{\infty} x^{\frac{s}{2}-1} e^{-\pi n^2 x} - dx + \,dx && \text{Zeta durch Summenbildung } \sum_{n=1}^{\infty} \\ \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} @@ -79,7 +82,7 @@ Nun substituieren wir $t$ mit $t = \pi n^2 x$ und $dt=\pi n^2 dx$ und erhalten x^{\frac{s}{2}-1} \sum_{n=1}^{\infty} e^{-\pi n^2 x} - dx. \label{zeta:equation:integral1} + \,dx. \label{zeta:equation:integral1} \end{align} Die Summe kürzen wir ab als $\psi(x) = \sum_{n=1}^{\infty} e^{-\pi n^2 x}$. %TODO Wieso folgendes -> aus Fourier Signal @@ -97,82 +100,103 @@ Zunächst teilen wir nun das Integral aus \eqref{zeta:equation:integral1} auf al \int_0^{\infty} x^{\frac{s}{2}-1} \psi(x) - dx + \,dx = + \underbrace{ \int_0^{1} x^{\frac{s}{2}-1} \psi(x) - dx + \,dx + }_{I_1} + + \underbrace{ \int_1^{\infty} x^{\frac{s}{2}-1} \psi(x) - dx, + \,dx + }_{I_2} + = + I_1 + I_2, \end{equation} -wobei wir uns nun auf den ersten Teil konzentrieren werden. -Dabei setzen wir das Wissen aus \eqref{zeta:equation:psi} ein und erhalten +wobei wir uns nun auf den ersten Teil $I_1$ konzentrieren werden. +Dabei setzen wir die Definition von $\psi(x)$ aus \eqref{zeta:equation:psi} ein und erhalten \begin{align} + I_1 + = \int_0^{1} x^{\frac{s}{2}-1} \psi(x) - dx + \,dx &= \int_0^{1} x^{\frac{s}{2}-1} \left( - \frac{1}{2} + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}} - + \frac{1}{2 \sqrt{x}}. + + \frac{1}{2 \sqrt{x}} \right) - dx + \,dx \\ &= \int_0^{1} x^{\frac{s}{2}-\frac{3}{2}} \psi \left( \frac{1}{x} \right) + \frac{1}{2} - \left( + \biggl( x^{\frac{s}{2}-\frac{3}{2}} - x^{\frac{s}{2}-1} - \right) - dx + \biggl) + \,dx \\ &= + \underbrace{ \int_0^{1} x^{\frac{s}{2}-\frac{3}{2}} \psi \left( \frac{1}{x} \right) - dx - + \frac{1}{2} + \,dx + }_{I_3} + + + \underbrace{ + \frac{1}{2} \int_0^1 x^{\frac{s}{2}-\frac{3}{2}} - x^{\frac{s}{2}-1} - dx. \label{zeta:equation:integral3} + \,dx + }_{I_4}. \label{zeta:equation:integral3} \end{align} -Dabei kann das zweite Integral gelöst werden als +Dabei kann das zweite Integral $I_4$ gelöst werden als \begin{equation} + I_4 + = \frac{1}{2} \int_0^1 x^{\frac{s}{2}-\frac{3}{2}} - x^{\frac{s}{2}-1} - dx + \,dx = \frac{1}{s(s-1)}. \end{equation} -Das erste Integral aus \eqref{zeta:equation:integral3} mit $\psi \left(\frac{1}{x} \right)$ ist nicht lösbar in dieser Form. +Das erste Integral $I_3$ aus \eqref{zeta:equation:integral3} mit $\psi \left(\frac{1}{x} \right)$ ist nicht lösbar in dieser Form. Deshalb substituieren wir $x = \frac{1}{u}$ und $dx = -\frac{1}{u^2}du$. Die untere Integralgrenze wechselt ebenfalls zu $x_0 = 0 \rightarrow u_0 = \infty$. Dies ergibt \begin{align} + I_3 + = \int_{\infty}^{1} - {\frac{1}{u}}^{\frac{s}{2}-\frac{3}{2}} + \left( + \frac{1}{u} + \right)^{\frac{s}{2}-\frac{3}{2}} \psi(u) \frac{-du}{u^2} &= \int_{1}^{\infty} - {\frac{1}{u}}^{\frac{s}{2}-\frac{3}{2}} + \left( + \frac{1}{u} + \right)^{\frac{s}{2}-\frac{3}{2}} \psi(u) \frac{du}{u^2} \\ @@ -180,21 +204,23 @@ Dies ergibt \int_{1}^{\infty} x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} \psi(x) - dx, + \,dx, \end{align} wobei wir durch Multiplikation mit $(-1)$ die Integralgrenzen tauschen dürfen. Es ist zu beachten das diese Grenzen nun identisch mit den Grenzen des zweiten Integrals von \eqref{zeta:equation:integral2} sind. Wir setzen beide Lösungen ein in Gleichung \eqref{zeta:equation:integral3} und erhalten \begin{equation} + I_1 + = \int_0^{1} x^{\frac{s}{2}-1} \psi(x) - dx + \,dx = \int_{1}^{\infty} x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} \psi(x) - dx + \,dx + \frac{1}{s(s-1)}. \end{equation} @@ -206,12 +232,12 @@ Dieses Resultat setzen wir wiederum ein in \eqref{zeta:equation:integral2}, um s \int_0^{1} x^{\frac{s}{2}-1} \psi(x) - dx + \,dx + \int_1^{\infty} x^{\frac{s}{2}-1} \psi(x) - dx + \,dx \nonumber \\ &= @@ -220,12 +246,12 @@ Dieses Resultat setzen wir wiederum ein in \eqref{zeta:equation:integral2}, um s \int_{1}^{\infty} x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} \psi(x) - dx + \,dx + \int_1^{\infty} x^{\frac{s}{2}-1} \psi(x) - dx + \,dx \\ &= \frac{1}{s(s-1)} @@ -237,7 +263,7 @@ Dieses Resultat setzen wir wiederum ein in \eqref{zeta:equation:integral2}, um s x^{\frac{s}{2}-1} \right) \psi(x) - dx + \,dx \\ &= \frac{-1}{s(1-s)} @@ -249,7 +275,7 @@ Dieses Resultat setzen wir wiederum ein in \eqref{zeta:equation:integral2}, um s x^{\frac{s}{2}} \right) \frac{\psi(x)}{x} - dx, + \,dx, \end{align} zu erhalten. Wenn wir dieses Resultat genau anschauen, erkennen wir dass sich nichts verändert wenn $s$ mit $1-s$ ersetzt wird. @@ -261,4 +287,4 @@ Somit haben wir die analytische Fortsetzung gefunden als \frac{\Gamma \left( \frac{1-s}{2} \right)}{\pi^{\frac{1-s}{2}}} \zeta(1-s). \end{equation} - +%TODO Definitionen und Gleichungen klarer unterscheiden diff --git a/buch/papers/zeta/zeta_gamma.tex b/buch/papers/zeta/zeta_gamma.tex index bed4262..49fea74 100644 --- a/buch/papers/zeta/zeta_gamma.tex +++ b/buch/papers/zeta/zeta_gamma.tex @@ -5,7 +5,7 @@ In diesem Abschnitt wird gezeigt, wie sich die Zetafunktion durch die Gammafunkt Dieser Zusammenhang der Art $\zeta(s) = f(\Gamma(s))$ wird später für die Herleitung der analytischen Fortsetzung gebraucht. %TODO ref Gamma -Wir erinnern uns an die Definition der Gammafunktion in \ref{buch:rekursion:gamma:integralbeweis} +Wir erinnern uns an die Definition der Gammafunktion in \eqref{buch:rekursion:gamma:integralbeweis} \begin{equation*} \Gamma(s) = -- cgit v1.2.1 From 5fa246097347d82af591aa186f1b7fe32fbd1cf3 Mon Sep 17 00:00:00 2001 From: Runterer <37069007+Runterer@users.noreply.github.com> Date: Mon, 16 May 2022 15:35:30 +0200 Subject: added tikz -> kudos nic --- buch/papers/zeta/continuation_overview.tikz.tex | 17 +++++++++++++++++ 1 file changed, 17 insertions(+) create mode 100644 buch/papers/zeta/continuation_overview.tikz.tex (limited to 'buch') diff --git a/buch/papers/zeta/continuation_overview.tikz.tex b/buch/papers/zeta/continuation_overview.tikz.tex new file mode 100644 index 0000000..03224ff --- /dev/null +++ b/buch/papers/zeta/continuation_overview.tikz.tex @@ -0,0 +1,17 @@ +\begin{tikzpicture}[>=stealth', auto, node distance=0.9cm, scale=2, + dot/.style={fill, circle, inner sep=0, minimum size=0.1cm}] + + \draw[->] (-2,0) -- (-1,0) node[dot]{} node[anchor=north]{$-1$} -- (0,0) node[anchor=north west]{$0$} -- (1,0) node[anchor=north west]{$1$} -- (2,0) node[anchor=west]{Re$(s)$}; 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charset=UTF-8 Content-Transfer-Encoding: 8bit --- buch/papers/nav/bilder/dreieck.png | Bin 0 -> 91703 bytes buch/papers/nav/bilder/kugel1.png | Bin 0 -> 9051 bytes buch/papers/nav/bilder/kugel2.png | Bin 0 -> 9103 bytes buch/papers/nav/bilder/kugel3.png | Bin 0 -> 215188 bytes buch/papers/nav/bilder/projektion.png | Bin 0 -> 41289 bytes buch/papers/nav/einleitung.tex | 17 +++ buch/papers/nav/flatearth.tex | 31 ++++++ buch/papers/nav/geschichte.tex | 22 ++++ buch/papers/nav/main.log | 109 +++++++++++++++++++ buch/papers/nav/main.tex | 29 ++---- buch/papers/nav/nautischesdreieck.tex | 190 ++++++++++++++++++++++++++++++++++ buch/papers/nav/packages.tex | 6 ++ buch/papers/nav/trigo.tex | 51 +++++++++ 13 files changed, 433 insertions(+), 22 deletions(-) create mode 100644 buch/papers/nav/bilder/dreieck.png create mode 100644 buch/papers/nav/bilder/kugel1.png create mode 100644 buch/papers/nav/bilder/kugel2.png create mode 100644 buch/papers/nav/bilder/kugel3.png create mode 100644 buch/papers/nav/bilder/projektion.png create mode 100644 buch/papers/nav/einleitung.tex create mode 100644 buch/papers/nav/flatearth.tex create mode 100644 buch/papers/nav/geschichte.tex create mode 100644 buch/papers/nav/main.log create mode 100644 buch/papers/nav/nautischesdreieck.tex create mode 100644 buch/papers/nav/trigo.tex (limited to 'buch') diff --git a/buch/papers/nav/bilder/dreieck.png b/buch/papers/nav/bilder/dreieck.png new file mode 100644 index 0000000..2b02105 Binary files /dev/null and b/buch/papers/nav/bilder/dreieck.png differ diff --git a/buch/papers/nav/bilder/kugel1.png b/buch/papers/nav/bilder/kugel1.png new file mode 100644 index 0000000..b3188b7 Binary files /dev/null and b/buch/papers/nav/bilder/kugel1.png differ diff --git a/buch/papers/nav/bilder/kugel2.png b/buch/papers/nav/bilder/kugel2.png new file mode 100644 index 0000000..057740f Binary files /dev/null and b/buch/papers/nav/bilder/kugel2.png differ diff --git a/buch/papers/nav/bilder/kugel3.png b/buch/papers/nav/bilder/kugel3.png new file mode 100644 index 0000000..97066a2 Binary files /dev/null and b/buch/papers/nav/bilder/kugel3.png differ diff --git a/buch/papers/nav/bilder/projektion.png b/buch/papers/nav/bilder/projektion.png new file mode 100644 index 0000000..5dcc0c8 Binary files /dev/null and b/buch/papers/nav/bilder/projektion.png differ diff --git a/buch/papers/nav/einleitung.tex b/buch/papers/nav/einleitung.tex new file mode 100644 index 0000000..42f4b6c --- /dev/null +++ b/buch/papers/nav/einleitung.tex @@ -0,0 +1,17 @@ +\documentclass[12pt]{scrartcl} +\usepackage{ucs} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{graphicx} + +\begin{document} +\section{Einleitung} +Heut zu Tage ist die Navigation ein Teil des Lebens. +Man versendet dem Kollegen seinen Standort, um sich das ewige Erklären zu sparen oder gibt die Adresse des Ziels ein um sich die Sucherei zu schenken. +Dies wird durch Technologien wie Funknavigation, welches ein auf Langzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt. +Vor all diesen technologischen Fortschritten gab es lediglich die Astronavigation, welche heute noch auf kleineren Schiffen benötigt wird im Falle eines Stromausfalls. +Aber wie funktioniert die Navigation mit den Sternen? Welche Hilfsmittel benötigt man, welche Rolle spielt die Mathematik und weshalb kann die Erde nicht flach sein? +In diesem Kapitel werden genau diese Fragen mithilfe des Nautischen Dreiecks, der Sphärischen Trigonometrie und einigen Hilfsmitteln und Messgeräten beantwortet. + + +\end{document} \ No newline at end of file diff --git a/buch/papers/nav/flatearth.tex b/buch/papers/nav/flatearth.tex new file mode 100644 index 0000000..b14dd4b --- /dev/null +++ b/buch/papers/nav/flatearth.tex @@ -0,0 +1,31 @@ +\documentclass[12pt]{scrartcl} +\usepackage{ucs} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{graphicx} + +\begin{document} + \section{Warum ist die Erde nicht flach?} + + \begin{figure}[h] + \begin{center} + \includegraphics[width=10cm]{bilder/projektion.png} + \caption{Mercator Projektion} + \end{center} + \end{figure} + +Es gibt heut zu Tage viele Beweise dafür, dass die Erde eine Kugel ist. +Die Fotos von unserem Planeten oder die Berichte der Astronauten. + Aber schon vor ca. 2300 Jahren hat Aristotoles bemerkt, dass Schiffe im Horizont verschwinden und die einzige Erklärung dafür die Kugelgestalt der Erde ist oder der Erdschatten bei einer Mondfinsternis immer rund ist. + Eratosthenes konnte etwa 100 Jahre später den Erdumfang berechnen. + Er beobachtete, dass die Sonne in Syene mittags im Zenit steht und gleichzeitig in Alexandria unter einem Winkel einfällt. + Mithilfe der Geometrie konnte er mit dem Abstand der Städte und dem Einfallswinkel den Umfang berechnen. + Auch in der Navigation würden grobe Fehler passieren, wenn man davon ausgeht, dass die Erde eine Scheibe ist. +Man sieht es zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. Grönland ist auf der Weltkarte so gross wie Afrika. +In der Anwendung Google Earth jedoch ist Grönland etwa so gross wie Algerien. +Das liegt daran, das man die 3D – Weltkarte nicht einfach auslegen kann. +Der Kartograph Gerhard Mercator projizierte die Erdkugel auf ein Papier und erstellte so eine winkeltreue Karte. Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. +Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen. + + +\end{document} \ No newline at end of file diff --git a/buch/papers/nav/geschichte.tex b/buch/papers/nav/geschichte.tex new file mode 100644 index 0000000..a20eb6d --- /dev/null +++ b/buch/papers/nav/geschichte.tex @@ -0,0 +1,22 @@ +\documentclass[12pt]{scrartcl} +\usepackage{ucs} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{graphicx} + +\begin{document} +\section{Geschichte der sphärischen Navigation} +Die Orientierung mit Hilfe der Sterne und der sphärischen Trigonometrie bewegt die Menschheit schon seit mehreren tausend Jahren. +Nach Hinweisen und Schätzungen von Forscher haben schon vor 4000 Jahren die Ägypter und Gelehrten aus Babylon mit Hilfe der Astronomie den Lauf der Gestirne (Himmelskörper) zu berechnen versucht, jedoch ohne Erfolg. +Etwa 350 vor Christus waren es die Griechen, welche den damaligen Astronomen Hilfestellungen mittels Kugel-Geometrien leisten konnten. +Aus diesen Geometrien wurden erste mathematische Sätze aufgestellt und ein paar Jahrhunderte später kamen zu diesem Thema auch Berechnungen dazu. +Ebenso wurden Kartenmaterial mit Sternenbilder angefertigt. +Die Sinusfunktion war noch nicht bekannt, jedoch kamen zu dieser Zeit die ersten Ansätze der Cosinusfunktion aus Indien. +Von diesen Hilfen darauf aufbauend konnte um 900 die Araber der Sinussatz entwickeln. +Doch ein paar weitere Jahrhunderte vergingen bis zu diesem Thema wieder verstärkt Forschung betrieben wurde. +Dies aus dem Grund, da im 15. Jahrhundert grosse Entdeckungsreisen, hauptsächlich per Schiff, erfolgten und die Orientierung vermehrt an Wichtigkeit gewann. +Auch die Verwendung der Tangens- und Sinusfunktion sowie der neu entwickelte Seitencosinussatz trugen zu einer Verbesserung der Orientierung herbei. +Im 16. Jahrhundert wurde dann ein weiterer trigonometrischer Satz, der Winkelcosinussatz hergeleitet. Stück für Stück wurden infolge der Entdeckung des Logarithmus im 17. Jahrhundert viele neue Methoden entwickelt. +Auch eine Verbesserung der kartographischen Verwendung der Kugelgeometrie wurde vorgenommen. +Es folgten weitere Entwicklungen in nicht euklidische Geometrien und im 19. Jahrhundert sowie auch im 20. Jahrhundert wurde zudem für die Relativitätstheorie auch die sphärische Trigonometrie beigezogen. +\end{document} \ No newline at end of file diff --git a/buch/papers/nav/main.log b/buch/papers/nav/main.log new file mode 100644 index 0000000..d7aa0a9 --- /dev/null +++ b/buch/papers/nav/main.log @@ -0,0 +1,109 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (MiKTeX 22.3) (preloaded format=pdflatex 2022.4.16) 16 MAY 2022 20:27 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**./main.tex +(main.tex +LaTeX2e <2021-11-15> patch level 1 +L3 programming layer <2022-02-24> +! Undefined control sequence. +l.6 \chapter + {Thema\label{chapter:nav}} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.6 \chapter{T + hema\label{chapter:nav}} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +Missing character: There is no T in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +! Undefined control sequence. +l.7 \lhead + {Thema} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! + +! LaTeX Error: Environment refsection undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.8 \begin{refsection} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Undefined control sequence. +l.9 \chapterauthor + {Hans Muster} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no H in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no M in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 6--10 +[][] + [] + + +! LaTeX Error: File `papers/nav/einleitung.tex' not found. + +Type X to quit or to proceed, +or enter new name. (Default extension: tex) + +Enter file name: +! Emergency stop. + + +l.13 \input{papers/nav/einleitung.tex} + ^^M +*** (cannot \read from terminal in nonstop modes) + + +Here is how much of TeX's memory you used: + 22 strings out of 478582 + 530 string characters out of 2856069 + 288951 words of memory out of 3000000 + 18307 multiletter control sequences out of 15000+600000 + 469259 words of font info for 28 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 16i,0n,26p,84b,28s stack positions out of 10000i,1000n,20000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/buch/papers/nav/main.tex b/buch/papers/nav/main.tex index e11e2c0..1ad16da 100644 --- a/buch/papers/nav/main.tex +++ b/buch/papers/nav/main.tex @@ -8,29 +8,14 @@ \begin{refsection} \chapterauthor{Hans Muster} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} -\input{papers/nav/teil0.tex} -\input{papers/nav/teil1.tex} -\input{papers/nav/teil2.tex} -\input{papers/nav/teil3.tex} + +\input{papers/nav/einleitung.tex} +\input{papers/nav/geschichte.tex} +\input{papers/nav/flatearth.tex} +\input{papers/nav/trigo.tex} +\input{papers/nav/nautischesdreieck.tex} + \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex new file mode 100644 index 0000000..0bb213c --- /dev/null +++ b/buch/papers/nav/nautischesdreieck.tex @@ -0,0 +1,190 @@ +\documentclass[12pt]{scrartcl} +\usepackage{ucs} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{graphicx} + \usepackage{xcolor, soul} + \sethlcolor{yellow} +\begin{document} + \setlength{\parindent}{0em} +\section{Das Nautische Dreieck} +\subsection{Definition des Nautischen Dreiecks} +Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der \textbf{Himmelskugel}. +Die Himmelskugel ist eine gedachte Kugel, welche die Erde und dessen Beobachter umgibt und als Rechenfläche für Koordinaten in der Astronomie und Geodäsie dient.\\ +Das Nautische Dreieck definiert sich durch folgende Ecken: +\begin{itemize} + \item Zenit + \item Gestirn + \item Himmelspol +\end{itemize} +Der Zenit ist jener Punkt, der vom Erdmittelpunkt durch denn eigenen Standort an die Himmelskugel verlängert wird. +Ein Gestirn ist ein Planet oder ein Fixstern, zu welchen es diverse Jahrbücher mit allen astronomischen Eigenschaften gibt. +Der Himmelspol ist der Nordpol an die Himmelskugel projeziert. +\\ +Zur Anwendung der Formeln der sphärischen Trigonometrie gelten folgende einfache Zusammenhänge: +\begin{itemize} + \item Seitenlänge Zenit zu Himmelspol $= \frac{\pi}{2} - \phi $ + \item Seitenlänge Himmelspol zu Gestirn $= \frac{\pi}{2} - \delta$ + \item Seitenlänge Zenit zu Gestirn $= \frac{\pi}{2} - h$ + \item Winkel von Zenit zu Himmelsnordpol zu Gestirn$=\pi - \alpha$ + \item Winkel von Himmelsnordpol zu Zenit und Gestirn$= \tau$ +\end{itemize} +Um mit diesen Zusammenhängen zu rechnen benötigt man folgende Legende: + +$\alpha \ \widehat{=} \ Rektaszension $ + +$\delta \ \widehat{=} \ Deklination =$ Breitengrad des Gestirns + +$\theta \ \widehat{=} \ Sternzeit$ + +$\phi \ \widehat{=} \ Geographische \ Breite $ + +$\tau = \theta-\alpha \ \widehat{=} \ Stundenwinkel =$ Längengrad des Gestirns + +$a \ \widehat{=} \ Azimut $ + +$h \ \widehat{=} \ Hoehe$ + + + +\subsection{Zusammenhang des Nautischen Dreiecks und des Kugeldreiecks auf der Erdkugel} + + \begin{center} + \includegraphics[height=5cm,width=5cm]{Bilder/kugel3.png} + \end{center} +Wie man im oberen Bild sieht und auch am Anfang dieses Kapitels bereits erwähnt wurde, liegt das Nautische Dreieck auf der Himmelskugel mit den Ecken Zenit, Gestirn und Himmelsnordpol. +Das selbe Dreieck kann man aber auch auf die Erdkugel projezieren und hat dann die Ecken Standort, Bildpunkt und Nordpol. +Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. + +\subsection{Varianten vom Nautischen Dreieck} +\section{Standortbestimmung ohne elektronische Hilfsmittel} +Um den eigenen Standort herauszufinden, wird in diesem Kapitel die Projektion Nautische Dreieck auf der Erdkugel zur Hilfe genommen. +Mithilfe einiger Hilfsmittel und der Sphärischen Trigonometrie kann man dann die Längen- und Breitengrade des eigenen Standortes bestimmen. + + \begin{center} + \includegraphics[width=6cm]{Bilder/dreieck.png} + \end{center} + + + +\subsection{Ecke P - Unser Standort} +Unser eigener Standort ist der gesuchte Punkt A. + +\subsection{Ecke A - Nordpol} +Der Vorteil ander Idee des Nautischen Dreiecks ist, dass eine Ecke immer der Nordpol (in der Himmelskugel der Himmelsnordpol) ist. +Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so simpel. + +\subsection{Ecke B und C - Bildpunkt XXX und YYY} +Für die Standortermittlung benötigt man als weiteren Punkt ein Gestirn bzw. seinen Bildpunkt auf der Erdkugel. +Damit das trigonometrische Rechnen einfacher wird, werden hier zwei Gestirne zur Hilfe genommen. +\\ +Es gibt diverse Gestirne, die man nutzen kann. +\begin{itemize} + \item Sonne + \item Mond + \item Die vier Navigationsplaneten Venus, Mars, Jupiter und Saturn +\end{itemize} + +Zu all diesen Gestirnen gibt es Ephemeriden (Jahrbücher). +Dort findet man unter Anderem die Rektaszension und Deklination, welche für jeden Tag und Stunde beschrieben ist. Für Minuten genaue Angaben muss man dann zwischen den Stunden interpolieren. +Mithilfe dieser beiden Angaben kann man die Längen- und Breitengrade diverser Gestirne berechnen. + +\subsubsection{Sternzeit und Rektaszension} +Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt steht. +Der Frühlungspunkt ist der Nullpunkt auf dem Himmelsäquator. +Die Tatsache, dass sich die Himmelskugel ca. vier Minuten schneller um die eigene Achse dreht als die Erdkugel, stellt hier ein kleines Problem dar. +Die Lösung ist die Sternzeit. +Am Frühlingspunkt (21. März) 12:00 Uhr ist die Sternzeit +$\theta = 0$. + +Die Sternzeit geht vom Frühlungspunkt aus, an welchem die Sonne den Himmelsäquator schneidet. +Für die Standortermittlung auf der Erdkugel ist es am einfachsten, wenn man die Sternzeit von Greenwich berechnet. +Für die Sternzeit von Greenwich braucht man als erstes das Julianische Datum vom aktuellen Tag, welches sich leicht recherchieren oder berechnen lässt: \hl{$JD=....$} + +Nun berechnet man $T=\frac{JD-2451545}{36525}$ und damit die mittlere Sternzeit von Greenwich + + $T_{Greenwich} = 6^h 41^m 50^s,54841 + 8640184^s,812866 * T + 0^s,093104*T^2 - 0^s,0000062 * T^3$. + + Wenn mann die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ mithilfe der Rektaszension und Sternzeit bestimmen. + Dies gilt analog auch für das zweite Gestirn. + + \subsubsection{Deklination} + Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und ergibt schlussendlich den Breitengrad $\psi = \delta$. + + + +\subsection{Bestimmung des eigenen Standortes P} +Nun hat man die Koordinaten der beiden Gestirne und man weiss die Koordinaten des Nordpols. +Somit können wir ein erstes Kugeldreieck auf der Erde aufspannen. + + + \begin{center} + \includegraphics[width=5cm]{Bilder/dreieck.png} + \end{center} + + +\subsubsection{Bestimmung des ersten Dreiecks} + Mithilfe des sphärischen Trigonometrie und den darausfolgenden Zusammenhängen des Nautischen Dreiecks können wir nun alle Seiten des Dreiecks $ABC$ berechnen. + + Die Seitenlänge der Seite "Nordpol zum Bildpunkt XXX" sei $c$. + Dann ist $c = \frac{\pi}{2} - \delta_1$. + + Die Seitenlänge der Seite "Nordpol zum Bildpunkt YYY" sei $b$. + Dann ist $b = \frac{\pi}{2} - \delta_2$. + + Der Innenwinkel beim der Ecke "Nordpol" sei $\alpha$. + Dann ist $ \alpha = |\lambda_1 - \lambda_2|$. + +mit + + $\delta_1 =$ Deklination Bildpunkt XXX + +$\delta_2 =$ Deklination Bildpunk YYY + +$\lambda_1 =$ Längengrad Bildpunkt XXX + +$\lambda_2 =$ Längengrad Bildpunkt YYY + + Wichtig ist: Die Differenz der Längengrade ist gleich der Innenwinkel Alpha, deswegen der Betrag! + +Nun haben wir die beiden Seiten $c\ und\ b$ und den Winkel $\alpha$, der sich zwischen diesen Seiten befindet. +Mithilfe des Seiten-Kosinussatzes $cos(a) = cos(b)*cos(c) + sin(b) * sin(c)*cos(\alpha)$ können wir nun die dritte Seitenlänge bestimmen. +Es ist darauf zu achten, dass hier natürlich die Seitenlängen in Bogenmass sind und dementsprechend der Kosinus und Sinus verwendet wird. +Jetzt fehlen noch die beiden anderen Innenwinkel $\beta \ und\ \gamma$. + +Dieser bestimmen wir mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$. +Hier muss man aufpassen, dass man Seite von Winkel unterscheiden kann. Im Zähler sind die Seiten, im Nenner die Winkel. Somit ist $sin(\beta) = sin(b) * \frac{sin(\alpha)}{sin(a)} $. + +Schlussendlich haben wir die Seiten $a,b\ und \ c$, die Ecken A,B und C und die Winkel $\alpha, \beta \ und \ \gamma$ bestimmt und somit das ganze erste Kugeldreieck berechnet. + +\subsubsection{Bestimmung des zweiten Dreiecks} +Wir bilden nun ein zweites Dreieck, welches die Ecken B und C des ersten Dreiecks besitzt. +Die dritte Ecke ist der eigene Standort P. +Unser Standort definiere sich aus einer geographischen Breite $\delta$ und einer geographischen Länge $\lambda$. + +Die Seite von P zu B sei $pb$ und die Seite von P zu C sei $pc$. +Die beiden Seitenlängen kann man mit dem Sextant messen und durch eine einfache Formel bestimmen, nämlich $pb=\frac{\pi}{2} - h_{B}$ und $pc=\frac{\pi}{2} - h_{C}$ + +mit $h_B=$ Höhe von Gestirn in B und $h_C=$ Höhe von Gestirn in C mit Sextant gemessen. +\\ + +Nun muss man eine Verbindungslinie ziehen zwischen P und A. Die Länge $l$ dieser Linie entspricht der gesuchten geographischen Breite $\delta$. Diese lässt sich mithilfe des Dreiecks $ABP$ mit den bekannten Seiten $c\ und \ pb$ und des Seiten-Kosinussatzes + +$cos(l) = cos(c)*cos(pb) + sin(c) * sin(pb)*cos(\beta)$ berechnen. + +Es fehlt uns noch $\beta1$. +Da wir aber $pc$, $pb$ und $a$ kennen, kann man mit dem Seiten-Kosinussatz den Winkel $\beta1$ berechnen +\\ + +Somit ist $\delta = cos(l) = cos(c)*cos(pb) + sin(c) * sin(pb)*cos(\beta)$. +\\ + +Für die Geographische Länge $\lambda$ des eigenen Standortes muss man den Winkel $\omega$, welcher sich im Dreieck $ABP$ în der Ecke bei $A$ befindet mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$ bestimmen. +\\ + +Somit ist $\omega=sin(pb)*\frac{sin(\beta)}{sin(l)}$ und unsere gesuchte geographische Länge schlussendlich +$\lambda=\lambda_1 - \omega$ + + + +\end{document} \ No newline at end of file diff --git a/buch/papers/nav/packages.tex b/buch/papers/nav/packages.tex index 9faa48d..15c7fdc 100644 --- a/buch/papers/nav/packages.tex +++ b/buch/papers/nav/packages.tex @@ -8,3 +8,9 @@ % following example %\usepackage{packagename} +\documentclass[12pt]{scrartcl} +\usepackage{ucs} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{graphicx} +\usepackage{xcolor, soul} diff --git a/buch/papers/nav/trigo.tex b/buch/papers/nav/trigo.tex new file mode 100644 index 0000000..0dbd7a1 --- /dev/null +++ b/buch/papers/nav/trigo.tex @@ -0,0 +1,51 @@ +\documentclass[12pt]{scrartcl} +\usepackage{ucs} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{graphicx} + + +\begin{document} + \section{Sphärische Trigonometrie} + \subsection{Das Kugeldreieck} + +Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden, so entsteht ein Kugeldreieck ABC. +A, B und C sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten. +Da die Länge der Grosskreisbögen wegen der Abhängigkeit vom Kugelradius ungeeignet ist, wird die Grösse einer Seite mit dem zugehörigen Mittelpunktwinkel des Grosskreisbogens angegeben. +Laut dieser Definition ist die Seite c der Winkel AMB. +Für ein Kugeldreieck gilt, dass die Summe der drei Seiten kleiner als $2\pi$ aber grösser als 0 ist. +Man kann bei Kugeldreiecken nicht so einfach unterscheiden, was Innen oder Aussen ist. +Wenn man drei Eckpunkte miteinander verbindet, ergeben sich immer 16 Kugeldreiecke. +Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha, \beta, \gamma < \pi$ nennt man Eulersche Dreiecke. +\begin{figure}[h] + \begin{center} + \includegraphics[width=6cm]{Bilder/kugel1.png} + \end{center} + +\end{figure} + +\subsection{Rechtwinkliges Dreieck und Rechtseitiges Dreieck} +Wie auch im uns bekannten Dreieck gibt es beim Kugeldreieck auch ein Rechtwinkliges Kugeldreieck, bei dem ein Winkel $\frac{\pi}{2}$ ist. +Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss. + \newpage +\subsection{Winkelangabe} + + \begin{center} + \includegraphics[width=8cm]{Bilder/kugel2.png} + \end{center} + +Die Winkel eines Kugeldreiecks sind die, welche die Halbtangenten in den Eckpunkten einschliessen. +Für die Summe der Innenwinkel gilt $\alpha+\beta+\gamma = \frac{A}{r^2} + \pi$ und +$\alpha+\beta+\gamma > \pi$. +Der sphärische Exzess $\epsilon = \alpha+\beta+\gamma - \pi$ beschreibt die Abweichung der Innenwinkelsumme von $\pi$ und ist proportional zum Flächeninhalt des Kugeldreiecks. + +\subsection{Sphärischer Sinussatz} +In jedem Dreieck ist das Verhältnis des Sinus einer Seite zum Sinus des Gegenwinkels konstant. +Das bedeutet, dass $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} $ auch beim Kugeldreieck gilt. + +\subsection{Sphärischer Satz des Pythagoras für das rechtwinklige Kugeldreieck} +Es gibt in der sphärischen Trigonometrie eigentlich garkeinen "Satz des Pythagoras", wie man ihn aus der zweidimensionalen Geometrie kennt. +In der sphärischen Trigonometrie gibt es aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks in eine Beziehung bringt. +Es gilt nämlich: $\cos c = \cos a * \cos b$ wenn $\alpha \lor \beta \lor \gamma = \frac{\pi}{2} $. + +\end{document} \ No newline at end of file -- cgit v1.2.1 From e898a9c36fb707474ee869f6ec47119d0592e59f Mon Sep 17 00:00:00 2001 From: "ENEZ-PC\\erdem" Date: Mon, 16 May 2022 20:32:38 +0200 Subject: =?UTF-8?q?Revert=20"Ich=20habe=20nun=20alle=20Kapitel=20als=20Tex?= =?UTF-8?q?tfile=20seperat=20eingef=C3=BCgt,=20einen=20zus=C3=A4tzlichen?= =?UTF-8?q?=20unterordner=20gemacht=20f=C3=BCr=20die=20bilder,=20dann=20im?= =?UTF-8?q?=20main.tex=20die=20input=20befehle=20angepasst=20und=20committ?= =?UTF-8?q?e=20nun."?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit This reverts commit d7bff7e403a0e54880cb04b350a91a2f664b2708. --- buch/papers/nav/bilder/dreieck.png | Bin 91703 -> 0 bytes buch/papers/nav/bilder/kugel1.png | Bin 9051 -> 0 bytes buch/papers/nav/bilder/kugel2.png | Bin 9103 -> 0 bytes buch/papers/nav/bilder/kugel3.png | Bin 215188 -> 0 bytes buch/papers/nav/bilder/projektion.png | Bin 41289 -> 0 bytes buch/papers/nav/einleitung.tex | 17 --- buch/papers/nav/flatearth.tex | 31 ------ buch/papers/nav/geschichte.tex | 22 ---- buch/papers/nav/main.log | 109 ------------------- buch/papers/nav/main.tex | 29 ++++-- buch/papers/nav/nautischesdreieck.tex | 190 ---------------------------------- buch/papers/nav/packages.tex | 6 -- buch/papers/nav/trigo.tex | 51 --------- 13 files changed, 22 insertions(+), 433 deletions(-) delete mode 100644 buch/papers/nav/bilder/dreieck.png delete mode 100644 buch/papers/nav/bilder/kugel1.png delete mode 100644 buch/papers/nav/bilder/kugel2.png delete mode 100644 buch/papers/nav/bilder/kugel3.png delete mode 100644 buch/papers/nav/bilder/projektion.png delete mode 100644 buch/papers/nav/einleitung.tex delete mode 100644 buch/papers/nav/flatearth.tex delete mode 100644 buch/papers/nav/geschichte.tex delete mode 100644 buch/papers/nav/main.log delete mode 100644 buch/papers/nav/nautischesdreieck.tex delete mode 100644 buch/papers/nav/trigo.tex (limited to 'buch') diff --git a/buch/papers/nav/bilder/dreieck.png b/buch/papers/nav/bilder/dreieck.png deleted file mode 100644 index 2b02105..0000000 Binary files a/buch/papers/nav/bilder/dreieck.png and /dev/null differ diff --git a/buch/papers/nav/bilder/kugel1.png b/buch/papers/nav/bilder/kugel1.png deleted file mode 100644 index b3188b7..0000000 Binary files a/buch/papers/nav/bilder/kugel1.png and /dev/null differ diff --git a/buch/papers/nav/bilder/kugel2.png b/buch/papers/nav/bilder/kugel2.png deleted file mode 100644 index 057740f..0000000 Binary files a/buch/papers/nav/bilder/kugel2.png and /dev/null differ diff --git a/buch/papers/nav/bilder/kugel3.png b/buch/papers/nav/bilder/kugel3.png deleted file mode 100644 index 97066a2..0000000 Binary files a/buch/papers/nav/bilder/kugel3.png and /dev/null differ diff --git a/buch/papers/nav/bilder/projektion.png b/buch/papers/nav/bilder/projektion.png deleted file mode 100644 index 5dcc0c8..0000000 Binary files a/buch/papers/nav/bilder/projektion.png and /dev/null differ diff --git a/buch/papers/nav/einleitung.tex b/buch/papers/nav/einleitung.tex deleted file mode 100644 index 42f4b6c..0000000 --- a/buch/papers/nav/einleitung.tex +++ /dev/null @@ -1,17 +0,0 @@ -\documentclass[12pt]{scrartcl} -\usepackage{ucs} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{graphicx} - -\begin{document} -\section{Einleitung} -Heut zu Tage ist die Navigation ein Teil des Lebens. -Man versendet dem Kollegen seinen Standort, um sich das ewige Erklären zu sparen oder gibt die Adresse des Ziels ein um sich die Sucherei zu schenken. -Dies wird durch Technologien wie Funknavigation, welches ein auf Langzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt. -Vor all diesen technologischen Fortschritten gab es lediglich die Astronavigation, welche heute noch auf kleineren Schiffen benötigt wird im Falle eines Stromausfalls. -Aber wie funktioniert die Navigation mit den Sternen? Welche Hilfsmittel benötigt man, welche Rolle spielt die Mathematik und weshalb kann die Erde nicht flach sein? -In diesem Kapitel werden genau diese Fragen mithilfe des Nautischen Dreiecks, der Sphärischen Trigonometrie und einigen Hilfsmitteln und Messgeräten beantwortet. - - -\end{document} \ No newline at end of file diff --git a/buch/papers/nav/flatearth.tex b/buch/papers/nav/flatearth.tex deleted file mode 100644 index b14dd4b..0000000 --- a/buch/papers/nav/flatearth.tex +++ /dev/null @@ -1,31 +0,0 @@ -\documentclass[12pt]{scrartcl} -\usepackage{ucs} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{graphicx} - -\begin{document} - \section{Warum ist die Erde nicht flach?} - - \begin{figure}[h] - \begin{center} - \includegraphics[width=10cm]{bilder/projektion.png} - \caption{Mercator Projektion} - \end{center} - \end{figure} - -Es gibt heut zu Tage viele Beweise dafür, dass die Erde eine Kugel ist. -Die Fotos von unserem Planeten oder die Berichte der Astronauten. - Aber schon vor ca. 2300 Jahren hat Aristotoles bemerkt, dass Schiffe im Horizont verschwinden und die einzige Erklärung dafür die Kugelgestalt der Erde ist oder der Erdschatten bei einer Mondfinsternis immer rund ist. - Eratosthenes konnte etwa 100 Jahre später den Erdumfang berechnen. - Er beobachtete, dass die Sonne in Syene mittags im Zenit steht und gleichzeitig in Alexandria unter einem Winkel einfällt. - Mithilfe der Geometrie konnte er mit dem Abstand der Städte und dem Einfallswinkel den Umfang berechnen. - Auch in der Navigation würden grobe Fehler passieren, wenn man davon ausgeht, dass die Erde eine Scheibe ist. -Man sieht es zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. Grönland ist auf der Weltkarte so gross wie Afrika. -In der Anwendung Google Earth jedoch ist Grönland etwa so gross wie Algerien. -Das liegt daran, das man die 3D – Weltkarte nicht einfach auslegen kann. -Der Kartograph Gerhard Mercator projizierte die Erdkugel auf ein Papier und erstellte so eine winkeltreue Karte. Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. -Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen. - - -\end{document} \ No newline at end of file diff --git a/buch/papers/nav/geschichte.tex b/buch/papers/nav/geschichte.tex deleted file mode 100644 index a20eb6d..0000000 --- a/buch/papers/nav/geschichte.tex +++ /dev/null @@ -1,22 +0,0 @@ -\documentclass[12pt]{scrartcl} -\usepackage{ucs} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{graphicx} - -\begin{document} -\section{Geschichte der sphärischen Navigation} -Die Orientierung mit Hilfe der Sterne und der sphärischen Trigonometrie bewegt die Menschheit schon seit mehreren tausend Jahren. -Nach Hinweisen und Schätzungen von Forscher haben schon vor 4000 Jahren die Ägypter und Gelehrten aus Babylon mit Hilfe der Astronomie den Lauf der Gestirne (Himmelskörper) zu berechnen versucht, jedoch ohne Erfolg. -Etwa 350 vor Christus waren es die Griechen, welche den damaligen Astronomen Hilfestellungen mittels Kugel-Geometrien leisten konnten. -Aus diesen Geometrien wurden erste mathematische Sätze aufgestellt und ein paar Jahrhunderte später kamen zu diesem Thema auch Berechnungen dazu. -Ebenso wurden Kartenmaterial mit Sternenbilder angefertigt. -Die Sinusfunktion war noch nicht bekannt, jedoch kamen zu dieser Zeit die ersten Ansätze der Cosinusfunktion aus Indien. -Von diesen Hilfen darauf aufbauend konnte um 900 die Araber der Sinussatz entwickeln. -Doch ein paar weitere Jahrhunderte vergingen bis zu diesem Thema wieder verstärkt Forschung betrieben wurde. -Dies aus dem Grund, da im 15. Jahrhundert grosse Entdeckungsreisen, hauptsächlich per Schiff, erfolgten und die Orientierung vermehrt an Wichtigkeit gewann. -Auch die Verwendung der Tangens- und Sinusfunktion sowie der neu entwickelte Seitencosinussatz trugen zu einer Verbesserung der Orientierung herbei. -Im 16. Jahrhundert wurde dann ein weiterer trigonometrischer Satz, der Winkelcosinussatz hergeleitet. Stück für Stück wurden infolge der Entdeckung des Logarithmus im 17. Jahrhundert viele neue Methoden entwickelt. -Auch eine Verbesserung der kartographischen Verwendung der Kugelgeometrie wurde vorgenommen. -Es folgten weitere Entwicklungen in nicht euklidische Geometrien und im 19. Jahrhundert sowie auch im 20. Jahrhundert wurde zudem für die Relativitätstheorie auch die sphärische Trigonometrie beigezogen. -\end{document} \ No newline at end of file diff --git a/buch/papers/nav/main.log b/buch/papers/nav/main.log deleted file mode 100644 index d7aa0a9..0000000 --- a/buch/papers/nav/main.log +++ /dev/null @@ -1,109 +0,0 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (MiKTeX 22.3) (preloaded format=pdflatex 2022.4.16) 16 MAY 2022 20:27 -entering extended mode - restricted \write18 enabled. - %&-line parsing enabled. -**./main.tex -(main.tex -LaTeX2e <2021-11-15> patch level 1 -L3 programming layer <2022-02-24> -! Undefined control sequence. -l.6 \chapter - {Thema\label{chapter:nav}} -The control sequence at the end of the top line -of your error message was never \def'ed. If you have -misspelled it (e.g., `\hobx'), type `I' and the correct -spelling (e.g., `I\hbox'). Otherwise just continue, -and I'll forget about whatever was undefined. - - -! 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Otherwise just continue, -and I'll forget about whatever was undefined. - -Missing character: There is no T in font nullfont! -Missing character: There is no h in font nullfont! -Missing character: There is no e in font nullfont! -Missing character: There is no m in font nullfont! -Missing character: There is no a in font nullfont! - -! LaTeX Error: Environment refsection undefined. - -See the LaTeX manual or LaTeX Companion for explanation. -Type H for immediate help. - ... - -l.8 \begin{refsection} - -Your command was ignored. -Type I to replace it with another command, -or to continue without it. - -! Undefined control sequence. -l.9 \chapterauthor - {Hans Muster} -The control sequence at the end of the top line -of your error message was never \def'ed. If you have -misspelled it (e.g., `\hobx'), type `I' and the correct -spelling (e.g., `I\hbox'). Otherwise just continue, -and I'll forget about whatever was undefined. - -Missing character: There is no H in font nullfont! -Missing character: There is no a in font nullfont! -Missing character: There is no n in font nullfont! -Missing character: There is no s in font nullfont! -Missing character: There is no M in font nullfont! -Missing character: There is no u in font nullfont! -Missing character: There is no s in font nullfont! -Missing character: There is no t in font nullfont! -Missing character: There is no e in font nullfont! -Missing character: There is no r in font nullfont! - -Overfull \hbox (20.0pt too wide) in paragraph at lines 6--10 -[][] - [] - - -! LaTeX Error: File `papers/nav/einleitung.tex' not found. - -Type X to quit or to proceed, -or enter new name. (Default extension: tex) - -Enter file name: -! Emergency stop. - - -l.13 \input{papers/nav/einleitung.tex} - ^^M -*** (cannot \read from terminal in nonstop modes) - - -Here is how much of TeX's memory you used: - 22 strings out of 478582 - 530 string characters out of 2856069 - 288951 words of memory out of 3000000 - 18307 multiletter control sequences out of 15000+600000 - 469259 words of font info for 28 fonts, out of 8000000 for 9000 - 1141 hyphenation exceptions out of 8191 - 16i,0n,26p,84b,28s stack positions out of 10000i,1000n,20000p,200000b,80000s -! ==> Fatal error occurred, no output PDF file produced! diff --git a/buch/papers/nav/main.tex b/buch/papers/nav/main.tex index 1ad16da..e11e2c0 100644 --- a/buch/papers/nav/main.tex +++ b/buch/papers/nav/main.tex @@ -8,14 +8,29 @@ \begin{refsection} \chapterauthor{Hans Muster} +Ein paar Hinweise für die korrekte Formatierung des Textes +\begin{itemize} +\item +Absätze werden gebildet, indem man eine Leerzeile einfügt. +Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. +\item +Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende +Optionen werden gelöscht. +Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. +\item +Beginnen Sie jeden Satz auf einer neuen Zeile. +Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen +in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt +anzuwenden. +\item +Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren +Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. +\end{itemize} - -\input{papers/nav/einleitung.tex} -\input{papers/nav/geschichte.tex} -\input{papers/nav/flatearth.tex} -\input{papers/nav/trigo.tex} -\input{papers/nav/nautischesdreieck.tex} - +\input{papers/nav/teil0.tex} +\input{papers/nav/teil1.tex} +\input{papers/nav/teil2.tex} +\input{papers/nav/teil3.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex deleted file mode 100644 index 0bb213c..0000000 --- a/buch/papers/nav/nautischesdreieck.tex +++ /dev/null @@ -1,190 +0,0 @@ -\documentclass[12pt]{scrartcl} -\usepackage{ucs} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{graphicx} - \usepackage{xcolor, soul} - \sethlcolor{yellow} -\begin{document} - \setlength{\parindent}{0em} -\section{Das Nautische Dreieck} -\subsection{Definition des Nautischen Dreiecks} -Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der \textbf{Himmelskugel}. -Die Himmelskugel ist eine gedachte Kugel, welche die Erde und dessen Beobachter umgibt und als Rechenfläche für Koordinaten in der Astronomie und Geodäsie dient.\\ -Das Nautische Dreieck definiert sich durch folgende Ecken: -\begin{itemize} - \item Zenit - \item Gestirn - \item Himmelspol -\end{itemize} -Der Zenit ist jener Punkt, der vom Erdmittelpunkt durch denn eigenen Standort an die Himmelskugel verlängert wird. -Ein Gestirn ist ein Planet oder ein Fixstern, zu welchen es diverse Jahrbücher mit allen astronomischen Eigenschaften gibt. -Der Himmelspol ist der Nordpol an die Himmelskugel projeziert. -\\ -Zur Anwendung der Formeln der sphärischen Trigonometrie gelten folgende einfache Zusammenhänge: -\begin{itemize} - \item Seitenlänge Zenit zu Himmelspol $= \frac{\pi}{2} - \phi $ - \item Seitenlänge Himmelspol zu Gestirn $= \frac{\pi}{2} - \delta$ - \item Seitenlänge Zenit zu Gestirn $= \frac{\pi}{2} - h$ - \item Winkel von Zenit zu Himmelsnordpol zu Gestirn$=\pi - \alpha$ - \item Winkel von Himmelsnordpol zu Zenit und Gestirn$= \tau$ -\end{itemize} -Um mit diesen Zusammenhängen zu rechnen benötigt man folgende Legende: - -$\alpha \ \widehat{=} \ Rektaszension $ - -$\delta \ \widehat{=} \ Deklination =$ Breitengrad des Gestirns - -$\theta \ \widehat{=} \ Sternzeit$ - -$\phi \ \widehat{=} \ Geographische \ Breite $ - -$\tau = \theta-\alpha \ \widehat{=} \ Stundenwinkel =$ Längengrad des Gestirns - -$a \ \widehat{=} \ Azimut $ - -$h \ \widehat{=} \ Hoehe$ - - - -\subsection{Zusammenhang des Nautischen Dreiecks und des Kugeldreiecks auf der Erdkugel} - - \begin{center} - \includegraphics[height=5cm,width=5cm]{Bilder/kugel3.png} - \end{center} -Wie man im oberen Bild sieht und auch am Anfang dieses Kapitels bereits erwähnt wurde, liegt das Nautische Dreieck auf der Himmelskugel mit den Ecken Zenit, Gestirn und Himmelsnordpol. -Das selbe Dreieck kann man aber auch auf die Erdkugel projezieren und hat dann die Ecken Standort, Bildpunkt und Nordpol. -Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. - -\subsection{Varianten vom Nautischen Dreieck} -\section{Standortbestimmung ohne elektronische Hilfsmittel} -Um den eigenen Standort herauszufinden, wird in diesem Kapitel die Projektion Nautische Dreieck auf der Erdkugel zur Hilfe genommen. -Mithilfe einiger Hilfsmittel und der Sphärischen Trigonometrie kann man dann die Längen- und Breitengrade des eigenen Standortes bestimmen. - - \begin{center} - \includegraphics[width=6cm]{Bilder/dreieck.png} - \end{center} - - - -\subsection{Ecke P - Unser Standort} -Unser eigener Standort ist der gesuchte Punkt A. - -\subsection{Ecke A - Nordpol} -Der Vorteil ander Idee des Nautischen Dreiecks ist, dass eine Ecke immer der Nordpol (in der Himmelskugel der Himmelsnordpol) ist. -Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so simpel. - -\subsection{Ecke B und C - Bildpunkt XXX und YYY} -Für die Standortermittlung benötigt man als weiteren Punkt ein Gestirn bzw. seinen Bildpunkt auf der Erdkugel. -Damit das trigonometrische Rechnen einfacher wird, werden hier zwei Gestirne zur Hilfe genommen. -\\ -Es gibt diverse Gestirne, die man nutzen kann. -\begin{itemize} - \item Sonne - \item Mond - \item Die vier Navigationsplaneten Venus, Mars, Jupiter und Saturn -\end{itemize} - -Zu all diesen Gestirnen gibt es Ephemeriden (Jahrbücher). -Dort findet man unter Anderem die Rektaszension und Deklination, welche für jeden Tag und Stunde beschrieben ist. Für Minuten genaue Angaben muss man dann zwischen den Stunden interpolieren. -Mithilfe dieser beiden Angaben kann man die Längen- und Breitengrade diverser Gestirne berechnen. - -\subsubsection{Sternzeit und Rektaszension} -Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt steht. -Der Frühlungspunkt ist der Nullpunkt auf dem Himmelsäquator. -Die Tatsache, dass sich die Himmelskugel ca. vier Minuten schneller um die eigene Achse dreht als die Erdkugel, stellt hier ein kleines Problem dar. -Die Lösung ist die Sternzeit. -Am Frühlingspunkt (21. März) 12:00 Uhr ist die Sternzeit -$\theta = 0$. - -Die Sternzeit geht vom Frühlungspunkt aus, an welchem die Sonne den Himmelsäquator schneidet. -Für die Standortermittlung auf der Erdkugel ist es am einfachsten, wenn man die Sternzeit von Greenwich berechnet. -Für die Sternzeit von Greenwich braucht man als erstes das Julianische Datum vom aktuellen Tag, welches sich leicht recherchieren oder berechnen lässt: \hl{$JD=....$} - -Nun berechnet man $T=\frac{JD-2451545}{36525}$ und damit die mittlere Sternzeit von Greenwich - - $T_{Greenwich} = 6^h 41^m 50^s,54841 + 8640184^s,812866 * T + 0^s,093104*T^2 - 0^s,0000062 * T^3$. - - Wenn mann die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ mithilfe der Rektaszension und Sternzeit bestimmen. - Dies gilt analog auch für das zweite Gestirn. - - \subsubsection{Deklination} - Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und ergibt schlussendlich den Breitengrad $\psi = \delta$. - - - -\subsection{Bestimmung des eigenen Standortes P} -Nun hat man die Koordinaten der beiden Gestirne und man weiss die Koordinaten des Nordpols. -Somit können wir ein erstes Kugeldreieck auf der Erde aufspannen. - - - \begin{center} - \includegraphics[width=5cm]{Bilder/dreieck.png} - \end{center} - - -\subsubsection{Bestimmung des ersten Dreiecks} - Mithilfe des sphärischen Trigonometrie und den darausfolgenden Zusammenhängen des Nautischen Dreiecks können wir nun alle Seiten des Dreiecks $ABC$ berechnen. - - Die Seitenlänge der Seite "Nordpol zum Bildpunkt XXX" sei $c$. - Dann ist $c = \frac{\pi}{2} - \delta_1$. - - Die Seitenlänge der Seite "Nordpol zum Bildpunkt YYY" sei $b$. - Dann ist $b = \frac{\pi}{2} - \delta_2$. - - Der Innenwinkel beim der Ecke "Nordpol" sei $\alpha$. - Dann ist $ \alpha = |\lambda_1 - \lambda_2|$. - -mit - - $\delta_1 =$ Deklination Bildpunkt XXX - -$\delta_2 =$ Deklination Bildpunk YYY - -$\lambda_1 =$ Längengrad Bildpunkt XXX - -$\lambda_2 =$ Längengrad Bildpunkt YYY - - Wichtig ist: Die Differenz der Längengrade ist gleich der Innenwinkel Alpha, deswegen der Betrag! - -Nun haben wir die beiden Seiten $c\ und\ b$ und den Winkel $\alpha$, der sich zwischen diesen Seiten befindet. -Mithilfe des Seiten-Kosinussatzes $cos(a) = cos(b)*cos(c) + sin(b) * sin(c)*cos(\alpha)$ können wir nun die dritte Seitenlänge bestimmen. -Es ist darauf zu achten, dass hier natürlich die Seitenlängen in Bogenmass sind und dementsprechend der Kosinus und Sinus verwendet wird. -Jetzt fehlen noch die beiden anderen Innenwinkel $\beta \ und\ \gamma$. - -Dieser bestimmen wir mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$. -Hier muss man aufpassen, dass man Seite von Winkel unterscheiden kann. Im Zähler sind die Seiten, im Nenner die Winkel. Somit ist $sin(\beta) = sin(b) * \frac{sin(\alpha)}{sin(a)} $. - -Schlussendlich haben wir die Seiten $a,b\ und \ c$, die Ecken A,B und C und die Winkel $\alpha, \beta \ und \ \gamma$ bestimmt und somit das ganze erste Kugeldreieck berechnet. - -\subsubsection{Bestimmung des zweiten Dreiecks} -Wir bilden nun ein zweites Dreieck, welches die Ecken B und C des ersten Dreiecks besitzt. -Die dritte Ecke ist der eigene Standort P. -Unser Standort definiere sich aus einer geographischen Breite $\delta$ und einer geographischen Länge $\lambda$. - -Die Seite von P zu B sei $pb$ und die Seite von P zu C sei $pc$. -Die beiden Seitenlängen kann man mit dem Sextant messen und durch eine einfache Formel bestimmen, nämlich $pb=\frac{\pi}{2} - h_{B}$ und $pc=\frac{\pi}{2} - h_{C}$ - -mit $h_B=$ Höhe von Gestirn in B und $h_C=$ Höhe von Gestirn in C mit Sextant gemessen. -\\ - -Nun muss man eine Verbindungslinie ziehen zwischen P und A. Die Länge $l$ dieser Linie entspricht der gesuchten geographischen Breite $\delta$. Diese lässt sich mithilfe des Dreiecks $ABP$ mit den bekannten Seiten $c\ und \ pb$ und des Seiten-Kosinussatzes - -$cos(l) = cos(c)*cos(pb) + sin(c) * sin(pb)*cos(\beta)$ berechnen. - -Es fehlt uns noch $\beta1$. -Da wir aber $pc$, $pb$ und $a$ kennen, kann man mit dem Seiten-Kosinussatz den Winkel $\beta1$ berechnen -\\ - -Somit ist $\delta = cos(l) = cos(c)*cos(pb) + sin(c) * sin(pb)*cos(\beta)$. -\\ - -Für die Geographische Länge $\lambda$ des eigenen Standortes muss man den Winkel $\omega$, welcher sich im Dreieck $ABP$ în der Ecke bei $A$ befindet mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$ bestimmen. -\\ - -Somit ist $\omega=sin(pb)*\frac{sin(\beta)}{sin(l)}$ und unsere gesuchte geographische Länge schlussendlich -$\lambda=\lambda_1 - \omega$ - - - -\end{document} \ No newline at end of file diff --git a/buch/papers/nav/packages.tex b/buch/papers/nav/packages.tex index 15c7fdc..9faa48d 100644 --- a/buch/papers/nav/packages.tex +++ b/buch/papers/nav/packages.tex @@ -8,9 +8,3 @@ % following example %\usepackage{packagename} -\documentclass[12pt]{scrartcl} -\usepackage{ucs} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{graphicx} -\usepackage{xcolor, soul} diff --git a/buch/papers/nav/trigo.tex b/buch/papers/nav/trigo.tex deleted file mode 100644 index 0dbd7a1..0000000 --- a/buch/papers/nav/trigo.tex +++ /dev/null @@ -1,51 +0,0 @@ -\documentclass[12pt]{scrartcl} -\usepackage{ucs} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{graphicx} - - -\begin{document} - \section{Sphärische Trigonometrie} - \subsection{Das Kugeldreieck} - -Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden, so entsteht ein Kugeldreieck ABC. -A, B und C sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten. -Da die Länge der Grosskreisbögen wegen der Abhängigkeit vom Kugelradius ungeeignet ist, wird die Grösse einer Seite mit dem zugehörigen Mittelpunktwinkel des Grosskreisbogens angegeben. -Laut dieser Definition ist die Seite c der Winkel AMB. -Für ein Kugeldreieck gilt, dass die Summe der drei Seiten kleiner als $2\pi$ aber grösser als 0 ist. -Man kann bei Kugeldreiecken nicht so einfach unterscheiden, was Innen oder Aussen ist. -Wenn man drei Eckpunkte miteinander verbindet, ergeben sich immer 16 Kugeldreiecke. -Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha, \beta, \gamma < \pi$ nennt man Eulersche Dreiecke. -\begin{figure}[h] - \begin{center} - \includegraphics[width=6cm]{Bilder/kugel1.png} - \end{center} - -\end{figure} - -\subsection{Rechtwinkliges Dreieck und Rechtseitiges Dreieck} -Wie auch im uns bekannten Dreieck gibt es beim Kugeldreieck auch ein Rechtwinkliges Kugeldreieck, bei dem ein Winkel $\frac{\pi}{2}$ ist. -Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss. - \newpage -\subsection{Winkelangabe} - - \begin{center} - \includegraphics[width=8cm]{Bilder/kugel2.png} - \end{center} - -Die Winkel eines Kugeldreiecks sind die, welche die Halbtangenten in den Eckpunkten einschliessen. -Für die Summe der Innenwinkel gilt $\alpha+\beta+\gamma = \frac{A}{r^2} + \pi$ und -$\alpha+\beta+\gamma > \pi$. -Der sphärische Exzess $\epsilon = \alpha+\beta+\gamma - \pi$ beschreibt die Abweichung der Innenwinkelsumme von $\pi$ und ist proportional zum Flächeninhalt des Kugeldreiecks. - -\subsection{Sphärischer Sinussatz} -In jedem Dreieck ist das Verhältnis des Sinus einer Seite zum Sinus des Gegenwinkels konstant. -Das bedeutet, dass $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} $ auch beim Kugeldreieck gilt. - -\subsection{Sphärischer Satz des Pythagoras für das rechtwinklige Kugeldreieck} -Es gibt in der sphärischen Trigonometrie eigentlich garkeinen "Satz des Pythagoras", wie man ihn aus der zweidimensionalen Geometrie kennt. -In der sphärischen Trigonometrie gibt es aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks in eine Beziehung bringt. -Es gilt nämlich: $\cos c = \cos a * \cos b$ wenn $\alpha \lor \beta \lor \gamma = \frac{\pi}{2} $. - -\end{document} \ No newline at end of file -- cgit v1.2.1 From 309284c1f79df5b8553b0b8875db188ff7d930af Mon Sep 17 00:00:00 2001 From: "ENEZ-PC\\erdem" Date: Mon, 16 May 2022 20:43:09 +0200 Subject: no message --- buch/papers/nav/bilder/dreieck.png | Bin 0 -> 91703 bytes buch/papers/nav/bilder/kugel1.png | Bin 0 -> 9051 bytes buch/papers/nav/bilder/kugel2.png | Bin 0 -> 9103 bytes buch/papers/nav/bilder/kugel3.png | Bin 0 -> 215188 bytes buch/papers/nav/bilder/projektion.png | Bin 0 -> 41289 bytes buch/papers/nav/einleitung.tex | 17 +++ buch/papers/nav/flatearth.tex | 31 ++++++ buch/papers/nav/geschichte.tex | 22 ++++ buch/papers/nav/main.tex | 28 ++--- buch/papers/nav/nautischesdreieck.tex | 190 ++++++++++++++++++++++++++++++++++ buch/papers/nav/packages.tex | 5 + buch/papers/nav/teil0.tex | 22 ---- buch/papers/nav/teil1.tex | 55 ---------- buch/papers/nav/teil2.tex | 40 ------- buch/papers/nav/teil3.tex | 40 ------- buch/papers/nav/trigo.tex | 51 +++++++++ 16 files changed, 322 insertions(+), 179 deletions(-) create mode 100644 buch/papers/nav/bilder/dreieck.png create mode 100644 buch/papers/nav/bilder/kugel1.png create mode 100644 buch/papers/nav/bilder/kugel2.png create mode 100644 buch/papers/nav/bilder/kugel3.png create mode 100644 buch/papers/nav/bilder/projektion.png create mode 100644 buch/papers/nav/einleitung.tex create mode 100644 buch/papers/nav/flatearth.tex create mode 100644 buch/papers/nav/geschichte.tex create mode 100644 buch/papers/nav/nautischesdreieck.tex delete mode 100644 buch/papers/nav/teil0.tex delete mode 100644 buch/papers/nav/teil1.tex delete mode 100644 buch/papers/nav/teil2.tex delete mode 100644 buch/papers/nav/teil3.tex create mode 100644 buch/papers/nav/trigo.tex (limited to 'buch') diff --git a/buch/papers/nav/bilder/dreieck.png b/buch/papers/nav/bilder/dreieck.png new file mode 100644 index 0000000..2b02105 Binary files /dev/null and b/buch/papers/nav/bilder/dreieck.png differ diff --git a/buch/papers/nav/bilder/kugel1.png b/buch/papers/nav/bilder/kugel1.png new file mode 100644 index 0000000..b3188b7 Binary files /dev/null and b/buch/papers/nav/bilder/kugel1.png differ diff --git a/buch/papers/nav/bilder/kugel2.png b/buch/papers/nav/bilder/kugel2.png new file mode 100644 index 0000000..057740f Binary files /dev/null and b/buch/papers/nav/bilder/kugel2.png differ diff --git a/buch/papers/nav/bilder/kugel3.png b/buch/papers/nav/bilder/kugel3.png new file mode 100644 index 0000000..97066a2 Binary files /dev/null and b/buch/papers/nav/bilder/kugel3.png differ diff --git a/buch/papers/nav/bilder/projektion.png b/buch/papers/nav/bilder/projektion.png new file mode 100644 index 0000000..5dcc0c8 Binary files /dev/null and b/buch/papers/nav/bilder/projektion.png differ diff --git a/buch/papers/nav/einleitung.tex b/buch/papers/nav/einleitung.tex new file mode 100644 index 0000000..42f4b6c --- /dev/null +++ b/buch/papers/nav/einleitung.tex @@ -0,0 +1,17 @@ +\documentclass[12pt]{scrartcl} +\usepackage{ucs} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{graphicx} + +\begin{document} +\section{Einleitung} +Heut zu Tage ist die Navigation ein Teil des Lebens. +Man versendet dem Kollegen seinen Standort, um sich das ewige Erklären zu sparen oder gibt die Adresse des Ziels ein um sich die Sucherei zu schenken. +Dies wird durch Technologien wie Funknavigation, welches ein auf Langzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt. +Vor all diesen technologischen Fortschritten gab es lediglich die Astronavigation, welche heute noch auf kleineren Schiffen benötigt wird im Falle eines Stromausfalls. +Aber wie funktioniert die Navigation mit den Sternen? Welche Hilfsmittel benötigt man, welche Rolle spielt die Mathematik und weshalb kann die Erde nicht flach sein? +In diesem Kapitel werden genau diese Fragen mithilfe des Nautischen Dreiecks, der Sphärischen Trigonometrie und einigen Hilfsmitteln und Messgeräten beantwortet. + + +\end{document} \ No newline at end of file diff --git a/buch/papers/nav/flatearth.tex b/buch/papers/nav/flatearth.tex new file mode 100644 index 0000000..b14dd4b --- /dev/null +++ b/buch/papers/nav/flatearth.tex @@ -0,0 +1,31 @@ +\documentclass[12pt]{scrartcl} +\usepackage{ucs} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{graphicx} + +\begin{document} + \section{Warum ist die Erde nicht flach?} + + \begin{figure}[h] + \begin{center} + \includegraphics[width=10cm]{bilder/projektion.png} + \caption{Mercator Projektion} + \end{center} + \end{figure} + +Es gibt heut zu Tage viele Beweise dafür, dass die Erde eine Kugel ist. +Die Fotos von unserem Planeten oder die Berichte der Astronauten. + Aber schon vor ca. 2300 Jahren hat Aristotoles bemerkt, dass Schiffe im Horizont verschwinden und die einzige Erklärung dafür die Kugelgestalt der Erde ist oder der Erdschatten bei einer Mondfinsternis immer rund ist. + Eratosthenes konnte etwa 100 Jahre später den Erdumfang berechnen. + Er beobachtete, dass die Sonne in Syene mittags im Zenit steht und gleichzeitig in Alexandria unter einem Winkel einfällt. + Mithilfe der Geometrie konnte er mit dem Abstand der Städte und dem Einfallswinkel den Umfang berechnen. + Auch in der Navigation würden grobe Fehler passieren, wenn man davon ausgeht, dass die Erde eine Scheibe ist. +Man sieht es zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. Grönland ist auf der Weltkarte so gross wie Afrika. +In der Anwendung Google Earth jedoch ist Grönland etwa so gross wie Algerien. +Das liegt daran, das man die 3D – Weltkarte nicht einfach auslegen kann. +Der Kartograph Gerhard Mercator projizierte die Erdkugel auf ein Papier und erstellte so eine winkeltreue Karte. Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. +Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen. + + +\end{document} \ No newline at end of file diff --git a/buch/papers/nav/geschichte.tex b/buch/papers/nav/geschichte.tex new file mode 100644 index 0000000..a20eb6d --- /dev/null +++ b/buch/papers/nav/geschichte.tex @@ -0,0 +1,22 @@ +\documentclass[12pt]{scrartcl} +\usepackage{ucs} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{graphicx} + +\begin{document} +\section{Geschichte der sphärischen Navigation} +Die Orientierung mit Hilfe der Sterne und der sphärischen Trigonometrie bewegt die Menschheit schon seit mehreren tausend Jahren. +Nach Hinweisen und Schätzungen von Forscher haben schon vor 4000 Jahren die Ägypter und Gelehrten aus Babylon mit Hilfe der Astronomie den Lauf der Gestirne (Himmelskörper) zu berechnen versucht, jedoch ohne Erfolg. +Etwa 350 vor Christus waren es die Griechen, welche den damaligen Astronomen Hilfestellungen mittels Kugel-Geometrien leisten konnten. +Aus diesen Geometrien wurden erste mathematische Sätze aufgestellt und ein paar Jahrhunderte später kamen zu diesem Thema auch Berechnungen dazu. +Ebenso wurden Kartenmaterial mit Sternenbilder angefertigt. +Die Sinusfunktion war noch nicht bekannt, jedoch kamen zu dieser Zeit die ersten Ansätze der Cosinusfunktion aus Indien. +Von diesen Hilfen darauf aufbauend konnte um 900 die Araber der Sinussatz entwickeln. +Doch ein paar weitere Jahrhunderte vergingen bis zu diesem Thema wieder verstärkt Forschung betrieben wurde. +Dies aus dem Grund, da im 15. Jahrhundert grosse Entdeckungsreisen, hauptsächlich per Schiff, erfolgten und die Orientierung vermehrt an Wichtigkeit gewann. +Auch die Verwendung der Tangens- und Sinusfunktion sowie der neu entwickelte Seitencosinussatz trugen zu einer Verbesserung der Orientierung herbei. +Im 16. Jahrhundert wurde dann ein weiterer trigonometrischer Satz, der Winkelcosinussatz hergeleitet. Stück für Stück wurden infolge der Entdeckung des Logarithmus im 17. Jahrhundert viele neue Methoden entwickelt. +Auch eine Verbesserung der kartographischen Verwendung der Kugelgeometrie wurde vorgenommen. +Es folgten weitere Entwicklungen in nicht euklidische Geometrien und im 19. Jahrhundert sowie auch im 20. Jahrhundert wurde zudem für die Relativitätstheorie auch die sphärische Trigonometrie beigezogen. +\end{document} \ No newline at end of file diff --git a/buch/papers/nav/main.tex b/buch/papers/nav/main.tex index e11e2c0..9758de9 100644 --- a/buch/papers/nav/main.tex +++ b/buch/papers/nav/main.tex @@ -8,29 +8,13 @@ \begin{refsection} \chapterauthor{Hans Muster} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} -\input{papers/nav/teil0.tex} -\input{papers/nav/teil1.tex} -\input{papers/nav/teil2.tex} -\input{papers/nav/teil3.tex} + +\input{papers/nav/einleitung.tex} +\input{papers/nav/geschichte.tex} +\input{papers/nav/flatearth.tex} +\input{papers/nav/trigo.tex} +\input{papers/nav/nautischesdreieck.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex new file mode 100644 index 0000000..0bb213c --- /dev/null +++ b/buch/papers/nav/nautischesdreieck.tex @@ -0,0 +1,190 @@ +\documentclass[12pt]{scrartcl} +\usepackage{ucs} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{graphicx} + \usepackage{xcolor, soul} + \sethlcolor{yellow} +\begin{document} + \setlength{\parindent}{0em} +\section{Das Nautische Dreieck} +\subsection{Definition des Nautischen Dreiecks} +Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der \textbf{Himmelskugel}. +Die Himmelskugel ist eine gedachte Kugel, welche die Erde und dessen Beobachter umgibt und als Rechenfläche für Koordinaten in der Astronomie und Geodäsie dient.\\ +Das Nautische Dreieck definiert sich durch folgende Ecken: +\begin{itemize} + \item Zenit + \item Gestirn + \item Himmelspol +\end{itemize} +Der Zenit ist jener Punkt, der vom Erdmittelpunkt durch denn eigenen Standort an die Himmelskugel verlängert wird. +Ein Gestirn ist ein Planet oder ein Fixstern, zu welchen es diverse Jahrbücher mit allen astronomischen Eigenschaften gibt. +Der Himmelspol ist der Nordpol an die Himmelskugel projeziert. +\\ +Zur Anwendung der Formeln der sphärischen Trigonometrie gelten folgende einfache Zusammenhänge: +\begin{itemize} + \item Seitenlänge Zenit zu Himmelspol $= \frac{\pi}{2} - \phi $ + \item Seitenlänge Himmelspol zu Gestirn $= \frac{\pi}{2} - \delta$ + \item Seitenlänge Zenit zu Gestirn $= \frac{\pi}{2} - h$ + \item Winkel von Zenit zu Himmelsnordpol zu Gestirn$=\pi - \alpha$ + \item Winkel von Himmelsnordpol zu Zenit und Gestirn$= \tau$ +\end{itemize} +Um mit diesen Zusammenhängen zu rechnen benötigt man folgende Legende: + +$\alpha \ \widehat{=} \ Rektaszension $ + +$\delta \ \widehat{=} \ Deklination =$ Breitengrad des Gestirns + +$\theta \ \widehat{=} \ Sternzeit$ + +$\phi \ \widehat{=} \ Geographische \ Breite $ + +$\tau = \theta-\alpha \ \widehat{=} \ Stundenwinkel =$ Längengrad des Gestirns + +$a \ \widehat{=} \ Azimut $ + +$h \ \widehat{=} \ Hoehe$ + + + +\subsection{Zusammenhang des Nautischen Dreiecks und des Kugeldreiecks auf der Erdkugel} + + \begin{center} + \includegraphics[height=5cm,width=5cm]{Bilder/kugel3.png} + \end{center} +Wie man im oberen Bild sieht und auch am Anfang dieses Kapitels bereits erwähnt wurde, liegt das Nautische Dreieck auf der Himmelskugel mit den Ecken Zenit, Gestirn und Himmelsnordpol. +Das selbe Dreieck kann man aber auch auf die Erdkugel projezieren und hat dann die Ecken Standort, Bildpunkt und Nordpol. +Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. + +\subsection{Varianten vom Nautischen Dreieck} +\section{Standortbestimmung ohne elektronische Hilfsmittel} +Um den eigenen Standort herauszufinden, wird in diesem Kapitel die Projektion Nautische Dreieck auf der Erdkugel zur Hilfe genommen. +Mithilfe einiger Hilfsmittel und der Sphärischen Trigonometrie kann man dann die Längen- und Breitengrade des eigenen Standortes bestimmen. + + \begin{center} + \includegraphics[width=6cm]{Bilder/dreieck.png} + \end{center} + + + +\subsection{Ecke P - Unser Standort} +Unser eigener Standort ist der gesuchte Punkt A. + +\subsection{Ecke A - Nordpol} +Der Vorteil ander Idee des Nautischen Dreiecks ist, dass eine Ecke immer der Nordpol (in der Himmelskugel der Himmelsnordpol) ist. +Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so simpel. + +\subsection{Ecke B und C - Bildpunkt XXX und YYY} +Für die Standortermittlung benötigt man als weiteren Punkt ein Gestirn bzw. seinen Bildpunkt auf der Erdkugel. +Damit das trigonometrische Rechnen einfacher wird, werden hier zwei Gestirne zur Hilfe genommen. +\\ +Es gibt diverse Gestirne, die man nutzen kann. +\begin{itemize} + \item Sonne + \item Mond + \item Die vier Navigationsplaneten Venus, Mars, Jupiter und Saturn +\end{itemize} + +Zu all diesen Gestirnen gibt es Ephemeriden (Jahrbücher). +Dort findet man unter Anderem die Rektaszension und Deklination, welche für jeden Tag und Stunde beschrieben ist. Für Minuten genaue Angaben muss man dann zwischen den Stunden interpolieren. +Mithilfe dieser beiden Angaben kann man die Längen- und Breitengrade diverser Gestirne berechnen. + +\subsubsection{Sternzeit und Rektaszension} +Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt steht. +Der Frühlungspunkt ist der Nullpunkt auf dem Himmelsäquator. +Die Tatsache, dass sich die Himmelskugel ca. vier Minuten schneller um die eigene Achse dreht als die Erdkugel, stellt hier ein kleines Problem dar. +Die Lösung ist die Sternzeit. +Am Frühlingspunkt (21. März) 12:00 Uhr ist die Sternzeit +$\theta = 0$. + +Die Sternzeit geht vom Frühlungspunkt aus, an welchem die Sonne den Himmelsäquator schneidet. +Für die Standortermittlung auf der Erdkugel ist es am einfachsten, wenn man die Sternzeit von Greenwich berechnet. +Für die Sternzeit von Greenwich braucht man als erstes das Julianische Datum vom aktuellen Tag, welches sich leicht recherchieren oder berechnen lässt: \hl{$JD=....$} + +Nun berechnet man $T=\frac{JD-2451545}{36525}$ und damit die mittlere Sternzeit von Greenwich + + $T_{Greenwich} = 6^h 41^m 50^s,54841 + 8640184^s,812866 * T + 0^s,093104*T^2 - 0^s,0000062 * T^3$. + + Wenn mann die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ mithilfe der Rektaszension und Sternzeit bestimmen. + Dies gilt analog auch für das zweite Gestirn. + + \subsubsection{Deklination} + Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und ergibt schlussendlich den Breitengrad $\psi = \delta$. + + + +\subsection{Bestimmung des eigenen Standortes P} +Nun hat man die Koordinaten der beiden Gestirne und man weiss die Koordinaten des Nordpols. +Somit können wir ein erstes Kugeldreieck auf der Erde aufspannen. + + + \begin{center} + \includegraphics[width=5cm]{Bilder/dreieck.png} + \end{center} + + +\subsubsection{Bestimmung des ersten Dreiecks} + Mithilfe des sphärischen Trigonometrie und den darausfolgenden Zusammenhängen des Nautischen Dreiecks können wir nun alle Seiten des Dreiecks $ABC$ berechnen. + + Die Seitenlänge der Seite "Nordpol zum Bildpunkt XXX" sei $c$. + Dann ist $c = \frac{\pi}{2} - \delta_1$. + + Die Seitenlänge der Seite "Nordpol zum Bildpunkt YYY" sei $b$. + Dann ist $b = \frac{\pi}{2} - \delta_2$. + + Der Innenwinkel beim der Ecke "Nordpol" sei $\alpha$. + Dann ist $ \alpha = |\lambda_1 - \lambda_2|$. + +mit + + $\delta_1 =$ Deklination Bildpunkt XXX + +$\delta_2 =$ Deklination Bildpunk YYY + +$\lambda_1 =$ Längengrad Bildpunkt XXX + +$\lambda_2 =$ Längengrad Bildpunkt YYY + + Wichtig ist: Die Differenz der Längengrade ist gleich der Innenwinkel Alpha, deswegen der Betrag! + +Nun haben wir die beiden Seiten $c\ und\ b$ und den Winkel $\alpha$, der sich zwischen diesen Seiten befindet. +Mithilfe des Seiten-Kosinussatzes $cos(a) = cos(b)*cos(c) + sin(b) * sin(c)*cos(\alpha)$ können wir nun die dritte Seitenlänge bestimmen. +Es ist darauf zu achten, dass hier natürlich die Seitenlängen in Bogenmass sind und dementsprechend der Kosinus und Sinus verwendet wird. +Jetzt fehlen noch die beiden anderen Innenwinkel $\beta \ und\ \gamma$. + +Dieser bestimmen wir mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$. +Hier muss man aufpassen, dass man Seite von Winkel unterscheiden kann. Im Zähler sind die Seiten, im Nenner die Winkel. Somit ist $sin(\beta) = sin(b) * \frac{sin(\alpha)}{sin(a)} $. + +Schlussendlich haben wir die Seiten $a,b\ und \ c$, die Ecken A,B und C und die Winkel $\alpha, \beta \ und \ \gamma$ bestimmt und somit das ganze erste Kugeldreieck berechnet. + +\subsubsection{Bestimmung des zweiten Dreiecks} +Wir bilden nun ein zweites Dreieck, welches die Ecken B und C des ersten Dreiecks besitzt. +Die dritte Ecke ist der eigene Standort P. +Unser Standort definiere sich aus einer geographischen Breite $\delta$ und einer geographischen Länge $\lambda$. + +Die Seite von P zu B sei $pb$ und die Seite von P zu C sei $pc$. +Die beiden Seitenlängen kann man mit dem Sextant messen und durch eine einfache Formel bestimmen, nämlich $pb=\frac{\pi}{2} - h_{B}$ und $pc=\frac{\pi}{2} - h_{C}$ + +mit $h_B=$ Höhe von Gestirn in B und $h_C=$ Höhe von Gestirn in C mit Sextant gemessen. +\\ + +Nun muss man eine Verbindungslinie ziehen zwischen P und A. Die Länge $l$ dieser Linie entspricht der gesuchten geographischen Breite $\delta$. Diese lässt sich mithilfe des Dreiecks $ABP$ mit den bekannten Seiten $c\ und \ pb$ und des Seiten-Kosinussatzes + +$cos(l) = cos(c)*cos(pb) + sin(c) * sin(pb)*cos(\beta)$ berechnen. + +Es fehlt uns noch $\beta1$. +Da wir aber $pc$, $pb$ und $a$ kennen, kann man mit dem Seiten-Kosinussatz den Winkel $\beta1$ berechnen +\\ + +Somit ist $\delta = cos(l) = cos(c)*cos(pb) + sin(c) * sin(pb)*cos(\beta)$. +\\ + +Für die Geographische Länge $\lambda$ des eigenen Standortes muss man den Winkel $\omega$, welcher sich im Dreieck $ABP$ în der Ecke bei $A$ befindet mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$ bestimmen. +\\ + +Somit ist $\omega=sin(pb)*\frac{sin(\beta)}{sin(l)}$ und unsere gesuchte geographische Länge schlussendlich +$\lambda=\lambda_1 - \omega$ + + + +\end{document} \ No newline at end of file diff --git a/buch/papers/nav/packages.tex b/buch/papers/nav/packages.tex index 9faa48d..16d3a3c 100644 --- a/buch/papers/nav/packages.tex +++ b/buch/papers/nav/packages.tex @@ -8,3 +8,8 @@ % following example %\usepackage{packagename} +\usepackage{ucs} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{graphicx} +\usepackage{xcolor, soul} \ No newline at end of file diff --git a/buch/papers/nav/teil0.tex b/buch/papers/nav/teil0.tex deleted file mode 100644 index f3323a9..0000000 --- a/buch/papers/nav/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{nav:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{nav:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/nav/teil1.tex b/buch/papers/nav/teil1.tex deleted file mode 100644 index 996202f..0000000 --- a/buch/papers/nav/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{nav:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{nav:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{nav:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{nav:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{nav:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/nav/teil2.tex b/buch/papers/nav/teil2.tex deleted file mode 100644 index 5a52e03..0000000 --- a/buch/papers/nav/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{nav:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{nav:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/nav/teil3.tex b/buch/papers/nav/teil3.tex deleted file mode 100644 index 2b5d2d5..0000000 --- a/buch/papers/nav/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{nav:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{nav:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/nav/trigo.tex b/buch/papers/nav/trigo.tex new file mode 100644 index 0000000..0dbd7a1 --- /dev/null +++ b/buch/papers/nav/trigo.tex @@ -0,0 +1,51 @@ +\documentclass[12pt]{scrartcl} +\usepackage{ucs} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{graphicx} + + +\begin{document} + \section{Sphärische Trigonometrie} + \subsection{Das Kugeldreieck} + +Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden, so entsteht ein Kugeldreieck ABC. +A, B und C sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten. +Da die Länge der Grosskreisbögen wegen der Abhängigkeit vom Kugelradius ungeeignet ist, wird die Grösse einer Seite mit dem zugehörigen Mittelpunktwinkel des Grosskreisbogens angegeben. +Laut dieser Definition ist die Seite c der Winkel AMB. +Für ein Kugeldreieck gilt, dass die Summe der drei Seiten kleiner als $2\pi$ aber grösser als 0 ist. +Man kann bei Kugeldreiecken nicht so einfach unterscheiden, was Innen oder Aussen ist. +Wenn man drei Eckpunkte miteinander verbindet, ergeben sich immer 16 Kugeldreiecke. +Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha, \beta, \gamma < \pi$ nennt man Eulersche Dreiecke. +\begin{figure}[h] + \begin{center} + \includegraphics[width=6cm]{Bilder/kugel1.png} + \end{center} + +\end{figure} + +\subsection{Rechtwinkliges Dreieck und Rechtseitiges Dreieck} +Wie auch im uns bekannten Dreieck gibt es beim Kugeldreieck auch ein Rechtwinkliges Kugeldreieck, bei dem ein Winkel $\frac{\pi}{2}$ ist. +Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss. + \newpage +\subsection{Winkelangabe} + + \begin{center} + \includegraphics[width=8cm]{Bilder/kugel2.png} + \end{center} + +Die Winkel eines Kugeldreiecks sind die, welche die Halbtangenten in den Eckpunkten einschliessen. +Für die Summe der Innenwinkel gilt $\alpha+\beta+\gamma = \frac{A}{r^2} + \pi$ und +$\alpha+\beta+\gamma > \pi$. +Der sphärische Exzess $\epsilon = \alpha+\beta+\gamma - \pi$ beschreibt die Abweichung der Innenwinkelsumme von $\pi$ und ist proportional zum Flächeninhalt des Kugeldreiecks. + +\subsection{Sphärischer Sinussatz} +In jedem Dreieck ist das Verhältnis des Sinus einer Seite zum Sinus des Gegenwinkels konstant. +Das bedeutet, dass $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} $ auch beim Kugeldreieck gilt. + +\subsection{Sphärischer Satz des Pythagoras für das rechtwinklige Kugeldreieck} +Es gibt in der sphärischen Trigonometrie eigentlich garkeinen "Satz des Pythagoras", wie man ihn aus der zweidimensionalen Geometrie kennt. +In der sphärischen Trigonometrie gibt es aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks in eine Beziehung bringt. +Es gilt nämlich: $\cos c = \cos a * \cos b$ wenn $\alpha \lor \beta \lor \gamma = \frac{\pi}{2} $. + +\end{document} \ No newline at end of file -- cgit v1.2.1 From 800ca10daf88dd073c239b6478bb34f81e48410f Mon Sep 17 00:00:00 2001 From: "ENEZ-PC\\erdem" Date: Tue, 17 May 2022 13:34:13 +0200 Subject: first commit nav --- buch/buch.aux | 22 + buch/buch.bbl | 0 buch/buch.blg | 48 + buch/buch.idx | 0 buch/buch.log | 2106 +++++++++++++++++++++++++++++++++ buch/papers/nav/einleitung.tex | 12 +- buch/papers/nav/flatearth.tex | 38 +- buch/papers/nav/main.tex | 5 +- buch/papers/nav/nautischesdreieck.tex | 139 ++- buch/papers/nav/packages.tex | 5 - 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Man versendet dem Kollegen seinen Standort, um sich das ewige Erklären zu sparen oder gibt die Adresse des Ziels ein um sich die Sucherei zu schenken. Dies wird durch Technologien wie Funknavigation, welches ein auf Langzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt. Vor all diesen technologischen Fortschritten gab es lediglich die Astronavigation, welche heute noch auf kleineren Schiffen benötigt wird im Falle eines Stromausfalls. Aber wie funktioniert die Navigation mit den Sternen? Welche Hilfsmittel benötigt man, welche Rolle spielt die Mathematik und weshalb kann die Erde nicht flach sein? -In diesem Kapitel werden genau diese Fragen mithilfe des Nautischen Dreiecks, der Sphärischen Trigonometrie und einigen Hilfsmitteln und Messgeräten beantwortet. - - -\end{document} \ No newline at end of file +In diesem Kapitel werden genau diese Fragen mithilfe des Nautischen Dreiecks, der Sphärischen Trigonometrie und einigen Hilfsmitteln und Messgeräten beantwortet. \ No newline at end of file diff --git a/buch/papers/nav/flatearth.tex b/buch/papers/nav/flatearth.tex index b14dd4b..fbabbde 100644 --- a/buch/papers/nav/flatearth.tex +++ b/buch/papers/nav/flatearth.tex @@ -1,31 +1,23 @@ -\documentclass[12pt]{scrartcl} -\usepackage{ucs} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{graphicx} -\begin{document} - \section{Warum ist die Erde nicht flach?} - - \begin{figure}[h] - \begin{center} - \includegraphics[width=10cm]{bilder/projektion.png} - \caption{Mercator Projektion} - \end{center} - \end{figure} + +\section{Warum ist die Erde nicht flach?} + +\begin{figure}[h] + \begin{center} + \includegraphics[width=10cm]{papers/nav/bilder/projektion.png} + \caption[Mercator Projektion]{Mercator Projektion} + \end{center} +\end{figure} Es gibt heut zu Tage viele Beweise dafür, dass die Erde eine Kugel ist. Die Fotos von unserem Planeten oder die Berichte der Astronauten. - Aber schon vor ca. 2300 Jahren hat Aristotoles bemerkt, dass Schiffe im Horizont verschwinden und die einzige Erklärung dafür die Kugelgestalt der Erde ist oder der Erdschatten bei einer Mondfinsternis immer rund ist. - Eratosthenes konnte etwa 100 Jahre später den Erdumfang berechnen. - Er beobachtete, dass die Sonne in Syene mittags im Zenit steht und gleichzeitig in Alexandria unter einem Winkel einfällt. - Mithilfe der Geometrie konnte er mit dem Abstand der Städte und dem Einfallswinkel den Umfang berechnen. - Auch in der Navigation würden grobe Fehler passieren, wenn man davon ausgeht, dass die Erde eine Scheibe ist. +Aber schon vor ca. 2300 Jahren hat Aristotoles bemerkt, dass Schiffe im Horizont verschwinden und die einzige Erklärung dafür die Kugelgestalt der Erde ist oder der Erdschatten bei einer Mondfinsternis immer rund ist. +Eratosthenes konnte etwa 100 Jahre später den Erdumfang berechnen. +Er beobachtete, dass die Sonne in Syene mittags im Zenit steht und gleichzeitig in Alexandria unter einem Winkel einfällt. +Mithilfe der Geometrie konnte er mit dem Abstand der Städte und dem Einfallswinkel den Umfang berechnen. +Auch in der Navigation würden grobe Fehler passieren, wenn man davon ausgeht, dass die Erde eine Scheibe ist. Man sieht es zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. Grönland ist auf der Weltkarte so gross wie Afrika. In der Anwendung Google Earth jedoch ist Grönland etwa so gross wie Algerien. Das liegt daran, das man die 3D – Weltkarte nicht einfach auslegen kann. Der Kartograph Gerhard Mercator projizierte die Erdkugel auf ein Papier und erstellte so eine winkeltreue Karte. Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. -Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen. - - -\end{document} \ No newline at end of file +Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen. \ No newline at end of file diff --git a/buch/papers/nav/main.tex b/buch/papers/nav/main.tex index 9758de9..8688421 100644 --- a/buch/papers/nav/main.tex +++ b/buch/papers/nav/main.tex @@ -4,9 +4,9 @@ % (c) 2020 Hochschule Rapperswil % \chapter{Thema\label{chapter:nav}} -\lhead{Thema} +\lhead{Sphärische Navigation} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Enez Erdem, Marc Kühne} @@ -15,6 +15,7 @@ \input{papers/nav/flatearth.tex} \input{papers/nav/trigo.tex} \input{papers/nav/nautischesdreieck.tex} +\input{papers/nav/sincos.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex index 0bb213c..d6e1388 100644 --- a/buch/papers/nav/nautischesdreieck.tex +++ b/buch/papers/nav/nautischesdreieck.tex @@ -1,12 +1,3 @@ -\documentclass[12pt]{scrartcl} -\usepackage{ucs} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{graphicx} - \usepackage{xcolor, soul} - \sethlcolor{yellow} -\begin{document} - \setlength{\parindent}{0em} \section{Das Nautische Dreieck} \subsection{Definition des Nautischen Dreiecks} Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der \textbf{Himmelskugel}. @@ -19,7 +10,7 @@ Das Nautische Dreieck definiert sich durch folgende Ecken: \end{itemize} Der Zenit ist jener Punkt, der vom Erdmittelpunkt durch denn eigenen Standort an die Himmelskugel verlängert wird. Ein Gestirn ist ein Planet oder ein Fixstern, zu welchen es diverse Jahrbücher mit allen astronomischen Eigenschaften gibt. -Der Himmelspol ist der Nordpol an die Himmelskugel projeziert. +Der Himmelspol ist der Nordpol an die Himmelskugel projiziert. \\ Zur Anwendung der Formeln der sphärischen Trigonometrie gelten folgende einfache Zusammenhänge: \begin{itemize} @@ -35,7 +26,7 @@ $\alpha \ \widehat{=} \ Rektaszension $ $\delta \ \widehat{=} \ Deklination =$ Breitengrad des Gestirns -$\theta \ \widehat{=} \ Sternzeit$ +$\theta \ \widehat{=} \ Sternzeit\ von\ Greenwich$ $\phi \ \widehat{=} \ Geographische \ Breite $ @@ -46,24 +37,31 @@ $a \ \widehat{=} \ Azimut $ $h \ \widehat{=} \ Hoehe$ - +\newpage \subsection{Zusammenhang des Nautischen Dreiecks und des Kugeldreiecks auf der Erdkugel} - +\begin{figure}[h] \begin{center} - \includegraphics[height=5cm,width=5cm]{Bilder/kugel3.png} + \includegraphics[height=5cm,width=5cm]{papers/nav/bilder/kugel3.png} + \caption[Nautisches Dreieck]{Nautisches Dreieck} \end{center} +\end{figure} + Wie man im oberen Bild sieht und auch am Anfang dieses Kapitels bereits erwähnt wurde, liegt das Nautische Dreieck auf der Himmelskugel mit den Ecken Zenit, Gestirn und Himmelsnordpol. Das selbe Dreieck kann man aber auch auf die Erdkugel projezieren und hat dann die Ecken Standort, Bildpunkt und Nordpol. Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. -\subsection{Varianten vom Nautischen Dreieck} + \section{Standortbestimmung ohne elektronische Hilfsmittel} Um den eigenen Standort herauszufinden, wird in diesem Kapitel die Projektion Nautische Dreieck auf der Erdkugel zur Hilfe genommen. Mithilfe einiger Hilfsmittel und der Sphärischen Trigonometrie kann man dann die Längen- und Breitengrade des eigenen Standortes bestimmen. +\begin{figure}[h] \begin{center} - \includegraphics[width=6cm]{Bilder/dreieck.png} - \end{center} + \includegraphics[width=6cm]{papers/nav/bilder/dreieck.png} + \caption[Dreieck für die Standortbestimmung]{Dreieck für die Standortbestimmung} + \end{center} +\end{figure} + @@ -73,8 +71,8 @@ Unser eigener Standort ist der gesuchte Punkt A. \subsection{Ecke A - Nordpol} Der Vorteil ander Idee des Nautischen Dreiecks ist, dass eine Ecke immer der Nordpol (in der Himmelskugel der Himmelsnordpol) ist. Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so simpel. - -\subsection{Ecke B und C - Bildpunkt XXX und YYY} +\newpage +\subsection{Ecke B und C - Bildpunkt X und Y} Für die Standortermittlung benötigt man als weiteren Punkt ein Gestirn bzw. seinen Bildpunkt auf der Erdkugel. Damit das trigonometrische Rechnen einfacher wird, werden hier zwei Gestirne zur Hilfe genommen. \\ @@ -96,64 +94,80 @@ Die Tatsache, dass sich die Himmelskugel ca. vier Minuten schneller um die eige Die Lösung ist die Sternzeit. Am Frühlingspunkt (21. März) 12:00 Uhr ist die Sternzeit $\theta = 0$. - + Die Sternzeit geht vom Frühlungspunkt aus, an welchem die Sonne den Himmelsäquator schneidet. Für die Standortermittlung auf der Erdkugel ist es am einfachsten, wenn man die Sternzeit von Greenwich berechnet. -Für die Sternzeit von Greenwich braucht man als erstes das Julianische Datum vom aktuellen Tag, welches sich leicht recherchieren oder berechnen lässt: \hl{$JD=....$} - -Nun berechnet man $T=\frac{JD-2451545}{36525}$ und damit die mittlere Sternzeit von Greenwich +Für die Sternzeit von Greenwich braucht man als erstes das Julianische Datum $T$ vom aktuellen Tag, welches sich leicht recherchieren lässt. +Im Anschluss berechnet man die Sternzeit von Greenwich +\\ +\\ +$T_{Greenwich} = 6^h 41^m 50^s,54841 + 8640184^s,812866 \cdot T + 0^s,093104 \cdot T^2 - 0^s,0000062 \cdot T^3$. +\\ +\\ +Wenn mann die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ mithilfe der Rektaszension und Sternzeit bestimmen. +Dies gilt analog auch für das zweite Gestirn. - $T_{Greenwich} = 6^h 41^m 50^s,54841 + 8640184^s,812866 * T + 0^s,093104*T^2 - 0^s,0000062 * T^3$. - - Wenn mann die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ mithilfe der Rektaszension und Sternzeit bestimmen. - Dies gilt analog auch für das zweite Gestirn. - - \subsubsection{Deklination} - Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und ergibt schlussendlich den Breitengrad $\psi = \delta$. - +\subsubsection{Deklination} +Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und ergibt schlussendlich den Breitengrad. +\newpage \subsection{Bestimmung des eigenen Standortes P} Nun hat man die Koordinaten der beiden Gestirne und man weiss die Koordinaten des Nordpols. Somit können wir ein erstes Kugeldreieck auf der Erde aufspannen. +\begin{figure}[h] \begin{center} - \includegraphics[width=5cm]{Bilder/dreieck.png} - \end{center} + \includegraphics[width=4.5cm]{papers/nav/bilder/dreieck.png} + \caption[Dreieck für die Standortbestimmung]{Dreieck für die Standortbestimmung} + \end{center} +\end{figure} \subsubsection{Bestimmung des ersten Dreiecks} - Mithilfe des sphärischen Trigonometrie und den darausfolgenden Zusammenhängen des Nautischen Dreiecks können wir nun alle Seiten des Dreiecks $ABC$ berechnen. - - Die Seitenlänge der Seite "Nordpol zum Bildpunkt XXX" sei $c$. - Dann ist $c = \frac{\pi}{2} - \delta_1$. - - Die Seitenlänge der Seite "Nordpol zum Bildpunkt YYY" sei $b$. - Dann ist $b = \frac{\pi}{2} - \delta_2$. - - Der Innenwinkel beim der Ecke "Nordpol" sei $\alpha$. - Dann ist $ \alpha = |\lambda_1 - \lambda_2|$. - + +$A=$ Nordpol + +$B=$ Bildpunkt des Gestirns XXX + +$C=$ Bildpunkt des Gestirns YYY +\\ +\\ +Mithilfe des sphärischen Trigonometrie und den darausfolgenden Zusammenhängen des Nautischen Dreiecks können wir nun alle Seiten des Dreiecks $ABC$ berechnen. + +Die Seitenlänge der Seite "Nordpol zum Bildpunkt XXX" sei $c$. +Dann ist $c = \frac{\pi}{2} - \delta_1$. + +Die Seitenlänge der Seite "Nordpol zum Bildpunkt YYY" sei $b$. +Dann ist $b = \frac{\pi}{2} - \delta_2$. + +Der Innenwinkel beim der Ecke "Nordpol" sei $\alpha$. +Dann ist $ \alpha = |\lambda_1 - \lambda_2|$. + mit - - $\delta_1 =$ Deklination Bildpunkt XXX - + +$\delta_1 =$ Deklination Bildpunkt XXX + $\delta_2 =$ Deklination Bildpunk YYY $\lambda_1 =$ Längengrad Bildpunkt XXX $\lambda_2 =$ Längengrad Bildpunkt YYY - Wichtig ist: Die Differenz der Längengrade ist gleich der Innenwinkel Alpha, deswegen der Betrag! + +Wichtig ist: Die Differenz der Längengrade ist gleich der Innenwinkel Alpha, deswegen der Betrag! Nun haben wir die beiden Seiten $c\ und\ b$ und den Winkel $\alpha$, der sich zwischen diesen Seiten befindet. -Mithilfe des Seiten-Kosinussatzes $cos(a) = cos(b)*cos(c) + sin(b) * sin(c)*cos(\alpha)$ können wir nun die dritte Seitenlänge bestimmen. +Mithilfe des Seiten-Kosinussatzes + +$cos(a) = \cos(b)\cdot \cos(c) + \sin(b) \cdot \sin(c)\cdot \cos(\alpha)$ können wir nun die dritte Seitenlänge bestimmen. + Es ist darauf zu achten, dass hier natürlich die Seitenlängen in Bogenmass sind und dementsprechend der Kosinus und Sinus verwendet wird. Jetzt fehlen noch die beiden anderen Innenwinkel $\beta \ und\ \gamma$. Dieser bestimmen wir mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$. -Hier muss man aufpassen, dass man Seite von Winkel unterscheiden kann. Im Zähler sind die Seiten, im Nenner die Winkel. Somit ist $sin(\beta) = sin(b) * \frac{sin(\alpha)}{sin(a)} $. +Hier muss man aufpassen, dass man Seite von Winkel unterscheiden kann. Im Zähler sind die Seiten, im Nenner die Winkel. Somit ist $\beta =\sin^{-1} [\sin(b) \cdot \frac{\sin(\alpha)}{\sin(a)}] $. Schlussendlich haben wir die Seiten $a,b\ und \ c$, die Ecken A,B und C und die Winkel $\alpha, \beta \ und \ \gamma$ bestimmt und somit das ganze erste Kugeldreieck berechnet. @@ -168,23 +182,22 @@ Die beiden Seitenlängen kann man mit dem Sextant messen und durch eine einfache mit $h_B=$ Höhe von Gestirn in B und $h_C=$ Höhe von Gestirn in C mit Sextant gemessen. \\ -Nun muss man eine Verbindungslinie ziehen zwischen P und A. Die Länge $l$ dieser Linie entspricht der gesuchten geographischen Breite $\delta$. Diese lässt sich mithilfe des Dreiecks $ABP$ mit den bekannten Seiten $c\ und \ pb$ und des Seiten-Kosinussatzes +Nun muss man eine Verbindungslinie ziehen zwischen P und A. Die Länge $l$ dieser Linie entspricht der gesuchten geographischen Breite $\delta$. Diese lässt sich mithilfe des Dreiecks $ABP$, den bekannten Seiten $c\ und \ pb$ und des Seiten-Kosinussatzes berechnen. -$cos(l) = cos(c)*cos(pb) + sin(c) * sin(pb)*cos(\beta)$ berechnen. +Für den Seiten-Kosinussatz benötigt es noch $\kappa$. +Da wir aber $pc$, $pb$ und $a$ kennen, kann man mit dem Seiten-Kosinussatz den Winkel $\beta1$ berechnen und anschliessend $\beta + \beta1 =\kappa$. -Es fehlt uns noch $\beta1$. -Da wir aber $pc$, $pb$ und $a$ kennen, kann man mit dem Seiten-Kosinussatz den Winkel $\beta1$ berechnen -\\ +Somit ist $cos(l) = \cos(c)\cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)$ -Somit ist $\delta = cos(l) = cos(c)*cos(pb) + sin(c) * sin(pb)*cos(\beta)$. -\\ +und -Für die Geographische Länge $\lambda$ des eigenen Standortes muss man den Winkel $\omega$, welcher sich im Dreieck $ABP$ în der Ecke bei $A$ befindet mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$ bestimmen. +$\delta =\cos^{-1} [\cos(c) \cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)]$. \\ -Somit ist $\omega=sin(pb)*\frac{sin(\beta)}{sin(l)}$ und unsere gesuchte geographische Länge schlussendlich -$\lambda=\lambda_1 - \omega$ - - +Für die Geographische Länge $\lambda$ des eigenen Standortes muss man den Winkel $\omega$, welcher sich im Dreieck $ACP$ in der Ecke bei $A$ befindet mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$ bestimmen. +\\ -\end{document} \ No newline at end of file +Somit ist $\omega=\sin^{-1}[\sin(pc) \cdot \frac{\sin(\gamma)}{\sin(l)}]$ und unsere gesuchte geographische Länge schlussendlich +$\lambda=\lambda_1 - \omega$ mit $\lambda_1$=Längengrad Bildpunkt XXX. +\newpage +\listoffigures \ No newline at end of file diff --git a/buch/papers/nav/packages.tex b/buch/papers/nav/packages.tex index 16d3a3c..9faa48d 100644 --- a/buch/papers/nav/packages.tex +++ b/buch/papers/nav/packages.tex @@ -8,8 +8,3 @@ % following example %\usepackage{packagename} -\usepackage{ucs} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{graphicx} -\usepackage{xcolor, soul} \ No newline at end of file diff --git a/buch/papers/nav/sincos.tex b/buch/papers/nav/sincos.tex new file mode 100644 index 0000000..23e3303 --- /dev/null +++ b/buch/papers/nav/sincos.tex @@ -0,0 +1,16 @@ + + + +\section{Warum sind die Sinus- und Kosinusfunktionen spezielle Funktionen?} +Es gibt Hinweise, dass sich schon die Babylonier und Ägypter vor 4000 Jahren sich mit Problemen der sphärischen Trigonometrie beschäftigt haben um den Lauf von Gestirnen (Himmelskörper) zu berechnen. +Jedoch konnten sie sie nicht lösen. +Die Geschichte der sphärischen Trigonometrie ist daher eng mit der Astronomie verknüpft. Ca. 350 vor Christus dachten die Griechen über Kugelgeometrie nach und wurde zu einer Hilfswissenschaft der Astronomen. +In Folge werden auch die ersten Sätze aufgestellt und wenige Jahrhunderte später wurden Berechnungen zu diesem Thema angestellt. +In dieser Zeit wurden auch die ersten Sternenkarten angefertigt, jedoch kannte man damals die Sinusfunktion noch nicht. +Aus Indien stammten die ersten Ansätze zu den Kosinussätzen. +Aufbauend auf den indischen und griechischen Forschungen entwickeln die Araber um 900 den Sinussatz. +Zur Zeit der großen Entdeckungsreisen im 15. Jahrhundert wurden die Forschungen in sphärischer Trigonometrie wieder forciert. +Der Sinussatz, die Tangensfunktion und der neu entwickelte Seitenkosinussatz wurden in dieser Zeit bereits verwendet. +Im nächsten Jahrhundert folgte der Winkelkosinussatz. +Durch weitere mathematische Entwicklungen wie den Logarithmus wurden im Laufe des nächsten Jahrhunderts viele neue Methoden und kartographische Anwendungen der Kugelgeometrie entdeckt. +Im 19. und 20. Jahrhundert wurden weitere nicht-euklidische Geometrien entwickelt und die sphärische Trigonometrie fand auch ihre Anwendung in der Relativitätstheorie. \ No newline at end of file diff --git a/buch/papers/nav/trigo.tex b/buch/papers/nav/trigo.tex index 0dbd7a1..2edd651 100644 --- a/buch/papers/nav/trigo.tex +++ b/buch/papers/nav/trigo.tex @@ -1,14 +1,6 @@ -\documentclass[12pt]{scrartcl} -\usepackage{ucs} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{graphicx} +\section{Sphärische Trigonometrie} +\subsection{Das Kugeldreieck} - -\begin{document} - \section{Sphärische Trigonometrie} - \subsection{Das Kugeldreieck} - Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden, so entsteht ein Kugeldreieck ABC. A, B und C sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten. Da die Länge der Grosskreisbögen wegen der Abhängigkeit vom Kugelradius ungeeignet ist, wird die Grösse einer Seite mit dem zugehörigen Mittelpunktwinkel des Grosskreisbogens angegeben. @@ -19,7 +11,8 @@ Wenn man drei Eckpunkte miteinander verbindet, ergeben sich immer 16 Kugeldreiec Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha, \beta, \gamma < \pi$ nennt man Eulersche Dreiecke. \begin{figure}[h] \begin{center} - \includegraphics[width=6cm]{Bilder/kugel1.png} + \includegraphics[width=6cm]{papers/nav/bilder/kugel1.png} + \caption[Das Kugeldreieck]{Das Kugeldreieck} \end{center} \end{figure} @@ -27,12 +20,15 @@ Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha \subsection{Rechtwinkliges Dreieck und Rechtseitiges Dreieck} Wie auch im uns bekannten Dreieck gibt es beim Kugeldreieck auch ein Rechtwinkliges Kugeldreieck, bei dem ein Winkel $\frac{\pi}{2}$ ist. Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss. - \newpage +\newpage \subsection{Winkelangabe} - +\begin{figure}[h] + \begin{center} - \includegraphics[width=8cm]{Bilder/kugel2.png} + \includegraphics[width=8cm]{papers/nav/bilder/kugel2.png} + \caption[Winkelangabe im Kugeldreieck]{Winkelangabe im Kugeldreieck} \end{center} +\end{figure} Die Winkel eines Kugeldreiecks sind die, welche die Halbtangenten in den Eckpunkten einschliessen. Für die Summe der Innenwinkel gilt $\alpha+\beta+\gamma = \frac{A}{r^2} + \pi$ und @@ -46,6 +42,4 @@ Das bedeutet, dass $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta) \subsection{Sphärischer Satz des Pythagoras für das rechtwinklige Kugeldreieck} Es gibt in der sphärischen Trigonometrie eigentlich garkeinen "Satz des Pythagoras", wie man ihn aus der zweidimensionalen Geometrie kennt. In der sphärischen Trigonometrie gibt es aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks in eine Beziehung bringt. -Es gilt nämlich: $\cos c = \cos a * \cos b$ wenn $\alpha \lor \beta \lor \gamma = \frac{\pi}{2} $. - -\end{document} \ No newline at end of file +Es gilt nämlich: $\cos c = \cos a \cdot \cos b$ wenn $\alpha= \frac{\pi}{2} \lor \beta=\frac{\pi}{2} \lor \gamma = \frac{\pi}{2} $. \ No newline at end of file -- cgit v1.2.1 From c0f7b4bd46fa66526f8ddfb20ce9edbcfbb03d81 Mon Sep 17 00:00:00 2001 From: "ENEZ-PC\\erdem" Date: Tue, 17 May 2022 16:02:53 +0200 Subject: no message --- buch/papers/nav/main.tex | 5 +++-- buch/papers/nav/nautischesdreieck.tex | 37 +++++++++++++++++++---------------- buch/papers/nav/packages.tex | 1 + buch/papers/nav/trigo.tex | 36 +++++++++++++++++++++++++++------- 4 files changed, 53 insertions(+), 26 deletions(-) (limited to 'buch') diff --git a/buch/papers/nav/main.tex b/buch/papers/nav/main.tex index 8688421..de8d1d6 100644 --- a/buch/papers/nav/main.tex +++ b/buch/papers/nav/main.tex @@ -3,7 +3,7 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:nav}} +\chapter{Spährische Navigation\label{chapter:nav}} \lhead{Sphärische Navigation} \begin{refsection} \chapterauthor{Enez Erdem, Marc Kühne} @@ -11,11 +11,12 @@ \input{papers/nav/einleitung.tex} +\input{papers/nav/sincos.tex} \input{papers/nav/geschichte.tex} \input{papers/nav/flatearth.tex} \input{papers/nav/trigo.tex} \input{papers/nav/nautischesdreieck.tex} -\input{papers/nav/sincos.tex} + \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex index d6e1388..b61e908 100644 --- a/buch/papers/nav/nautischesdreieck.tex +++ b/buch/papers/nav/nautischesdreieck.tex @@ -37,6 +37,7 @@ $a \ \widehat{=} \ Azimut $ $h \ \widehat{=} \ Hoehe$ + \newpage \subsection{Zusammenhang des Nautischen Dreiecks und des Kugeldreiecks auf der Erdkugel} \begin{figure}[h] @@ -129,45 +130,47 @@ Somit können wir ein erstes Kugeldreieck auf der Erde aufspannen. $A=$ Nordpol -$B=$ Bildpunkt des Gestirns XXX +$B=$ Bildpunkt des Gestirns X -$C=$ Bildpunkt des Gestirns YYY +$C=$ Bildpunkt des Gestirns Y \\ \\ Mithilfe des sphärischen Trigonometrie und den darausfolgenden Zusammenhängen des Nautischen Dreiecks können wir nun alle Seiten des Dreiecks $ABC$ berechnen. -Die Seitenlänge der Seite "Nordpol zum Bildpunkt XXX" sei $c$. +Die Seitenlänge der Seite "Nordpol zum Bildpunkt X" sei $c$. Dann ist $c = \frac{\pi}{2} - \delta_1$. -Die Seitenlänge der Seite "Nordpol zum Bildpunkt YYY" sei $b$. +Die Seitenlänge der Seite "Nordpol zum Bildpunkt Y" sei $b$. Dann ist $b = \frac{\pi}{2} - \delta_2$. Der Innenwinkel beim der Ecke "Nordpol" sei $\alpha$. Dann ist $ \alpha = |\lambda_1 - \lambda_2|$. - +\\ +\\ mit -$\delta_1 =$ Deklination Bildpunkt XXX - -$\delta_2 =$ Deklination Bildpunk YYY +$\delta_1 =$ Deklination Bildpunkt X -$\lambda_1 =$ Längengrad Bildpunkt XXX +$\delta_2 =$ Deklination Bildpunk Y -$\lambda_2 =$ Längengrad Bildpunkt YYY +$\lambda_1 =$ Längengrad Bildpunkt X +$\lambda_2 =$ Längengrad Bildpunkt Y Wichtig ist: Die Differenz der Längengrade ist gleich der Innenwinkel Alpha, deswegen der Betrag! - +\\ +\\ Nun haben wir die beiden Seiten $c\ und\ b$ und den Winkel $\alpha$, der sich zwischen diesen Seiten befindet. Mithilfe des Seiten-Kosinussatzes - -$cos(a) = \cos(b)\cdot \cos(c) + \sin(b) \cdot \sin(c)\cdot \cos(\alpha)$ können wir nun die dritte Seitenlänge bestimmen. - +$cos(a) = \cos(b)\cdot \cos(c) + \sin(b) \cdot \sin(c)\cdot \cos(\alpha)$ +können wir nun die dritte Seitenlänge bestimmen. Es ist darauf zu achten, dass hier natürlich die Seitenlängen in Bogenmass sind und dementsprechend der Kosinus und Sinus verwendet wird. -Jetzt fehlen noch die beiden anderen Innenwinkel $\beta \ und\ \gamma$. -Dieser bestimmen wir mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$. -Hier muss man aufpassen, dass man Seite von Winkel unterscheiden kann. Im Zähler sind die Seiten, im Nenner die Winkel. Somit ist $\beta =\sin^{-1} [\sin(b) \cdot \frac{\sin(\alpha)}{\sin(a)}] $. +Jetzt fehlen noch die beiden anderen Innenwinkel $\beta \ und\ \gamma$. +Diese bestimmen wir mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$. +Hier muss man aufpassen, dass man Seite von Winkel unterscheiden kann. +Im Zähler sind die Seiten, im Nenner die Winkel. +Somit ist $\beta =\sin^{-1} [\sin(b) \cdot \frac{\sin(\alpha)}{\sin(a)}] $. Schlussendlich haben wir die Seiten $a,b\ und \ c$, die Ecken A,B und C und die Winkel $\alpha, \beta \ und \ \gamma$ bestimmt und somit das ganze erste Kugeldreieck berechnet. diff --git a/buch/papers/nav/packages.tex b/buch/papers/nav/packages.tex index 9faa48d..5b87303 100644 --- a/buch/papers/nav/packages.tex +++ b/buch/papers/nav/packages.tex @@ -8,3 +8,4 @@ % following example %\usepackage{packagename} +\usepackage{amsmath} \ No newline at end of file diff --git a/buch/papers/nav/trigo.tex b/buch/papers/nav/trigo.tex index 2edd651..8b4634f 100644 --- a/buch/papers/nav/trigo.tex +++ b/buch/papers/nav/trigo.tex @@ -1,3 +1,4 @@ +\setlength{\parindent}{0em} \section{Sphärische Trigonometrie} \subsection{Das Kugeldreieck} @@ -11,7 +12,7 @@ Wenn man drei Eckpunkte miteinander verbindet, ergeben sich immer 16 Kugeldreiec Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha, \beta, \gamma < \pi$ nennt man Eulersche Dreiecke. \begin{figure}[h] \begin{center} - \includegraphics[width=6cm]{papers/nav/bilder/kugel1.png} + %\includegraphics[width=6cm]{papers/nav/bilder/kugel1.png} \caption[Das Kugeldreieck]{Das Kugeldreieck} \end{center} @@ -25,21 +26,42 @@ Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine S \begin{figure}[h] \begin{center} - \includegraphics[width=8cm]{papers/nav/bilder/kugel2.png} + %\includegraphics[width=8cm]{papers/nav/bilder/kugel2.png} \caption[Winkelangabe im Kugeldreieck]{Winkelangabe im Kugeldreieck} \end{center} \end{figure} + Die Winkel eines Kugeldreiecks sind die, welche die Halbtangenten in den Eckpunkten einschliessen. -Für die Summe der Innenwinkel gilt $\alpha+\beta+\gamma = \frac{A}{r^2} + \pi$ und -$\alpha+\beta+\gamma > \pi$. -Der sphärische Exzess $\epsilon = \alpha+\beta+\gamma - \pi$ beschreibt die Abweichung der Innenwinkelsumme von $\pi$ und ist proportional zum Flächeninhalt des Kugeldreiecks. +Für die Summe der Innenwinkel gilt +\begin{align} + \alpha+\beta+\gamma &= \frac{A}{r^2} + \pi \ \text{und} \ \alpha+\beta+\gamma > \pi. \nonumber +\end{align} + +Der sphärische Exzess +\begin{align} + \epsilon = \alpha+\beta+\gamma - \pi \nonumber +\end{align} +beschreibt die Abweichung der Innenwinkelsumme von $\pi$ und ist proportional zum Flächeninhalt des Kugeldreiecks. \subsection{Sphärischer Sinussatz} In jedem Dreieck ist das Verhältnis des Sinus einer Seite zum Sinus des Gegenwinkels konstant. -Das bedeutet, dass $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} $ auch beim Kugeldreieck gilt. + +Das bedeutet, dass + +\begin{align} + \frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} \nonumber \ \text{auch beim Kugeldreieck gilt.} +\end{align} + + \subsection{Sphärischer Satz des Pythagoras für das rechtwinklige Kugeldreieck} Es gibt in der sphärischen Trigonometrie eigentlich garkeinen "Satz des Pythagoras", wie man ihn aus der zweidimensionalen Geometrie kennt. In der sphärischen Trigonometrie gibt es aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks in eine Beziehung bringt. -Es gilt nämlich: $\cos c = \cos a \cdot \cos b$ wenn $\alpha= \frac{\pi}{2} \lor \beta=\frac{\pi}{2} \lor \gamma = \frac{\pi}{2} $. \ No newline at end of file + +Es gilt nämlich: +\begin{align} + \cos c = \cos a \cdot \cos b \ \text{wenn} \nonumber & + \alpha = \frac{\pi}{2} \lor \beta =\frac{\pi}{2} \lor \gamma = \frac{\pi}{2}.\nonumber +\end{align} + \ No newline at end of file -- cgit v1.2.1 From 955047b8a63a3b08b27d9203030e2b5193e21dab Mon Sep 17 00:00:00 2001 From: Andrea Mozzini Vellen Date: Wed, 18 May 2022 13:55:56 +0200 Subject: Ersten Entwurf --- buch/papers/kreismembran/main.tex | 10 -- buch/papers/kreismembran/teil1.tex | 181 ++++++++++++++++++------------------- buch/papers/kreismembran/teil2.tex | 128 +++++++++++++++++++------- buch/papers/kreismembran/teil3.tex | 102 ++++++++++++++------- 4 files changed, 255 insertions(+), 166 deletions(-) (limited to 'buch') diff --git a/buch/papers/kreismembran/main.tex b/buch/papers/kreismembran/main.tex index eafec18..e63a118 100644 --- a/buch/papers/kreismembran/main.tex +++ b/buch/papers/kreismembran/main.tex @@ -7,16 +7,6 @@ \lhead{Schwingungen einer kreisförmligen Membran} \begin{refsection} \chapterauthor{Andrea Mozzini Vellen und Tim Tönz} -\begin{itemize} -\item -Tim ist ein snitch -\item -ich dachte wir sind gute Freunden -\item -du schuldest mir ein bier -\item -auch ein gin tonic -\end{itemize} \input{papers/kreismembran/teil0.tex} \input{papers/kreismembran/teil1.tex} diff --git a/buch/papers/kreismembran/teil1.tex b/buch/papers/kreismembran/teil1.tex index 29a47a6..aef5b79 100644 --- a/buch/papers/kreismembran/teil1.tex +++ b/buch/papers/kreismembran/teil1.tex @@ -3,101 +3,98 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -\section{Die Hankel Transformation \label{kreismembran:section:teil1}} -\rhead{Die Hankel Transformation} - -Hermann Hankel (1839-1873) war ein deutscher Mathematiker, der für seinen Beitrag zur mathematischen Analyse und insbesondere für seine namensgebende Transformation bekannt ist. -Diese Transformation tritt bei der Untersuchung von funktionen auf, die nur von der Enternung des Ursprungs abhängen. -Er studierte auch funktionen, jetzt Hankel- oder Bessel- Funktionen genannt, der dritten Art. -Die Hankel Transformation mit Bessel Funktionen al Kern taucht natürlich bei achsensymmetrischen Problemen auf, die in Zylindrischen Polarkoordinaten formuliert sind. -In diesem Kapitel werden die Theorie der Transformation und einige Eigenschaften der Grundoperationen erläutert. - -Wir führen die Definition der Hankel Transformation aus der zweidimensionalen Fourier Transformation und ihrer Umkehrung ein, die durch: -\begin{align} - \mathscr{F}\{f(x,y)\} & = F(k,l)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-i( \bm{\kappa}\cdot \mathbf{r})}f(x,y) dx dy,\label{equation:fourier_transform}\\ - \mathscr{F}^{-1}\{F(x,y)\} & = f(x,y)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{i(\bm{\kappa}\cdot \mathbf{r}))}F(k,l) dx dy \label{equation:inv_fourier_transform} -\end{align} -wo $\mathbf{r}=(x,y)$ und $\bm{\kappa}=(k,l)$. Wie bereits erwähnt, sind Polarkoordinaten für diese Art von Problemen am besten geeignet, also mit, $(x,y)=r(\cos\theta,\sin\theta)$ und $(k,l)=\kappa(\cos\phi,\sin\phi)$, findet man $\bm{\kappa}\cdot\mathbf{r}=\kappa r(\cos(\theta-\phi))$ und danach: -\begin{align} - F(k,\phi)=\frac{1}{2\pi}\int_{0}^{\infty}r dr \int_{0}^{2\pi}e^{-ikr\cos(\theta-\phi)}f(r,\theta) d\phi. - \label{equation:F_ohne_variable_wechsel} -\end{align} -Dann wird angenommen dass, $f(r,\theta)=e^{in\theta}f(r)$, was keine strenge Einschränkung ist, und es wird eine Änderung der Variabeln vorgenommen $\theta-\phi=\alpha-\frac{\pi}{2}$, um \ref{equation:F_ohne_variable_wechsel} zu reduzieren: -\begin{align} - F(k,\phi)=\frac{1}{2\pi}\int_{0}^{\infty}rf(r) dr \int_{\phi_{0}}^{2\pi+\phi_{0}}e^{in(\phi-\frac{\pi}{2})+i(n\alpha-kr\sin\alpha)} d\alpha, - \label{equation:F_ohne_bessel} -\end{align} -wo $\phi_{0}=(\frac{\pi}{2}-\phi)$. - -Unter Verwendung der Integral Darstellung der Besselfunktion vom Ordnung n -\begin{align} - J_n(\kappa r)=\frac{1}{2\pi}\int_{\phi_{0}}^{2\pi + \phi_{0}}e^{i(n\alpha-\kappa r \sin \alpha)} d\alpha - \label{equation:bessel_n_ordnung} -\end{align} -\eqref{equation:F_ohne_bessel} wird sie zu: -\begin{align} - F(k,\phi)&=e^{in(\phi-\frac{\pi}{2})}\int_{0}^{\infty}rJ_n(\kappa r) f(r) dr \label{equation:F_mit_bessel_step_1} \\ - &=e^{in(\phi-\frac{\pi}{2})}\tilde{f}_n(\kappa), - \label{equation:F_mit_bessel_step_2} -\end{align} -wo $\tilde{f}_n(\kappa)$ ist die \textit{Hankel Transformation} von $f(r)$ und ist formell definiert durch: -\begin{align} - \mathscr{H}_n\{f(r)\}=\tilde{f}_n(\kappa)=\int_{0}^{\infty}rJ_n(\kappa r) f(r) dr. - \label{equation:hankel} -\end{align} - -Ähnlich verhält es sich mit der inversen Fourier Transformation in Form von polaren Koordinaten unter der Annahme $f(r,\theta)=e^{in\theta}f(r)$ mit \ref{equation:F_mit_bessel_step_2}, wird die inverse Fourier Transformation \ref{equation:inv_fourier_transform}: +\section{Lösungsmethode 1: Separationsmethode  + \label{kreismembran:section:teil1}} +\rhead{Lösungsmethode 1: Separationsmethode} +An diesem Punkt bleibt also nur noch die Lösung der partiellen Differentialgleichung. In diesem Kapitel wird sie mit Hilfe der Separationsmetode gelöst. + +Wie im vorherigen Kapitel gezeigt, lautet die partielle Differentialgleichung, die die Schwingungen einer Membran beschreibt: +\begin{equation*} + \frac{1}{c^2}\frac{\partial^2u}{\partial t^2} = \Delta u +\end{equation*} +Da es sich um eine Kreisscheibe handelt, werden Polarkoordinaten verwendet, so dass sich der Laplaceoperator ergibt: +\begin{equation*} + \Delta + = + \frac{\partial^2}{\partial r^2} + + + \frac1r + \frac{\partial}{\partial r} + + + \frac{1}{r 2} + \frac{\partial^2}{\partial\varphi^2}. + \label{buch:pde:kreis:laplace} +\end{equation*} + +Es wird eine runde elastische Membran berücksichtigt, die den Gebietbereich $\Omega$ abdeckt und am Rand $\Gamma$ befestigt ist. +Es wird daher davon ausgegangen, dass die Membran aus einem homogenen Material von vernachlässigbarer Dicke gefertigt ist. +Die Membran kann verformt werden, aber innere elastische Kräfte wirken den Verformungen entgegen. Es wirken keine äusseren Kräfte. Es handelt sich somit von einer kreisförmligen eigespannten homogenen schwingenden Membran. + +Daher ist die Membranabweichung im Punkt $(r,\varphi)$ $\in$ $\overline{\rm \Omega}$ zum Zeitpunkt $t$: \begin{align*} - e^{in\theta}f(r)&=\frac{1}{2\pi}\int_{0}^{\infty}\kappa d\kappa \int_{0}^{2\pi}e^{i\kappa r \cos (\theta - \phi)}F(\kappa,\phi) d\phi\\ - &= \frac{1}{2\pi}\int_{0}^{\infty}\kappa \tilde{f}_n(\kappa) d\kappa \int_{0}^{2\pi}e^{in(\phi - \frac{\pi}{2})- i\kappa r \cos (\theta - \phi)} d\phi, + u: \overline{\rm \Omega} \times \mathbb{R}_{\geq 0} &\longrightarrow \mathbb{R}\\ + (r,\varphi,t) &\longmapsto u(r,\varphi,t) \end{align*} -was durch den Wechsel der Variablen $\theta-\phi=-(\alpha+\frac{\pi}{2})$ und $\theta_0=-(\theta+\frac{\pi}{2})$, - -\begin{align} - &= \frac{1}{2\pi}\int_{0}^{\infty}\kappa \tilde{f}_n(\kappa) d\kappa \int_{\theta_0}^{2\pi+\theta_0}e^{in(\theta + \alpha - i\kappa r \sin\alpha)} d\alpha \nonumber \\ - &= e^{in\theta}\int_{0}^{\infty}\kappa J_n(\kappa r) \tilde{f}_n(\kappa) d\kappa,\quad \text{von \eqref{equation:bessel_n_ordnung}} -\end{align} - -Also, die inverse \textit{Hankel Transformation} ist so definiert: -\begin{align} - \mathscr{H}^{-1}_n\{\tilde{f}_n(\kappa)\}=f(r)=\int_{0}^{\infty}\kappa J_n(\kappa r) \tilde{f}_n(\kappa) d\kappa. - \label{equation:inv_hankel} -\end{align} - -Anstelle von $\tilde{f}_n(\kappa)$, wird häufig für die Hankel Transformation verwendet, indem die Ordnung angegeben wird. -\eqref{equation:hankel} und \eqref{equation:inv_hankel} Integralen existieren für eine grosse Klasse von Funktionen, die normalerweise in physikalischen Anwendungen benötigt werden. -Alternativ kann auch die berühmte Hankel Transformationsformel verwendet werden, - -\begin{align} - f(r) = \int_{0}^{\infty}\kappa J_n(\kappa r) d\kappa \int_{0}^{\infty} p J_n(\kappa p)f(p) dp, - \label{equation:hankel_integral_formula} -\end{align} -um die Hankel Transformation \eqref{equation:hankel} und ihre Inverse \eqref{equation:inv_hankel} zu definieren. -Insbesondere die Hankel Transformation der nullten Ordnung ($n=0$) und der ersten Ordnung ($n=1$) sind häufig nützlich, um Lösungen für Probleme mit der Laplace Gleichung in einer achsensymmetrischen zylindrischen Geometrie zu finden. - -\subsection{Operative Eigenschaften der Hankel Transformation\label{sub:op_properties_hankel}} -In diesem Kapitel werden die operativen Eigenschaften der Hankel Transformation aufgeführt. Der Beweis für ihre Gültigkeit wird jedoch nicht analysiert. - -\subsubsection{Skalierung \label{subsub:skalierung}} -Wenn $\mathscr{H}_n\{f(r)\}=\tilde{f}_n(\kappa)$, dann: - -\begin{equation} - \mathscr{H}_n\{f(ar)\}=\frac{1}{a^{2}}\tilde{f}_n \left(\frac{\kappa}{a}\right), \quad a>0. -\end{equation} - -\subsubsection{Persevalsche Relation \label{subsub:perseval}} -Wenn $\tilde{f}(\kappa)=\mathscr{H}_n\{f(r)\}$ und $\tilde{g}(\kappa)=\mathscr{H}_n\{g(r)\}$, dann: - +Da die Membran am Rand befestigt ist, kann es keine Schwingungen geben, so dass die \textit{Dirichlet-Randbedingung} gilt: +\begin{equation*} + u\big|_{\Gamma} = 0 +\end{equation*} +Um eine eindeutige Lösung bestimmen zu können, werden die folgenden Anfangsbedingungen festgelegt: +\begin{align*} + u(r,\varphi, 0) &= f(r,\varphi)\\ + \frac{\partial}{\partial t} u(r,\varphi, 0) &= g(r,\varphi) +\end{align*} +Daher muss an dieser Stelle von einer Separation der Variablen ausgegangen werden: +\begin{equation*} + u(r,\varphi, t) = F(r)G(\varphi)T(t) +\end{equation*} +Dank der Randbedingungen kann also gefordert werden, dass $F(R)=0$ ist, und natürlich, dass $G(\varphi)$ $2\pi$ periodisch ist. Eingesetz in der Differenzialgleichung ergibt: +\begin{equation*} + \frac{1}{c^2}\frac{T''(t)}{T(t)}=\frac{F''(r)}{F(r)}+\frac{1}{r}\frac{F'(r)}{F(r)}+\frac{1}{r^2}\frac{G''(\varphi)}{G(\varphi)} +\end{equation*} +Da die linke Seite nur von $t$ und die rechte Seite nur von $r$ und $\varphi$ abhängt, müssen sie gleich einer reellen Zahl sein. Aus physikalischen Grunden suchen wir nach Lösungen, die weder exponentiell in der Zeit wachsen noch exponentiell abklingen. Dies bedeutet, dass die Konstante negativ sein muss, also schreibt man $k=-k^2$. Daraus ergeben sich die folgenden zwei Gleichungen: +\begin{gather*} + T''(t) + c^2\kappa^2T(t) = 0\\ + r^2\frac{F''(r)}{F(r)} + r \frac{F'(r)}{F(r)} +\kappa^2 r^2 = - \frac{G''(\varphi)}{G(\varphi)} +\end{gather*} +In der zweiten Gleichung hängt die linke Seite nur von $r$ ab, während die rechte Seite nur von $\varphi$ abhängt. Sie müssen also wiederum gleich einer reellen Zahl $\nu$ sein. Also das: +\begin{gather*} + r^2F''(r) + rF'(r) + (\kappa^2 r^2 - \nu)F(r) = 0 \\ + G''(\varphi) = \nu G(\varphi) +\end{gather*} +$G$ kann in einer Fourierreihe entwickelt werden, so dass man sieht, dass $\nu$ die Form $n^2$ mit einer positiven ganzen Zahl sein muss, also: +\begin{equation*} + G(\varphi) = C_n \cos(\varphi) + D_n \sin(\varphi) +\end{equation*} +Die Gleichung $F$ hat die Gestalt +\begin{equation*} + r^2F''(r) + rF'(r) + (\kappa^2 r^2 - n^2)F(r) = 0 \quad (*) +\end{equation*} +Wir bereits in der Vorlesung von Prof. Müller gezeigt, sind die Besselfunktionen +\begin{equation*} + J_{\nu}(x) = r^\nu \displaystyle\sum_{m=0}^{\infty} \frac{(-1)^m x^{2m}}{2^{2m+\nu}m! \Gamma (\nu + m+1)} +\end{equation*} +Lösungen der "Besselschen Differenzialgleichung" +\begin{equation*} + x^2 y'' + xy' + (x^2 - \nu^2)y = 0 +\end{equation*} +Die Funktionen $F(r) = J_n(\kappa r)$ lösen also die Differentialgleichung $(*)$. Die +Randbedingung $F(R)=0$ impliziert, dass $\kappa R$ eine Nullstelle der Besselfunktion +$J_n$ sein muss. Man kann zeigen, dass die Besselfunktionen $J_n, n \geq 0$, alle unendlich +viele Nullstellen +\begin{equation*} + \alpha_{1n} < \alpha_{2n} < ... +\end{equation*} +haben, und dass $\underset{\substack{m\to\infty}}{\text{lim}} \alpha_{mn}=\infty$. Somit ergit sich, dass $\kappa = \frac{\alpha_{mn}}{R}$ für ein $m\geq 1$, und dass +\begin{equation*} + F(r) = J_n (\kappa_{mn}r) \quad mit \quad \kappa_{mn}=\frac{\alpha_{mn}}{R} +\end{equation*} +Die Differenzialgleichung $T''(t) + c^2\kappa^2T(t) = 0$, wird auf ähnliche Weise gelöst wie $G(\varphi)$. Durch Überlagerung aller Ergebnisse erhält man die Lösung \begin{equation} - \int_{0}^{\infty}rf(r) dr = \int_{0}^{\infty}\kappa\tilde{f}(\kappa)\tilde{g}(\kappa) d\kappa. + u(r, \varphi, t) = \displaystyle\sum_{m=1}^{\infty}\displaystyle\sum_{n=0}^{\infty} J_n (k_{mn}r)\cos(n\varphi)[a_{mn}\cos(c \kappa_{mn} t)+b_{mn}\sin(c \kappa_{mn} t)] \end{equation} +Dabei sind m und n ganze Zahlen, wobei m für die Anzahl der Knotenkreise und n +für die Anzahl der Knotenlinien steht. Es gibt bestimmte Bereiche auf der Membran, in denen es keine Bewegung oder Vibration gibt. Wenn der nicht schwingende Bereich ein Kreis ist, nennt man ihn einen Knotenkreis, und wenn er eine Linie ist, nennt man ihn ebenfalls eine Knotenlinie. $Jn(\kappa_{mn}r)$ ist die Besselfunktion $n$-ter Ordnung, wobei kmn die Wellenzahl und $r$ der Radius ist. $a_{mn}$ und $b_{mn}$ sind die zu bestimmenden Konstanten. -\subsubsection{Hankel Transformationen von Ableitungen \label{subsub:ableitungen}} -Wenn $\tilde{f}_n(\kappa)=\mathscr{H}_n\{f(r)\}$, dann: - -\begin{align} - &\mathscr{H}_n\{f'(r)\}=\frac{\kappa}{2n}\left[(n-1)\tilde{f}_{n+1}(\kappa)-(n+1)\tilde{f}_{n-1}(\kappa)\right], \quad n\geq1, \\ - &\mathscr{H}_1\{f'(r)\}=-\kappa \tilde{f}_0(\kappa), -\end{align} -bereitgestellt dass $[rf(r)]$ verschwindet als $r\to0$ und $r\to\infty=0$. \ No newline at end of file +An diesem Punkt stellte sich die Frage, ob es möglich wäre, die partielle Differentialgleichung mit einer anderen Methode als der der Trennung der Variablen zu lösen. Nach einer kurzen Recherche und Diskussion mit Prof. Müller wurde festgestellt, dass die beste Methode die Transformationsmethode ist, genauer gesagt die Anwendung der Hankel-Transformation. Im nächsten Kapitel wird daher diese Integraltransformation vorgestellt und entwickelt, und es wird erläutert, warum sie für diese Art von Problem geeignet ist. diff --git a/buch/papers/kreismembran/teil2.tex b/buch/papers/kreismembran/teil2.tex index 45357f2..8afe817 100644 --- a/buch/papers/kreismembran/teil2.tex +++ b/buch/papers/kreismembran/teil2.tex @@ -2,48 +2,112 @@ % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Lösung der partiellen Differentialgleichung - \label{kreismembran:section:teil2}} -\rhead{Lösung der partiellen Differentialgleichung} +\section{Die Hankel Transformation \label{kreismembran:section:teil2}} +\rhead{Die Hankel Transformation} + +Hermann Hankel (1839-1873) war ein deutscher Mathematiker, der für seinen Beitrag zur mathematischen Analyse und insbesondere für seine namensgebende Transformation bekannt ist. +Diese Transformation tritt bei der Untersuchung von funktionen auf, die nur von der Enternung des Ursprungs abhängen. +Er studierte auch funktionen, jetzt Hankel- oder Bessel- Funktionen genannt, der dritten Art. +Die Hankel Transformation mit Bessel Funktionen al Kern taucht natürlich bei achsensymmetrischen Problemen auf, die in Zylindrischen Polarkoordinaten formuliert sind. +In diesem Kapitel werden die Theorie der Transformation und einige Eigenschaften der Grundoperationen erläutert. + + +Wir führen die Definition der Hankel Transformation aus der zweidimensionalen Fourier Transformation und ihrer Umkehrung ein, die durch: +\begin{align} + \mathscr{F}\{f(x,y)\} & = F(k,l)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-i( \bm{\kappa}\cdot \mathbf{r})}f(x,y) dx dy,\label{equation:fourier_transform}\\ + \mathscr{F}^{-1}\{F(x,y)\} & = f(x,y)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{i(\bm{\kappa}\cdot \mathbf{r}))}F(k,l) dx dy \label{equation:inv_fourier_transform} +\end{align} +wo $\mathbf{r}=(x,y)$ und $\bm{\kappa}=(k,l)$. Wie bereits erwähnt, sind Polarkoordinaten für diese Art von Problemen am besten geeignet, also mit, $(x,y)=r(\cos\theta,\sin\theta)$ und $(k,l)=\kappa(\cos\phi,\sin\phi)$, findet man $\bm{\kappa}\cdot\mathbf{r}=\kappa r(\cos(\theta-\phi))$ und danach: +\begin{align} + F(k,\phi)=\frac{1}{2\pi}\int_{0}^{\infty}r dr \int_{0}^{2\pi}e^{-ikr\cos(\theta-\phi)}f(r,\theta) d\phi. + \label{equation:F_ohne_variable_wechsel} +\end{align} +Dann wird angenommen dass, $f(r,\theta)=e^{in\theta}f(r)$, was keine strenge Einschränkung ist, und es wird eine Änderung der Variabeln vorgenommen $\theta-\phi=\alpha-\frac{\pi}{2}$, um \eqref{equation:F_ohne_variable_wechsel} zu reduzieren: +\begin{align} + F(k,\phi)=\frac{1}{2\pi}\int_{0}^{\infty}rf(r) dr \int_{\phi_{0}}^{2\pi+\phi_{0}}e^{in(\phi-\frac{\pi}{2})+i(n\alpha-kr\sin\alpha)} d\alpha, + \label{equation:F_ohne_bessel} +\end{align} +wo $\phi_{0}=(\frac{\pi}{2}-\phi)$. + +Unter Verwendung der Integral Darstellung der Besselfunktion vom Ordnung n +\begin{align} + J_n(\kappa r)=\frac{1}{2\pi}\int_{\phi_{0}}^{2\pi + \phi_{0}}e^{i(n\alpha-\kappa r \sin \alpha)} d\alpha + \label{equation:bessel_n_ordnung} +\end{align} +\eqref{equation:F_ohne_bessel} wird sie zu: +\begin{align} + F(k,\phi)&=e^{in(\phi-\frac{\pi}{2})}\int_{0}^{\infty}rJ_n(\kappa r) f(r) dr \label{equation:F_mit_bessel_step_1} \\ + &=e^{in(\phi-\frac{\pi}{2})}\tilde{f}_n(\kappa), + \label{equation:F_mit_bessel_step_2} +\end{align} +wo $\tilde{f}_n(\kappa)$ ist die \textit{Hankel Transformation} von $f(r)$ und ist formell definiert durch: +\begin{align} + \mathscr{H}_n\{f(r)\}=\tilde{f}_n(\kappa)=\int_{0}^{\infty}rJ_n(\kappa r) f(r) dr. + \label{equation:hankel} +\end{align} + +Ähnlich verhält es sich mit der inversen Fourier Transformation in Form von polaren Koordinaten unter der Annahme $f(r,\theta)=e^{in\theta}f(r)$ mit \eqref{equation:F_mit_bessel_step_2}, wird die inverse Fourier Transformation \eqref{equation:inv_fourier_transform}: + +\begin{align} + e^{in\theta}f(r)&=\frac{1}{2\pi}\int_{0}^{\infty}\kappa d\kappa \int_{0}^{2\pi}e^{i\kappa r \cos (\theta - \phi)}F(\kappa,\phi) d\phi\\ + &= \frac{1}{2\pi}\int_{0}^{\infty}\kappa \tilde{f}_n(\kappa) d\kappa \int_{0}^{2\pi}e^{in(\phi - \frac{\pi}{2})- i\kappa r \cos (\theta - \phi)} d\phi, +\end{align} +was durch den Wechsel der Variablen $\theta-\phi=-(\alpha+\frac{\pi}{2})$ und $\theta_0=-(\theta+\frac{\pi}{2})$, + +\begin{align} + &= \frac{1}{2\pi}\int_{0}^{\infty}\kappa \tilde{f}_n(\kappa) d\kappa \int_{\theta_0}^{2\pi+\theta_0}e^{in(\theta + \alpha - i\kappa r \sin\alpha)} d\alpha \nonumber \\ + &= e^{in\theta}\int_{0}^{\infty}\kappa J_n(\kappa r) \tilde{f}_n(\kappa) d\kappa,\quad \text{von \eqref{equation:bessel_n_ordnung}} +\end{align} + +Also, die inverse \textit{Hankel Transformation} ist so definiert: +\begin{align} + \mathscr{H}^{-1}_n\{\tilde{f}_n(\kappa)\}=f(r)=\int_{0}^{\infty}\kappa J_n(\kappa r) \tilde{f}_n(\kappa) d\kappa. + \label{equation:inv_hankel} +\end{align} + +Anstelle von $\tilde{f}_n(\kappa)$, wird häufig für die Hankel Transformation verwendet, indem die Ordnung angegeben wird. +\eqref{equation:hankel} und \eqref{equation:inv_hankel} Integralen existieren für eine grosse Klasse von Funktionen, die normalerweise in physikalischen Anwendungen benötigt werden. +Alternativ kann auch die berühmte Hankel Transformationsformel verwendet werden, + +\begin{align} + f(r) = \int_{0}^{\infty}\kappa J_n(\kappa r) d\kappa \int_{0}^{\infty} p J_n(\kappa p)f(p) dp, + \label{equation:hankel_integral_formula} +\end{align} +um die Hankel Transformation \eqref{equation:hankel} und ihre Inverse \eqref{equation:inv_hankel} zu definieren. +Insbesondere die Hankel Transformation der nullten Ordnung ($n=0$) und der ersten Ordnung ($n=1$) sind häufig nützlich, um Lösungen für Probleme mit der Laplace Gleichung in einer achsensymmetrischen zylindrischen Geometrie zu finden. + +\subsection{Operative Eigenschaften der Hankel Transformation\label{sub:op_properties_hankel}} +In diesem Kapitel werden die operativen Eigenschaften der Hankel Transformation aufgeführt. Der Beweis für ihre Gültigkeit wird jedoch nicht analysiert. + +\subsubsection{Theorem 1: Skalierung \label{subsub:skalierung}} +Wenn $\mathscr{H}_n\{f(r)\}=\tilde{f}_n(\kappa)$, dann: -Wie im vorherigen Kapitel gezeigt, lautet die partielle Differentialgleichung, die die Schwingungen einer Membran beschreibt: \begin{equation*} - \frac{1}{c^2}\frac{\partial^2u}{\partial t^2} = \Delta u + \mathscr{H}_n\{f(ar)\}=\frac{1}{a^{2}}\tilde{f}_n \left(\frac{\kappa}{a}\right), \quad a>0. \end{equation*} -Da es sich um eine Kreisscheibe handelt, werden Polarkoordinaten verwendet, so dass sich der Laplaceoperator ergibt: + +\subsubsection{Theorem 2: Persevalsche Relation \label{subsub:perseval}} +Wenn $\tilde{f}(\kappa)=\mathscr{H}_n\{f(r)\}$ und $\tilde{g}(\kappa)=\mathscr{H}_n\{g(r)\}$, dann: + \begin{equation*} - \Delta - = - \frac{\partial^2}{\partial r^2} - + - \frac1r - \frac{\partial}{\partial r} - + - \frac{1}{r 2} - \frac{\partial^2}{\partial\varphi^2}. - \label{buch:pde:kreis:laplace} + \int_{0}^{\infty}rf(r) dr = \int_{0}^{\infty}\kappa\tilde{f}(\kappa)\tilde{g}(\kappa) d\kappa. \end{equation*} -Es wird eine runde elastische Membran berücksichtigt, die den Gebietbereich $\Omega$ abdeckt und am Rand $\Gamma$ befestigt ist. -Es wird daher davon ausgegangen, dass die Membran aus einem homogenen Material von vernachlässigbarer Dicke gefertigt ist. -Die Membran kann verformt werden, aber innere elastische Kräfte wirken den Verformungen entgegen. Es wirken keine äusseren Kräfte. Es handelt sich somit von einer kreisförmligen eigespannten homogenen schwingenden Membran. +\subsubsection{Theorem 3: Hankel Transformationen von Ableitungen \label{subsub:ableitungen}} +Wenn $\tilde{f}_n(\kappa)=\mathscr{H}_n\{f(r)\}$, dann: -Daher ist die Membranabweichung im Punkt $(r,\theta)$ $\in$ $\overline{\rm \Omega}$ zum Zeitpunkt $t$: \begin{align*} - u: \overline{\rm \Omega} \times \mathbb{R}_{\geq 0} &\longrightarrow \mathbb{R}\\ - (r,\theta,t) &\longmapsto u(r,\theta,t) + &\mathscr{H}_n\{f'(r)\}=\frac{\kappa}{2n}\left[(n-1)\tilde{f}_{n+1}(\kappa)-(n+1)\tilde{f}_{n-1}(\kappa)\right], \quad n\geq1, \\ + &\mathscr{H}_1\{f'(r)\}=-\kappa \tilde{f}_0(\kappa), \end{align*} -Da die Membran am Rand befestigt ist, kann es keine Schwingungen geben, so dass die \textit{Dirichlet-Randbedingung} gilt: -\begin{equation*} - u\big|_{\Gamma} = 0 -\end{equation*} +bereitgestellt dass $[rf(r)]$ verschwindet als $r\to0$ und $r\to\infty$. +\subsubsection{Theorem 4 \label{subsub:thorem4}} +Wenn $\mathscr{H}_n\{f(r)\}=\tilde{f}_n(\kappa)$, dann: -Um eine eindeutige Lösung bestimmen zu können, werden die folgenden Anfangsbedingungen festgelegt: +\begin{equation*} + \mathscr{H}_n \left\{ \left( \nabla^2 - \frac{n^2}{r^2} f(r)\right)\right\}= \mathscr{H}_n\left\{\frac{1}{r}\frac{d}{dr}\left(r\frac{df}{dr}\right) - \frac{n^2}{r^2}f(r)\right\}=-\kappa^2\tilde{f}_{n}(\kappa), +\end{equation*} +bereitgestellt dass $rf'(r)$ und $rf(r)$ verschwinden als $r\to0$ und $r\to\infty$. -\begin{align*} - u(r,\theta, 0) &:= f(x,y)\\ - \frac{\partial}{\partial t} u(r,\theta, 0) &:= g(x,y) -\end{align*} -An dieser Stelle könnte man zum Beispiel die bereits in Kapitel (TODO:refKAPITEL) vorgestellte Methode der Separation anwenden. Da es sich in diesem Fall jedoch um einem achsensymmetrischen Problem handelt, das in Polarkoordinaten formuliert ist, wird man die Transformationsmethode verwenden, insbesondere die Hankel Transformation. diff --git a/buch/papers/kreismembran/teil3.tex b/buch/papers/kreismembran/teil3.tex index 73dee0f..bef8b5f 100644 --- a/buch/papers/kreismembran/teil3.tex +++ b/buch/papers/kreismembran/teil3.tex @@ -3,38 +3,76 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 3 +\section{Lösungsmethode 2: Transformationsmethode \label{kreismembran:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{kreismembran:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +\rhead{Lösungsmethode 2: Transformationsmethode} +Die Hankel-Transformation wird dann zur Lösung der Differentialgleichung verwendet. Es müssen jedoch einige Änderungen an dem Problem vorgenommen werden, damit es mit den Annahmen übereinstimmt, die für die Verwendung der Hankel-Transformation erforderlich sind. Das heisst, dass die Funktion u nur von der Entfernung zum Ausgangspunkt abhängt. Wir führen also das Konzept einer unendlichen und achsensymmetrischen Membran ein: +\begin{equation*} + \frac{\partial^2u}{\partial t^2} + = + c^2 \left(\frac{\partial^2 u}{\partial r^2} + + + \frac{1}{r} + \frac{\partial u}{\partial r} \right), \quad 00 + \label{eq:PDE_inf_membane} +\end{equation*} + +\begin{align} + u(r,0)=f(r), \quad \frac{\partial}{\partial t} u(r,0) = g(r), \quad \text{für} \quad 0 Date: Wed, 18 May 2022 14:20:41 +0200 Subject: =?UTF-8?q?Dreiecke=20f=C3=BCr=20Nav?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../050-differential/uebungsaufgaben/airy.cpp | 4 +- buch/chapters/110-elliptisch/ellintegral.tex | 2 +- buch/papers/nav/images/Makefile | 10 ++- buch/papers/nav/images/common.inc | 28 +++++- buch/papers/nav/images/dreieck3d1.pdf | Bin 0 -> 90451 bytes buch/papers/nav/images/dreieck3d1.pov | 12 +-- buch/papers/nav/images/dreieck3d2.pdf | Bin 0 -> 69523 bytes buch/papers/nav/images/dreieck3d2.pov | 6 +- buch/papers/nav/images/dreieck3d3.pdf | Bin 0 -> 82512 bytes buch/papers/nav/images/dreieck3d3.pov | 8 +- buch/papers/nav/images/dreieck3d4.pdf | Bin 0 -> 85037 bytes buch/papers/nav/images/dreieck3d4.pov | 8 +- buch/papers/nav/images/dreieck3d5.pdf | Bin 0 -> 70054 bytes buch/papers/nav/images/dreieck3d5.pov | 6 +- buch/papers/nav/images/dreieck3d6.pov | 2 +- buch/papers/nav/images/dreieck3d7.pov | 10 +-- buch/papers/nav/images/dreieck3d8.jpg | Bin 0 -> 93432 bytes buch/papers/nav/images/dreieck3d8.pdf | Bin 0 -> 107370 bytes buch/papers/nav/images/dreieck3d8.pov | 96 +++++++++++++++++++++ buch/papers/nav/images/dreieck3d8.tex | 57 ++++++++++++ 20 files changed, 216 insertions(+), 33 deletions(-) create mode 100644 buch/papers/nav/images/dreieck3d1.pdf create mode 100644 buch/papers/nav/images/dreieck3d2.pdf create mode 100644 buch/papers/nav/images/dreieck3d3.pdf create mode 100644 buch/papers/nav/images/dreieck3d4.pdf create mode 100644 buch/papers/nav/images/dreieck3d5.pdf create mode 100644 buch/papers/nav/images/dreieck3d8.jpg create mode 100644 buch/papers/nav/images/dreieck3d8.pdf create mode 100644 buch/papers/nav/images/dreieck3d8.pov create mode 100644 buch/papers/nav/images/dreieck3d8.tex (limited to 'buch') diff --git a/buch/chapters/050-differential/uebungsaufgaben/airy.cpp b/buch/chapters/050-differential/uebungsaufgaben/airy.cpp index e4df8e1..eb5c6be 100644 --- a/buch/chapters/050-differential/uebungsaufgaben/airy.cpp +++ b/buch/chapters/050-differential/uebungsaufgaben/airy.cpp @@ -44,8 +44,8 @@ double h0f1(double c, double x) { double f1(double x) { // unfortunately, gsl_sf_hyperg_0F1 does not work if c<1, because // it uses a relation to the bessel functions - //return gsl_sf_hyperg_0F1(2/3, x*x*x/9.); - return h0f1(2./3., x*x*x/9.); + return gsl_sf_hyperg_0F1(2/3, x*x*x/9.); + //return h0f1(2./3., x*x*x/9.); } double f2(double x) { diff --git a/buch/chapters/110-elliptisch/ellintegral.tex b/buch/chapters/110-elliptisch/ellintegral.tex index 4cb2ba3..3acce2f 100644 --- a/buch/chapters/110-elliptisch/ellintegral.tex +++ b/buch/chapters/110-elliptisch/ellintegral.tex @@ -651,7 +651,7 @@ werden, dass $1-k'^2=k^2$ ist. \begin{definition} Ist $0\le k\le 1$ der Modul eines elliptischen Integrals, dann heisst -$k' = \sqrt{1-k^2}$ er {\em Komplementärmodul} oder {\em Komplement +$k' = \sqrt{1-k^2}$ der {\em Komplementärmodul} oder {\em Komplement des Moduls}. Es ist $k^2+k'^2=1$. \end{definition} diff --git a/buch/papers/nav/images/Makefile b/buch/papers/nav/images/Makefile index c9dcacc..bbdea2f 100644 --- a/buch/papers/nav/images/Makefile +++ b/buch/papers/nav/images/Makefile @@ -50,7 +50,8 @@ DREIECKE3D = \ dreieck3d4.pdf \ dreieck3d5.pdf \ dreieck3d6.pdf \ - dreieck3d7.pdf + dreieck3d7.pdf \ + dreieck3d8.pdf dreiecke3d: $(DREIECKE3D) @@ -106,3 +107,10 @@ dreieck3d7.jpg: dreieck3d7.png dreieck3d7.pdf: dreieck3d7.tex dreieck3d7.jpg pdflatex dreieck3d7.tex +dreieck3d8.png: dreieck3d8.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d8.png dreieck3d8.pov +dreieck3d8.jpg: dreieck3d8.png + convert dreieck3d8.png -density 300 -units PixelsPerInch dreieck3d8.jpg +dreieck3d8.pdf: dreieck3d8.tex dreieck3d8.jpg + pdflatex dreieck3d8.tex + diff --git a/buch/papers/nav/images/common.inc b/buch/papers/nav/images/common.inc index 33d9384..e2a1ed0 100644 --- a/buch/papers/nav/images/common.inc +++ b/buch/papers/nav/images/common.inc @@ -97,13 +97,13 @@ union { } #end -#macro winkel(w, p, q, staerke) +#macro winkel(w, p, q, staerke, r) #declare n = vnormalize(w); #declare pp = vnormalize(p - vdot(n, p) * n); #declare qq = vnormalize(q - vdot(n, q) * n); intersection { sphere { <0, 0, 0>, 1 + staerke } - cone { <0, 0, 0>, 0, 1.2 * vnormalize(w), 0.4 } + cone { <0, 0, 0>, 0, 1.2 * vnormalize(w), r } plane { -vcross(n, qq) * vdot(vcross(n, qq), pp), 0 } plane { -vcross(n, pp) * vdot(vcross(n, pp), qq), 0 } } @@ -113,8 +113,30 @@ union { sphere { p, 1.5 * staerke } #end +#macro dreieck(p, q, r, farbe) + #declare n1 = vnormalize(vcross(p, q)); + #declare n2 = vnormalize(vcross(q, r)); + #declare n3 = vnormalize(vcross(r, p)); + intersection { + plane { n1, 0 } + plane { n2, 0 } + plane { n3, 0 } + sphere { <0, 0, 0>, 1 + 0.001 } + pigment { + color farbe + } + finish { + metallic + specular 0.4 + } + } +#end + #declare fett = 0.015; -#declare fine = 0.010; +#declare fein = 0.010; + +#declare klein = 0.3; +#declare gross = 0.4; #declare dreieckfarbe = rgb<0.6,0.6,0.6>; #declare rot = rgb<0.8,0.2,0.2>; diff --git a/buch/papers/nav/images/dreieck3d1.pdf b/buch/papers/nav/images/dreieck3d1.pdf new file mode 100644 index 0000000..015bce7 Binary files /dev/null and b/buch/papers/nav/images/dreieck3d1.pdf differ diff --git a/buch/papers/nav/images/dreieck3d1.pov b/buch/papers/nav/images/dreieck3d1.pov index 8afe60e..e491075 100644 --- a/buch/papers/nav/images/dreieck3d1.pov +++ b/buch/papers/nav/images/dreieck3d1.pov @@ -12,9 +12,9 @@ union { punkt(A, fett) punkt(B, fett) punkt(C, fett) - punkt(P, fine) - seite(B, P, fine) - seite(C, P, fine) + punkt(P, fein) + seite(B, P, fein) + seite(C, P, fein) pigment { color dreieckfarbe } @@ -25,7 +25,7 @@ union { } object { - winkel(A, B, C, fine) + winkel(A, B, C, fein, gross) pigment { color rot } @@ -36,7 +36,7 @@ object { } object { - winkel(B, C, A, fine) + winkel(B, C, A, fein, gross) pigment { color gruen } @@ -47,7 +47,7 @@ object { } object { - winkel(C, A, B, fine) + winkel(C, A, B, fein, gross) pigment { color blau } diff --git a/buch/papers/nav/images/dreieck3d2.pdf b/buch/papers/nav/images/dreieck3d2.pdf new file mode 100644 index 0000000..6b3f09d Binary files /dev/null and b/buch/papers/nav/images/dreieck3d2.pdf differ diff --git a/buch/papers/nav/images/dreieck3d2.pov b/buch/papers/nav/images/dreieck3d2.pov index c23a54c..c0625ce 100644 --- a/buch/papers/nav/images/dreieck3d2.pov +++ b/buch/papers/nav/images/dreieck3d2.pov @@ -12,9 +12,9 @@ union { punkt(A, fett) punkt(B, fett) punkt(C, fett) - punkt(P, fine) - seite(B, P, fine) - seite(C, P, fine) + punkt(P, fein) + seite(B, P, fein) + seite(C, P, fein) pigment { color dreieckfarbe } diff --git a/buch/papers/nav/images/dreieck3d3.pdf b/buch/papers/nav/images/dreieck3d3.pdf new file mode 100644 index 0000000..7d79455 Binary files /dev/null and b/buch/papers/nav/images/dreieck3d3.pdf differ diff --git a/buch/papers/nav/images/dreieck3d3.pov b/buch/papers/nav/images/dreieck3d3.pov index f2496b5..b6f64d5 100644 --- a/buch/papers/nav/images/dreieck3d3.pov +++ b/buch/papers/nav/images/dreieck3d3.pov @@ -12,9 +12,9 @@ union { punkt(A, fett) punkt(B, fett) punkt(C, fett) - punkt(P, fine) - seite(B, P, fine) - seite(C, P, fine) + punkt(P, fein) + seite(B, P, fein) + seite(C, P, fein) pigment { color dreieckfarbe } @@ -25,7 +25,7 @@ union { } object { - winkel(A, B, C, fine) + winkel(A, B, C, fein, gross) pigment { color rot } diff --git a/buch/papers/nav/images/dreieck3d4.pdf b/buch/papers/nav/images/dreieck3d4.pdf new file mode 100644 index 0000000..e1ea757 Binary files /dev/null and b/buch/papers/nav/images/dreieck3d4.pdf differ diff --git a/buch/papers/nav/images/dreieck3d4.pov b/buch/papers/nav/images/dreieck3d4.pov index bddcf7c..b6f17e3 100644 --- a/buch/papers/nav/images/dreieck3d4.pov +++ b/buch/papers/nav/images/dreieck3d4.pov @@ -6,9 +6,9 @@ #include "common.inc" union { - seite(A, B, fine) - seite(A, C, fine) - punkt(A, fine) + seite(A, B, fein) + seite(A, C, fein) + punkt(A, fein) punkt(B, fett) punkt(C, fett) punkt(P, fett) @@ -25,7 +25,7 @@ union { } object { - winkel(B, C, P, fine) + winkel(B, C, P, fein, gross) pigment { color rgb<0.6,0.4,0.2> } diff --git a/buch/papers/nav/images/dreieck3d5.pdf b/buch/papers/nav/images/dreieck3d5.pdf new file mode 100644 index 0000000..6848331 Binary files /dev/null and b/buch/papers/nav/images/dreieck3d5.pdf differ diff --git a/buch/papers/nav/images/dreieck3d5.pov b/buch/papers/nav/images/dreieck3d5.pov index 32fc9e6..188f181 100644 --- a/buch/papers/nav/images/dreieck3d5.pov +++ b/buch/papers/nav/images/dreieck3d5.pov @@ -6,9 +6,9 @@ #include "common.inc" union { - seite(A, B, fine) - seite(A, C, fine) - punkt(A, fine) + seite(A, B, fein) + seite(A, C, fein) + punkt(A, fein) punkt(B, fett) punkt(C, fett) punkt(P, fett) diff --git a/buch/papers/nav/images/dreieck3d6.pov b/buch/papers/nav/images/dreieck3d6.pov index 7611950..191a1e7 100644 --- a/buch/papers/nav/images/dreieck3d6.pov +++ b/buch/papers/nav/images/dreieck3d6.pov @@ -25,7 +25,7 @@ union { } object { - winkel(B, A, P, fine) + winkel(B, A, P, fein, gross) pigment { color rgb<0.6,0.2,0.6> } diff --git a/buch/papers/nav/images/dreieck3d7.pov b/buch/papers/nav/images/dreieck3d7.pov index fa48f5b..aae5c6c 100644 --- a/buch/papers/nav/images/dreieck3d7.pov +++ b/buch/papers/nav/images/dreieck3d7.pov @@ -10,13 +10,13 @@ union { seite(A, P, fett) seite(C, P, fett) - seite(A, B, fine) - seite(B, C, fine) - seite(B, P, fine) + seite(A, B, fein) + seite(B, C, fein) + seite(B, P, fein) punkt(A, fett) punkt(C, fett) punkt(P, fett) - punkt(B, fine) + punkt(B, fein) pigment { color dreieckfarbe } @@ -27,7 +27,7 @@ union { } object { - winkel(A, P, C, fine) + winkel(A, P, C, fein, gross) pigment { color rgb<0.4,0.4,1> } diff --git a/buch/papers/nav/images/dreieck3d8.jpg b/buch/papers/nav/images/dreieck3d8.jpg new file mode 100644 index 0000000..52bd25e Binary files /dev/null and b/buch/papers/nav/images/dreieck3d8.jpg differ diff --git a/buch/papers/nav/images/dreieck3d8.pdf b/buch/papers/nav/images/dreieck3d8.pdf new file mode 100644 index 0000000..9d630aa Binary files /dev/null and b/buch/papers/nav/images/dreieck3d8.pdf differ diff --git a/buch/papers/nav/images/dreieck3d8.pov b/buch/papers/nav/images/dreieck3d8.pov new file mode 100644 index 0000000..9e9921a --- /dev/null +++ b/buch/papers/nav/images/dreieck3d8.pov @@ -0,0 +1,96 @@ +// +// dreiecke3d8.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +union { + seite(A, B, fett) + seite(B, C, fett) + seite(A, C, fett) + seite(A, P, fein) + seite(B, P, fett) + seite(C, P, fett) + punkt(A, fett) + punkt(B, fett) + punkt(C, fett) + punkt(P, fett) + pigment { + color dreieckfarbe + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, B, C, fein, klein) + pigment { + color rot + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, C, A, fein, klein) + pigment { + color gruen + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(C, A, B, fein, gross) + pigment { + color blau + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(A, P, C, fein/2, gross) + pigment { + color rgb<0.8,0,0.8> + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, P, C, fein, klein) + pigment { + color rgb<1,0.8,0> + } + finish { + specular 0.95 + metallic + } +} + +object { + winkel(B, P, A, fein/2, gross) + pigment { + color rgb<0.4,0.6,0.8> + } + finish { + specular 0.95 + metallic + } +} + +dreieck(A, B, C, White) + + diff --git a/buch/papers/nav/images/dreieck3d8.tex b/buch/papers/nav/images/dreieck3d8.tex new file mode 100644 index 0000000..c59c7b0 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d8.tex @@ -0,0 +1,57 @@ +% +% dreieck3d8.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=8cm]{dreieck3d8.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (0.7,3.8) {$A$}; +\node at (-3.4,-0.8) {$B$}; +\node at (3.3,-2.1) {$C$}; +\node at (-1.4,-3.5) {$P$}; + +\node at (-1.9,2.1) {$c$}; +\node at (-0.2,-1.2) {$a$}; +\node at (2.6,1.5) {$b$}; +\node at (-0.8,0) {$l$}; + +\node at (-2.6,-2.2) {$p_b$}; +\node at (1,-2.9) {$p_c$}; + +\node at (0.7,3.3) {$\alpha$}; +\node at (0.8,2.85) {$\omega$}; +\node at (-2.6,-0.6) {$\beta$}; +\node at (2.3,-1.2) {$\gamma$}; +\node at (-2.6,-1.3) {$\beta_1$}; +\node at (-2.1,-0.8) {$\kappa$}; + +\end{tikzpicture} + +\end{document} + -- cgit v1.2.1 From 525ff82400b685dc6dd0d6376253545720471be0 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 18 May 2022 14:25:26 +0200 Subject: remove bad files --- buch/buch.aux | 22 - buch/buch.bbl | 0 buch/buch.blg | 48 - buch/buch.idx | 0 buch/buch.log | 2106 --------------------------------- buch/papers/nav/images/dreieck3d5.pdf | Bin 70054 -> 70045 bytes 6 files changed, 2176 deletions(-) delete mode 100644 buch/buch.aux delete mode 100644 buch/buch.bbl delete mode 100644 buch/buch.blg delete mode 100644 buch/buch.idx delete mode 100644 buch/buch.log (limited to 'buch') diff --git a/buch/buch.aux b/buch/buch.aux deleted file mode 100644 index 6730af9..0000000 --- a/buch/buch.aux +++ /dev/null @@ -1,22 +0,0 @@ -\relax -\providecommand\hyper@newdestlabel[2]{} -\providecommand\babel@aux[2]{} -\@nameuse{bbl@beforestart} -\catcode `"\active -\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} -\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined -\global\let\oldcontentsline\contentsline -\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} -\global\let\oldnewlabel\newlabel -\gdef\newlabel#1#2{\newlabelxx{#1}#2} -\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} -\AtEndDocument{\ifx\hyper@anchor\@undefined -\let\contentsline\oldcontentsline -\let\newlabel\oldnewlabel -\fi} -\fi} -\global\let\hyper@last\relax -\gdef\HyperFirstAtBeginDocument#1{#1} -\providecommand\HyField@AuxAddToFields[1]{} -\providecommand\HyField@AuxAddToCoFields[2]{} -\providecommand\BKM@entry[2]{} diff --git a/buch/buch.bbl b/buch/buch.bbl deleted file mode 100644 index e69de29..0000000 diff --git a/buch/buch.blg b/buch/buch.blg deleted file mode 100644 index 706b1d8..0000000 --- a/buch/buch.blg +++ /dev/null @@ -1,48 +0,0 @@ -This is BibTeX, Version 0.99d -Capacity: max_strings=200000, hash_size=200000, hash_prime=170003 -The top-level auxiliary file: buch.aux -I found no \citation commands---while reading file buch.aux -I found no \bibdata command---while reading file buch.aux -I found no \bibstyle command---while reading file buch.aux -You've used 0 entries, - 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TeX capacity exceeded, sorry [main memory size=3000000]. - \XKV@resb - -l.575 } - -If you really absolutely need more capacity, -you can ask a wizard to enlarge me. - - -Here is how much of TeX's memory you used: - 81845 strings out of 478582 - 2061696 string characters out of 2856069 - 3000001 words of memory out of 3000000 - 98832 multiletter control sequences out of 15000+600000 - 484206 words of font info for 75 fonts, out of 8000000 for 9000 - 1143 hyphenation exceptions out of 8191 - 105i,3n,99p,3369b,2422s stack positions out of 10000i,1000n,20000p,200000b,80000s -! ==> Fatal error occurred, no output PDF file produced! diff --git a/buch/papers/nav/images/dreieck3d5.pdf b/buch/papers/nav/images/dreieck3d5.pdf index 6848331..0c86d36 100644 Binary files a/buch/papers/nav/images/dreieck3d5.pdf and b/buch/papers/nav/images/dreieck3d5.pdf differ -- cgit v1.2.1 From 93bdfca3b41397e43537ee334e57883a9ef79279 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 18 May 2022 14:30:12 +0200 Subject: fix nav/makefile.inc --- buch/papers/nav/Makefile.inc | 12 +++++++----- 1 file changed, 7 insertions(+), 5 deletions(-) (limited to 'buch') diff --git a/buch/papers/nav/Makefile.inc b/buch/papers/nav/Makefile.inc index b30377e..24ab4ee 100644 --- a/buch/papers/nav/Makefile.inc +++ b/buch/papers/nav/Makefile.inc @@ -6,9 +6,11 @@ dependencies-nav = \ papers/nav/packages.tex \ papers/nav/main.tex \ - papers/nav/references.bib \ - papers/nav/teil0.tex \ - papers/nav/teil1.tex \ - papers/nav/teil2.tex \ - papers/nav/teil3.tex + papers/nav/einleitung.tex \ + papers/nav/flatearth.tex \ + papers/nav/geschichte.tex \ + papers/nav/nautischesdreieck.tex \ + papers/nav/sincos.tex \ + papers/nav/trigo.tex \ + papers/nav/references.bib -- cgit v1.2.1 From 88cc6d9775114c70d8723a52a869179ce806d2f7 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 18 May 2022 14:43:23 +0200 Subject: fix paper offset --- buch/splitpapers | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'buch') diff --git a/buch/splitpapers b/buch/splitpapers index 9ae5aae..e1b6834 100755 --- a/buch/splitpapers +++ b/buch/splitpapers @@ -16,7 +16,7 @@ then fi awk 'BEGIN { - offsetpage = 10 + offsetpage = 12 startpage = 0 identifier = "" chapterno = 0 -- cgit v1.2.1 From b37f9519bbfd57b3a7d25cca887ff44ff2253921 Mon Sep 17 00:00:00 2001 From: "ENEZ-PC\\erdem" Date: Thu, 19 May 2022 14:40:25 +0200 Subject: Korrektur von Feedback --- buch/papers/nav/bilder/dreieck.pdf | Bin 0 -> 107370 bytes buch/papers/nav/bilder/ephe.png | Bin 0 -> 184799 bytes buch/papers/nav/einleitung.tex | 6 +- buch/papers/nav/flatearth.tex | 12 ++- buch/papers/nav/geschichte.tex | 22 ---- buch/papers/nav/nautischesdreieck.tex | 190 ++++++++++++++++------------------ buch/papers/nav/sincos.tex | 21 ++-- buch/papers/nav/trigo.tex | 57 +++++++--- 8 files changed, 158 insertions(+), 150 deletions(-) create mode 100644 buch/papers/nav/bilder/dreieck.pdf create mode 100644 buch/papers/nav/bilder/ephe.png delete mode 100644 buch/papers/nav/geschichte.tex (limited to 'buch') diff --git a/buch/papers/nav/bilder/dreieck.pdf b/buch/papers/nav/bilder/dreieck.pdf new file mode 100644 index 0000000..9d630aa Binary files /dev/null and b/buch/papers/nav/bilder/dreieck.pdf differ diff --git a/buch/papers/nav/bilder/ephe.png b/buch/papers/nav/bilder/ephe.png new file mode 100644 index 0000000..0aeef6f Binary files /dev/null and b/buch/papers/nav/bilder/ephe.png differ diff --git a/buch/papers/nav/einleitung.tex b/buch/papers/nav/einleitung.tex index e24f294..8d8c5c1 100644 --- a/buch/papers/nav/einleitung.tex +++ b/buch/papers/nav/einleitung.tex @@ -1,9 +1,9 @@ \section{Einleitung} -Heut zu Tage ist die Navigation ein Teil des Lebens. -Man versendet dem Kollegen seinen Standort, um sich das ewige Erklären zu sparen oder gibt die Adresse des Ziels ein um sich die Sucherei zu schenken. +Heutzutage ist die Navigation ein Teil des Lebens. +Man sendet dem Kollegen seinen eigenen Standort, um sich das ewige Erklären zu sparen oder gibt die Adresse des Ziels ein, damit man seinen Aufenthaltsort zum Beispiel auf einer riesigen Wiese am See findet. Dies wird durch Technologien wie Funknavigation, welches ein auf Langzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt. Vor all diesen technologischen Fortschritten gab es lediglich die Astronavigation, welche heute noch auf kleineren Schiffen benötigt wird im Falle eines Stromausfalls. Aber wie funktioniert die Navigation mit den Sternen? Welche Hilfsmittel benötigt man, welche Rolle spielt die Mathematik und weshalb kann die Erde nicht flach sein? -In diesem Kapitel werden genau diese Fragen mithilfe des Nautischen Dreiecks, der Sphärischen Trigonometrie und einigen Hilfsmitteln und Messgeräten beantwortet. \ No newline at end of file +In diesem Kapitel werden genau diese Fragen mithilfe des nautischen Dreiecks, der sphärischen Trigonometrie und einigen Hilfsmitteln und Messgeräten beantwortet. \ No newline at end of file diff --git a/buch/papers/nav/flatearth.tex b/buch/papers/nav/flatearth.tex index fbabbde..bec242e 100644 --- a/buch/papers/nav/flatearth.tex +++ b/buch/papers/nav/flatearth.tex @@ -2,7 +2,7 @@ \section{Warum ist die Erde nicht flach?} -\begin{figure}[h] +\begin{figure} \begin{center} \includegraphics[width=10cm]{papers/nav/bilder/projektion.png} \caption[Mercator Projektion]{Mercator Projektion} @@ -14,10 +14,14 @@ Die Fotos von unserem Planeten oder die Berichte der Astronauten. Aber schon vor ca. 2300 Jahren hat Aristotoles bemerkt, dass Schiffe im Horizont verschwinden und die einzige Erklärung dafür die Kugelgestalt der Erde ist oder der Erdschatten bei einer Mondfinsternis immer rund ist. Eratosthenes konnte etwa 100 Jahre später den Erdumfang berechnen. Er beobachtete, dass die Sonne in Syene mittags im Zenit steht und gleichzeitig in Alexandria unter einem Winkel einfällt. -Mithilfe der Geometrie konnte er mit dem Abstand der Städte und dem Einfallswinkel den Umfang berechnen. +Mithilfe der Trigonometrie konnte er mit dem Abstand der Städte und dem Einfallswinkel den Umfang berechnen. + Auch in der Navigation würden grobe Fehler passieren, wenn man davon ausgeht, dass die Erde eine Scheibe ist. -Man sieht es zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. Grönland ist auf der Weltkarte so gross wie Afrika. +Man sieht es zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. +Grönland ist auf der Weltkarte so gross wie Afrika. In der Anwendung Google Earth jedoch ist Grönland etwa so gross wie Algerien. Das liegt daran, das man die 3D – Weltkarte nicht einfach auslegen kann. -Der Kartograph Gerhard Mercator projizierte die Erdkugel auf ein Papier und erstellte so eine winkeltreue Karte. Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. + +Der Kartograph Gerhard Mercator projizierte die Erdkugel auf ein Papier und erstellte so eine winkeltreue Karte. +Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen. \ No newline at end of file diff --git a/buch/papers/nav/geschichte.tex b/buch/papers/nav/geschichte.tex deleted file mode 100644 index a20eb6d..0000000 --- a/buch/papers/nav/geschichte.tex +++ /dev/null @@ -1,22 +0,0 @@ -\documentclass[12pt]{scrartcl} -\usepackage{ucs} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{graphicx} - -\begin{document} -\section{Geschichte der sphärischen Navigation} -Die Orientierung mit Hilfe der Sterne und der sphärischen Trigonometrie bewegt die Menschheit schon seit mehreren tausend Jahren. -Nach Hinweisen und Schätzungen von Forscher haben schon vor 4000 Jahren die Ägypter und Gelehrten aus Babylon mit Hilfe der Astronomie den Lauf der Gestirne (Himmelskörper) zu berechnen versucht, jedoch ohne Erfolg. -Etwa 350 vor Christus waren es die Griechen, welche den damaligen Astronomen Hilfestellungen mittels Kugel-Geometrien leisten konnten. -Aus diesen Geometrien wurden erste mathematische Sätze aufgestellt und ein paar Jahrhunderte später kamen zu diesem Thema auch Berechnungen dazu. -Ebenso wurden Kartenmaterial mit Sternenbilder angefertigt. -Die Sinusfunktion war noch nicht bekannt, jedoch kamen zu dieser Zeit die ersten Ansätze der Cosinusfunktion aus Indien. -Von diesen Hilfen darauf aufbauend konnte um 900 die Araber der Sinussatz entwickeln. -Doch ein paar weitere Jahrhunderte vergingen bis zu diesem Thema wieder verstärkt Forschung betrieben wurde. -Dies aus dem Grund, da im 15. Jahrhundert grosse Entdeckungsreisen, hauptsächlich per Schiff, erfolgten und die Orientierung vermehrt an Wichtigkeit gewann. -Auch die Verwendung der Tangens- und Sinusfunktion sowie der neu entwickelte Seitencosinussatz trugen zu einer Verbesserung der Orientierung herbei. -Im 16. Jahrhundert wurde dann ein weiterer trigonometrischer Satz, der Winkelcosinussatz hergeleitet. Stück für Stück wurden infolge der Entdeckung des Logarithmus im 17. Jahrhundert viele neue Methoden entwickelt. -Auch eine Verbesserung der kartographischen Verwendung der Kugelgeometrie wurde vorgenommen. -Es folgten weitere Entwicklungen in nicht euklidische Geometrien und im 19. Jahrhundert sowie auch im 20. Jahrhundert wurde zudem für die Relativitätstheorie auch die sphärische Trigonometrie beigezogen. -\end{document} \ No newline at end of file diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex index b61e908..a85b119 100644 --- a/buch/papers/nav/nautischesdreieck.tex +++ b/buch/papers/nav/nautischesdreieck.tex @@ -1,17 +1,13 @@ \section{Das Nautische Dreieck} \subsection{Definition des Nautischen Dreiecks} -Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der \textbf{Himmelskugel}. -Die Himmelskugel ist eine gedachte Kugel, welche die Erde und dessen Beobachter umgibt und als Rechenfläche für Koordinaten in der Astronomie und Geodäsie dient.\\ -Das Nautische Dreieck definiert sich durch folgende Ecken: -\begin{itemize} - \item Zenit - \item Gestirn - \item Himmelspol -\end{itemize} +Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der Himmelskugel}. +Die Himmelskugel ist eine gedachte Kugel, welche die Erde und dessen Beobachter umgibt und als Rechenfläche für Koordinaten in der Astronomie und Geodäsie dient. +Das nautische Dreieck definiert sich durch folgende Ecken: Zenit, Gestirn und Himmelspol. + Der Zenit ist jener Punkt, der vom Erdmittelpunkt durch denn eigenen Standort an die Himmelskugel verlängert wird. Ein Gestirn ist ein Planet oder ein Fixstern, zu welchen es diverse Jahrbücher mit allen astronomischen Eigenschaften gibt. Der Himmelspol ist der Nordpol an die Himmelskugel projiziert. -\\ + Zur Anwendung der Formeln der sphärischen Trigonometrie gelten folgende einfache Zusammenhänge: \begin{itemize} \item Seitenlänge Zenit zu Himmelspol $= \frac{\pi}{2} - \phi $ @@ -21,34 +17,30 @@ Zur Anwendung der Formeln der sphärischen Trigonometrie gelten folgende einfach \item Winkel von Himmelsnordpol zu Zenit und Gestirn$= \tau$ \end{itemize} Um mit diesen Zusammenhängen zu rechnen benötigt man folgende Legende: - -$\alpha \ \widehat{=} \ Rektaszension $ - -$\delta \ \widehat{=} \ Deklination =$ Breitengrad des Gestirns - -$\theta \ \widehat{=} \ Sternzeit\ von\ Greenwich$ - -$\phi \ \widehat{=} \ Geographische \ Breite $ - -$\tau = \theta-\alpha \ \widehat{=} \ Stundenwinkel =$ Längengrad des Gestirns - -$a \ \widehat{=} \ Azimut $ - -$h \ \widehat{=} \ Hoehe$ - - - -\newpage -\subsection{Zusammenhang des Nautischen Dreiecks und des Kugeldreiecks auf der Erdkugel} -\begin{figure}[h] +\begin{center} + \begin{tabular}{ c c c } + Winkel && Name / Beziehung \\ + \hline + $\alpha$ && Rektaszension \\ + $\delta$ && Deklination \\ + $\theta$ && Sternzeit von Greenwich\\ + $\phi$ && Geographische Breite\\ + $\tau=\theta-\alpha$ && Stundenwinkel und Längengrad des Gestirns. \\ + $a$ && Azimut\\ + $h$ && Höhe + \end{tabular} +\end{center} + +\subsection{Zusammenhang des nautischen Dreiecks und des Kugeldreiecks auf der Erdkugel} +\begin{figure} \begin{center} - \includegraphics[height=5cm,width=5cm]{papers/nav/bilder/kugel3.png} + \includegraphics[height=5cm,width=8cm]{papers/nav/bilder/kugel3.png} \caption[Nautisches Dreieck]{Nautisches Dreieck} \end{center} \end{figure} -Wie man im oberen Bild sieht und auch am Anfang dieses Kapitels bereits erwähnt wurde, liegt das Nautische Dreieck auf der Himmelskugel mit den Ecken Zenit, Gestirn und Himmelsnordpol. -Das selbe Dreieck kann man aber auch auf die Erdkugel projezieren und hat dann die Ecken Standort, Bildpunkt und Nordpol. +Wie man im oberen Bild sieht, liegt das nautische Dreieck auf der Himmelskugel mit den Ecken Zenit, Gestirn und Himmelsnordpol. +Das selbe Dreieck kann man aber auch auf die Erdkugel projizieren und es hat dann die Ecken Standort, Bildpunkt und Nordpol. Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. @@ -56,9 +48,9 @@ Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem Um den eigenen Standort herauszufinden, wird in diesem Kapitel die Projektion Nautische Dreieck auf der Erdkugel zur Hilfe genommen. Mithilfe einiger Hilfsmittel und der Sphärischen Trigonometrie kann man dann die Längen- und Breitengrade des eigenen Standortes bestimmen. -\begin{figure}[h] +\begin{figure} \begin{center} - \includegraphics[width=6cm]{papers/nav/bilder/dreieck.png} + \includegraphics[width=10cm]{papers/nav/bilder/dreieck.pdf} \caption[Dreieck für die Standortbestimmung]{Dreieck für die Standortbestimmung} \end{center} \end{figure} @@ -66,75 +58,76 @@ Mithilfe einiger Hilfsmittel und der Sphärischen Trigonometrie kann man dann di -\subsection{Ecke P - Unser Standort} -Unser eigener Standort ist der gesuchte Punkt A. - -\subsection{Ecke A - Nordpol} -Der Vorteil ander Idee des Nautischen Dreiecks ist, dass eine Ecke immer der Nordpol (in der Himmelskugel der Himmelsnordpol) ist. +\subsection{Ecke $P$ und $A$} +Unser eigener Standort ist der gesuchte Ecke $P$ und die Ecke $A$ ist in unserem Fall der Nordpol. +Der Vorteil ander Idee des Nautischen Dreiecks ist, dass eine Ecke immer der Nordpol ist. Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so simpel. -\newpage -\subsection{Ecke B und C - Bildpunkt X und Y} + +\subsection{Ecke $B$ und $C$ - Bildpunkt X und Y} Für die Standortermittlung benötigt man als weiteren Punkt ein Gestirn bzw. seinen Bildpunkt auf der Erdkugel. Damit das trigonometrische Rechnen einfacher wird, werden hier zwei Gestirne zur Hilfe genommen. -\\ -Es gibt diverse Gestirne, die man nutzen kann. -\begin{itemize} - \item Sonne - \item Mond - \item Die vier Navigationsplaneten Venus, Mars, Jupiter und Saturn -\end{itemize} +Es gibt diverse Gestirne, die man nutzen kann wie zum Beispiel die Sonne, der Mond oder die vier Navigationsplaneten Venus, Mars, Jupiter und Saturn. + +\subsection{Ephemeriden} +Zu all diesen Gestirnen gibt es Ephemeriden, die man auch Jahrbücher nennt. +In diesen findet man Begriffe wie Rektaszension, Deklination und Sternzeit. +Da diese Angaben in Stundenabständen gegeben sind, muss man für die minutengenaue Bestimmung zwischen den Stunden interpolieren. +Was diese Begriffe bedeuten, wird in den kommenden beiden Abschnitten erklärt. -Zu all diesen Gestirnen gibt es Ephemeriden (Jahrbücher). -Dort findet man unter Anderem die Rektaszension und Deklination, welche für jeden Tag und Stunde beschrieben ist. Für Minuten genaue Angaben muss man dann zwischen den Stunden interpolieren. -Mithilfe dieser beiden Angaben kann man die Längen- und Breitengrade diverser Gestirne berechnen. +\begin{figure} + \begin{center} + \includegraphics[width=18cm]{papers/nav/bilder/ephe.png} + \caption[Astrodienst - Ephemeriden Januar 2022]{Astrodienst - Ephemeriden Januar 2022} + \end{center} +\end{figure} + +\subsubsection{Deklination} +Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und ergibt schlussendlich den Breitengrad. \subsubsection{Sternzeit und Rektaszension} -Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt steht. +Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt steht und geht vom Koordinatensystem der Himmelskugel aus. Der Frühlungspunkt ist der Nullpunkt auf dem Himmelsäquator. Die Tatsache, dass sich die Himmelskugel ca. vier Minuten schneller um die eigene Achse dreht als die Erdkugel, stellt hier ein kleines Problem dar. Die Lösung ist die Sternzeit. +Mit dieser können wir die schnellere Drehung der Himmelskugel ausgleichen und können die Am Frühlingspunkt (21. März) 12:00 Uhr ist die Sternzeit $\theta = 0$. Die Sternzeit geht vom Frühlungspunkt aus, an welchem die Sonne den Himmelsäquator schneidet. Für die Standortermittlung auf der Erdkugel ist es am einfachsten, wenn man die Sternzeit von Greenwich berechnet. -Für die Sternzeit von Greenwich braucht man als erstes das Julianische Datum $T$ vom aktuellen Tag, welches sich leicht recherchieren lässt. +Für die Sternzeit von Greenwich $\theta $braucht man als erstes das Julianische Datum $T$ vom aktuellen Tag, welches sich leicht recherchieren lässt. Im Anschluss berechnet man die Sternzeit von Greenwich -\\ -\\ -$T_{Greenwich} = 6^h 41^m 50^s,54841 + 8640184^s,812866 \cdot T + 0^s,093104 \cdot T^2 - 0^s,0000062 \cdot T^3$. -\\ -\\ -Wenn mann die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ mithilfe der Rektaszension und Sternzeit bestimmen. -Dies gilt analog auch für das zweite Gestirn. -\subsubsection{Deklination} -Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und ergibt schlussendlich den Breitengrad. +$\theta = 6^h 41^m 50^s,54841 + 8640184^s,812866 \cdot T + 0^s,093104 \cdot T^2 - 0^s,0000062 \cdot T^3$. +Wenn mann die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ mithilfe der Rektaszension und Sternzeit von Greenwich bestimmen. +Dies gilt analog auch für das zweite Gestirn. -\newpage \subsection{Bestimmung des eigenen Standortes P} Nun hat man die Koordinaten der beiden Gestirne und man weiss die Koordinaten des Nordpols. -Somit können wir ein erstes Kugeldreieck auf der Erde aufspannen. - +Damit wir unseren Standort bestimmen können, bilden wir zuerst das Dreieck $ABC$, dann das Dreieck $BPC$ und zum Schluss noch das Dreieck $ABP$. +Mithilfe dieser Dreiecken können wir die einfachen Sätze der sphärischen Trigonometrie anwenden und benötigen lediglich ein Ephemeride zu den Gestirnen und einen Sextant. -\begin{figure}[h] +\begin{figure} \begin{center} - \includegraphics[width=4.5cm]{papers/nav/bilder/dreieck.png} + \includegraphics[width=10cm]{papers/nav/bilder/dreieck.pdf} \caption[Dreieck für die Standortbestimmung]{Dreieck für die Standortbestimmung} \end{center} \end{figure} -\subsubsection{Bestimmung des ersten Dreiecks} - -$A=$ Nordpol +\subsubsection{Dreieck $ABC$} -$B=$ Bildpunkt des Gestirns X +\begin{center} + \begin{tabular}{ c c c } + Ecke && Name \\ + \hline + $A$ && Nordpol \\ + $B$ && Bildpunkt des Gestirns $X$ \\ + $C$&& Bildpunkt des Gestirns $Y$ + \end{tabular} +\end{center} -$C=$ Bildpunkt des Gestirns Y -\\ -\\ Mithilfe des sphärischen Trigonometrie und den darausfolgenden Zusammenhängen des Nautischen Dreiecks können wir nun alle Seiten des Dreiecks $ABC$ berechnen. Die Seitenlänge der Seite "Nordpol zum Bildpunkt X" sei $c$. @@ -145,24 +138,24 @@ Dann ist $b = \frac{\pi}{2} - \delta_2$. Der Innenwinkel beim der Ecke "Nordpol" sei $\alpha$. Dann ist $ \alpha = |\lambda_1 - \lambda_2|$. -\\ -\\ -mit - -$\delta_1 =$ Deklination Bildpunkt X -$\delta_2 =$ Deklination Bildpunk Y - -$\lambda_1 =$ Längengrad Bildpunkt X - -$\lambda_2 =$ Längengrad Bildpunkt Y +mit +\begin{center} + \begin{tabular}{ c c c } + Ecke && Name \\ + \hline + $\delta_1$ && Deklination Bildpunkt $X$ \\ + $\delta_2$ && Deklination Bildpunk $Y$ \\ + $\lambda_1 $&& Längengrad Bildpunkt $X$\\ + $\lambda_2$ && Längengrad Bildpunkt $Y$ + \end{tabular} +\end{center} Wichtig ist: Die Differenz der Längengrade ist gleich der Innenwinkel Alpha, deswegen der Betrag! -\\ -\\ + Nun haben wir die beiden Seiten $c\ und\ b$ und den Winkel $\alpha$, der sich zwischen diesen Seiten befindet. Mithilfe des Seiten-Kosinussatzes -$cos(a) = \cos(b)\cdot \cos(c) + \sin(b) \cdot \sin(c)\cdot \cos(\alpha)$ +$\cos(a) = \cos(b)\cdot \cos(c) + \sin(b) \cdot \sin(c)\cdot \cos(\alpha)$ können wir nun die dritte Seitenlänge bestimmen. Es ist darauf zu achten, dass hier natürlich die Seitenlängen in Bogenmass sind und dementsprechend der Kosinus und Sinus verwendet wird. @@ -174,7 +167,7 @@ Somit ist $\beta =\sin^{-1} [\sin(b) \cdot \frac{\sin(\alpha)}{\sin(a)}] $. Schlussendlich haben wir die Seiten $a,b\ und \ c$, die Ecken A,B und C und die Winkel $\alpha, \beta \ und \ \gamma$ bestimmt und somit das ganze erste Kugeldreieck berechnet. -\subsubsection{Bestimmung des zweiten Dreiecks} +\subsubsection{Dreieck $BPC$} Wir bilden nun ein zweites Dreieck, welches die Ecken B und C des ersten Dreiecks besitzt. Die dritte Ecke ist der eigene Standort P. Unser Standort definiere sich aus einer geographischen Breite $\delta$ und einer geographischen Länge $\lambda$. @@ -183,24 +176,23 @@ Die Seite von P zu B sei $pb$ und die Seite von P zu C sei $pc$. Die beiden Seitenlängen kann man mit dem Sextant messen und durch eine einfache Formel bestimmen, nämlich $pb=\frac{\pi}{2} - h_{B}$ und $pc=\frac{\pi}{2} - h_{C}$ mit $h_B=$ Höhe von Gestirn in B und $h_C=$ Höhe von Gestirn in C mit Sextant gemessen. -\\ +Zum Schluss müssen wir noch den Winkel $\beta1$ mithilfe des Seiten-Kosinussatzes mit den bekannten Seiten $pc$, $pb$ und $a$ bestimmen. +\subsubsection{Dreieck $ABP$} Nun muss man eine Verbindungslinie ziehen zwischen P und A. Die Länge $l$ dieser Linie entspricht der gesuchten geographischen Breite $\delta$. Diese lässt sich mithilfe des Dreiecks $ABP$, den bekannten Seiten $c\ und \ pb$ und des Seiten-Kosinussatzes berechnen. -Für den Seiten-Kosinussatz benötigt es noch $\kappa$. -Da wir aber $pc$, $pb$ und $a$ kennen, kann man mit dem Seiten-Kosinussatz den Winkel $\beta1$ berechnen und anschliessend $\beta + \beta1 =\kappa$. +Für den Seiten-Kosinussatz benötigt es noch $\kappa=\beta + \beta1$. -Somit ist $cos(l) = \cos(c)\cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)$ +Somit ist $\cos(l) = \cos(c)\cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)$ und -$\delta =\cos^{-1} [\cos(c) \cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)]$. -\\ +\[ +\delta =\cos^{-1} [\cos(c) \cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)]. +\] Für die Geographische Länge $\lambda$ des eigenen Standortes muss man den Winkel $\omega$, welcher sich im Dreieck $ACP$ in der Ecke bei $A$ befindet mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$ bestimmen. -\\ -Somit ist $\omega=\sin^{-1}[\sin(pc) \cdot \frac{\sin(\gamma)}{\sin(l)}]$ und unsere gesuchte geographische Länge schlussendlich -$\lambda=\lambda_1 - \omega$ mit $\lambda_1$=Längengrad Bildpunkt XXX. -\newpage -\listoffigures \ No newline at end of file +Somit ist \[ \omega=\sin^{-1}[\sin(pc) \cdot \frac{\sin(\gamma)}{\sin(l)}] \]und unsere gesuchte geographische Länge schlussendlich +\[\lambda=\lambda_1 - \omega\] +mit $\lambda_1$=Längengrad Bildpunkt XXX. diff --git a/buch/papers/nav/sincos.tex b/buch/papers/nav/sincos.tex index 23e3303..bb7f1e4 100644 --- a/buch/papers/nav/sincos.tex +++ b/buch/papers/nav/sincos.tex @@ -1,16 +1,19 @@ -\section{Warum sind die Sinus- und Kosinusfunktionen spezielle Funktionen?} -Es gibt Hinweise, dass sich schon die Babylonier und Ägypter vor 4000 Jahren sich mit Problemen der sphärischen Trigonometrie beschäftigt haben um den Lauf von Gestirnen (Himmelskörper) zu berechnen. -Jedoch konnten sie sie nicht lösen. -Die Geschichte der sphärischen Trigonometrie ist daher eng mit der Astronomie verknüpft. Ca. 350 vor Christus dachten die Griechen über Kugelgeometrie nach und wurde zu einer Hilfswissenschaft der Astronomen. -In Folge werden auch die ersten Sätze aufgestellt und wenige Jahrhunderte später wurden Berechnungen zu diesem Thema angestellt. +\section{Sphärische Navigation und Winkelfunktionen} +Es gibt Hinweise, dass sich schon die Babylonier und Ägypter vor 4000 Jahren sich mit Problemen der sphärischen Trigonometrie beschäftigt haben um den Lauf von Gestirnen zu berechnen. +Jedoch konnten sie dieses Problem nicht lösen. + +Die Geschichte der sphärischen Trigonometrie ist daher eng mit der Astronomie verknüpft. Ca. 350 vor Christus dachten die Griechen über Kugelgeometrie nach und sie wurde zu einer Hilfswissenschaft der Astronomen. +In Folge werden auch die ersten Sätze aufgestellt und wenige Jahrhunderte später wurden Berechnungen mithilfe des Sternkataloges von Hipparchos angestellt und darauffolgend Kartenmaterial erstellt. In dieser Zeit wurden auch die ersten Sternenkarten angefertigt, jedoch kannte man damals die Sinusfunktion noch nicht. Aus Indien stammten die ersten Ansätze zu den Kosinussätzen. -Aufbauend auf den indischen und griechischen Forschungen entwickeln die Araber um 900 den Sinussatz. -Zur Zeit der großen Entdeckungsreisen im 15. Jahrhundert wurden die Forschungen in sphärischer Trigonometrie wieder forciert. -Der Sinussatz, die Tangensfunktion und der neu entwickelte Seitenkosinussatz wurden in dieser Zeit bereits verwendet. -Im nächsten Jahrhundert folgte der Winkelkosinussatz. +Aufbauend auf den indischen und griechischen Forschungen entwickeln die Araber um das 9. Jahrhundert den Sinussatz. +Doch ein paar weitere Jahrhunderte vergingen bis zu diesem Thema wieder verstärkt Forschung betrieben wurde, da im 15. Jahrhundert grosse Entdeckungsreisen, hauptsächlich per Schiff, erfolgten und die Orientierung mit Sternen vermehrt an Wichtigkeit gewann. +Man nutzte für die Kartographie nun die Kugelgeometrie, um die Genauigkeit zu erhöhen. +Der Sinussatz, die Tangensfunktion und der neu entwickelte Seitenkosinussatz wurden in dieser Zeit bereits verwendet und im darauffolgenden Jahrhundert folgte der Winkelkosinussatz. + + Durch weitere mathematische Entwicklungen wie den Logarithmus wurden im Laufe des nächsten Jahrhunderts viele neue Methoden und kartographische Anwendungen der Kugelgeometrie entdeckt. Im 19. und 20. Jahrhundert wurden weitere nicht-euklidische Geometrien entwickelt und die sphärische Trigonometrie fand auch ihre Anwendung in der Relativitätstheorie. \ No newline at end of file diff --git a/buch/papers/nav/trigo.tex b/buch/papers/nav/trigo.tex index 8b4634f..cf2f242 100644 --- a/buch/papers/nav/trigo.tex +++ b/buch/papers/nav/trigo.tex @@ -1,18 +1,38 @@ -\setlength{\parindent}{0em} + \section{Sphärische Trigonometrie} +In der sphärischen Trigonometrie gibt es eine Symetrie zwischen Seiten und Winkel, also zu jedem Satz über Seiten und Winkel gibt es einen entsprechenden Satz, mit dem man Winkel durch Seiten und Seiten durch Winkel ersetzt hat. +Dabei gibt es folgenden Zusammenhang zwischen der ebenen- und sphärischen Trigonometrie: +\begin{center} + + +\begin{tabular}{ccc} + Eben & $\leftrightarrow$ & sphärisch \\ + \hline + $a$ & $\leftrightarrow$ & $\sin \ a$ \\ + + $a^2$ & $\leftrightarrow$ & $-\cos \ a$ \\ +\end{tabular} +\end{center} + \subsection{Das Kugeldreieck} -Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden, so entsteht ein Kugeldreieck ABC. -A, B und C sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten. +Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden, so entsteht ein Kugeldreieck $ABC$. +Für ein Kugeldreieck gilt, dass die Summe der drei Seiten kleiner als $2\pi$ aber grösser als 0 ist. +$A$, $B$ und $C$ sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten. +Ein Grosskreis ist ein größtmöglicher Kreis auf einer Kugeloberfläche. +Sein Mittelpunkt fällt immer mit dem Mittelpunkt der Kugel zusammen und ein Schnitt auf dem Großkreis teilt die Kugel in jedem Fall in zwei gleich grosse Hälften. + +Da es unendlich viele Möglichkeiten gibt, eine Kugel so zu zerschneiden, dass die Schnittebene den Kugelmittelpunkt trifft, gibt es auch unendlich viele Grosskreise. Da die Länge der Grosskreisbögen wegen der Abhängigkeit vom Kugelradius ungeeignet ist, wird die Grösse einer Seite mit dem zugehörigen Mittelpunktwinkel des Grosskreisbogens angegeben. -Laut dieser Definition ist die Seite c der Winkel AMB. -Für ein Kugeldreieck gilt, dass die Summe der drei Seiten kleiner als $2\pi$ aber grösser als 0 ist. +Laut dieser Definition ist die Seite $c$ der Winkel $AMB$. + Man kann bei Kugeldreiecken nicht so einfach unterscheiden, was Innen oder Aussen ist. Wenn man drei Eckpunkte miteinander verbindet, ergeben sich immer 16 Kugeldreiecke. Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha, \beta, \gamma < \pi$ nennt man Eulersche Dreiecke. -\begin{figure}[h] + +\begin{figure} \begin{center} - %\includegraphics[width=6cm]{papers/nav/bilder/kugel1.png} + \includegraphics[width=6cm]{papers/nav/bilder/kugel1.png} \caption[Das Kugeldreieck]{Das Kugeldreieck} \end{center} @@ -21,12 +41,12 @@ Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha \subsection{Rechtwinkliges Dreieck und Rechtseitiges Dreieck} Wie auch im uns bekannten Dreieck gibt es beim Kugeldreieck auch ein Rechtwinkliges Kugeldreieck, bei dem ein Winkel $\frac{\pi}{2}$ ist. Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss. -\newpage -\subsection{Winkelangabe} -\begin{figure}[h] + +\subsection{Winkelsumme} +\begin{figure} \begin{center} - %\includegraphics[width=8cm]{papers/nav/bilder/kugel2.png} + \includegraphics[width=8cm]{papers/nav/bilder/kugel2.png} \caption[Winkelangabe im Kugeldreieck]{Winkelangabe im Kugeldreieck} \end{center} \end{figure} @@ -37,13 +57,15 @@ Für die Summe der Innenwinkel gilt \begin{align} \alpha+\beta+\gamma &= \frac{A}{r^2} + \pi \ \text{und} \ \alpha+\beta+\gamma > \pi. \nonumber \end{align} - +\subsubsection{Sphärischer Exzess} Der sphärische Exzess \begin{align} \epsilon = \alpha+\beta+\gamma - \pi \nonumber \end{align} beschreibt die Abweichung der Innenwinkelsumme von $\pi$ und ist proportional zum Flächeninhalt des Kugeldreiecks. +\subsubsection{Flächeninnhalt} +Der Flächeninhalt $A$ lässt sich aus den Winkeln $\alpha,\ \beta, \ \gamma$ und dem Kugelradius $r$ berechnen. \subsection{Sphärischer Sinussatz} In jedem Dreieck ist das Verhältnis des Sinus einer Seite zum Sinus des Gegenwinkels konstant. @@ -53,7 +75,16 @@ Das bedeutet, dass \frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} \nonumber \ \text{auch beim Kugeldreieck gilt.} \end{align} +\subsection{Sphärischer Kosinussätze} +Auch in der sphärischen Trigonometrie gibt es den Seitenkosinussatz +\begin{align} + cos \ a = \cos b \cdot \cos c + \sin b \cdot \sin c \cdot \cos \alpha \nonumber +\end{align} %Seitenkosinussatz +und den Winkelkosinussatz +\begin{align} + \cos \gamma = -\cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \cdot \cos c \nonumber +\end{align} \subsection{Sphärischer Satz des Pythagoras für das rechtwinklige Kugeldreieck} Es gibt in der sphärischen Trigonometrie eigentlich garkeinen "Satz des Pythagoras", wie man ihn aus der zweidimensionalen Geometrie kennt. @@ -62,6 +93,6 @@ In der sphärischen Trigonometrie gibt es aber auch einen Satz, der alle drei Se Es gilt nämlich: \begin{align} \cos c = \cos a \cdot \cos b \ \text{wenn} \nonumber & - \alpha = \frac{\pi}{2} \lor \beta =\frac{\pi}{2} \lor \gamma = \frac{\pi}{2}.\nonumber + \alpha = \frac{\pi}{2} \nonumber \end{align} \ No newline at end of file -- cgit v1.2.1 From b3283eb05091a88841668c39d422da53d66e1cdd Mon Sep 17 00:00:00 2001 From: "ENEZ-PC\\erdem" Date: Thu, 19 May 2022 14:51:50 +0200 Subject: update korrektur --- buch/papers/nav/main.tex | 5 ++--- buch/papers/nav/nautischesdreieck.tex | 4 ++-- 2 files changed, 4 insertions(+), 5 deletions(-) (limited to 'buch') diff --git a/buch/papers/nav/main.tex b/buch/papers/nav/main.tex index de8d1d6..47764e8 100644 --- a/buch/papers/nav/main.tex +++ b/buch/papers/nav/main.tex @@ -6,14 +6,13 @@ \chapter{Spährische Navigation\label{chapter:nav}} \lhead{Sphärische Navigation} \begin{refsection} -\chapterauthor{Enez Erdem, Marc Kühne} +\chapterauthor{Enez Erdem und Marc Kühne} \input{papers/nav/einleitung.tex} -\input{papers/nav/sincos.tex} -\input{papers/nav/geschichte.tex} \input{papers/nav/flatearth.tex} +\input{papers/nav/sincos.tex} \input{papers/nav/trigo.tex} \input{papers/nav/nautischesdreieck.tex} diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex index a85b119..0a498f0 100644 --- a/buch/papers/nav/nautischesdreieck.tex +++ b/buch/papers/nav/nautischesdreieck.tex @@ -1,6 +1,6 @@ \section{Das Nautische Dreieck} \subsection{Definition des Nautischen Dreiecks} -Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der Himmelskugel}. +Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der Himmelskugel. Die Himmelskugel ist eine gedachte Kugel, welche die Erde und dessen Beobachter umgibt und als Rechenfläche für Koordinaten in der Astronomie und Geodäsie dient. Das nautische Dreieck definiert sich durch folgende Ecken: Zenit, Gestirn und Himmelspol. @@ -195,4 +195,4 @@ Für die Geographische Länge $\lambda$ des eigenen Standortes muss man den Wink Somit ist \[ \omega=\sin^{-1}[\sin(pc) \cdot \frac{\sin(\gamma)}{\sin(l)}] \]und unsere gesuchte geographische Länge schlussendlich \[\lambda=\lambda_1 - \omega\] -mit $\lambda_1$=Längengrad Bildpunkt XXX. +mit $\lambda_1$=Längengrad Bildpunkt $X -- cgit v1.2.1 From 32f1a1d818f0fe28b2ae97071e31a773ee2d028a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 19 May 2022 17:28:33 +0200 Subject: some local changes --- buch/papers/fresnel/Makefile | 14 -- buch/papers/fresnel/eulerspirale.m | 61 --------- buch/papers/fresnel/eulerspirale.pdf | Bin 22592 -> 0 bytes buch/papers/fresnel/eulerspirale.tex | 41 ------ buch/papers/fresnel/fresnelgraph.pdf | Bin 30018 -> 0 bytes buch/papers/fresnel/fresnelgraph.tex | 46 ------- buch/papers/fresnel/images/Makefile | 38 ++++++ buch/papers/fresnel/images/apfel.jpg | Bin 0 -> 139884 bytes buch/papers/fresnel/images/apfel.pdf | Bin 0 -> 157895 bytes buch/papers/fresnel/images/apfel.tex | 49 +++++++ buch/papers/fresnel/images/eulerspirale.m | 61 +++++++++ buch/papers/fresnel/images/eulerspirale.pdf | Bin 0 -> 22592 bytes buch/papers/fresnel/images/eulerspirale.tex | 41 ++++++ buch/papers/fresnel/images/fresnelgraph.pdf | Bin 0 -> 30018 bytes buch/papers/fresnel/images/fresnelgraph.tex | 46 +++++++ buch/papers/fresnel/images/kruemmung.pdf | Bin 0 -> 10179 bytes buch/papers/fresnel/images/kruemmung.tex | 51 ++++++++ buch/papers/fresnel/images/pfad.pdf | Bin 0 -> 19264 bytes buch/papers/fresnel/images/pfad.tex | 37 ++++++ buch/papers/fresnel/images/schale.pdf | Bin 0 -> 352570 bytes buch/papers/fresnel/images/schale.pov | 191 ++++++++++++++++++++++++++++ buch/papers/fresnel/images/schale.tex | 77 +++++++++++ buch/papers/fresnel/main.tex | 5 + buch/papers/fresnel/pfad.pdf | Bin 19126 -> 0 bytes buch/papers/fresnel/pfad.tex | 34 ----- buch/papers/fresnel/references.bib | 6 + buch/papers/fresnel/teil0.tex | 6 +- buch/papers/fresnel/teil1.tex | 11 +- buch/papers/fresnel/teil2.tex | 161 ++++++++++++++++++++++- buch/papers/fresnel/teil3.tex | 4 +- 30 files changed, 772 insertions(+), 208 deletions(-) delete mode 100644 buch/papers/fresnel/eulerspirale.m delete mode 100644 buch/papers/fresnel/eulerspirale.pdf delete mode 100644 buch/papers/fresnel/eulerspirale.tex delete mode 100644 buch/papers/fresnel/fresnelgraph.pdf delete mode 100644 buch/papers/fresnel/fresnelgraph.tex create mode 100644 buch/papers/fresnel/images/Makefile create mode 100644 buch/papers/fresnel/images/apfel.jpg create mode 100644 buch/papers/fresnel/images/apfel.pdf create mode 100644 buch/papers/fresnel/images/apfel.tex create mode 100644 buch/papers/fresnel/images/eulerspirale.m create mode 100644 buch/papers/fresnel/images/eulerspirale.pdf create mode 100644 buch/papers/fresnel/images/eulerspirale.tex create mode 100644 buch/papers/fresnel/images/fresnelgraph.pdf create mode 100644 buch/papers/fresnel/images/fresnelgraph.tex create mode 100644 buch/papers/fresnel/images/kruemmung.pdf create mode 100644 buch/papers/fresnel/images/kruemmung.tex create mode 100644 buch/papers/fresnel/images/pfad.pdf create mode 100644 buch/papers/fresnel/images/pfad.tex create mode 100644 buch/papers/fresnel/images/schale.pdf create mode 100644 buch/papers/fresnel/images/schale.pov create mode 100644 buch/papers/fresnel/images/schale.tex delete mode 100644 buch/papers/fresnel/pfad.pdf delete mode 100644 buch/papers/fresnel/pfad.tex (limited to 'buch') diff --git a/buch/papers/fresnel/Makefile b/buch/papers/fresnel/Makefile index 11af3a7..ed74861 100644 --- a/buch/papers/fresnel/Makefile +++ b/buch/papers/fresnel/Makefile @@ -3,20 +3,6 @@ # # (c) 2022 Prof Dr Andreas Mueller # -all: fresnelgraph.pdf eulerspirale.pdf pfad.pdf - images: @echo "no images to be created in fresnel" -eulerpath.tex: eulerspirale.m - octave eulerspirale.m - -fresnelgraph.pdf: fresnelgraph.tex eulerpath.tex - pdflatex fresnelgraph.tex - -eulerspirale.pdf: eulerspirale.tex eulerpath.tex - pdflatex eulerspirale.tex - -pfad.pdf: pfad.tex - pdflatex pfad.tex - diff --git a/buch/papers/fresnel/eulerspirale.m b/buch/papers/fresnel/eulerspirale.m deleted file mode 100644 index 84e3696..0000000 --- a/buch/papers/fresnel/eulerspirale.m +++ /dev/null @@ -1,61 +0,0 @@ -# -# eulerspirale.m -# -# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue -# -global n; -n = 1000; -global tmax; -tmax = 10; -global N; -N = round(n*5/tmax); - -function retval = f(x, t) - x = pi * t^2 / 2; - retval = [ cos(x); sin(x) ]; -endfunction - -x0 = [ 0; 0 ]; -t = tmax * (0:n) / n; - -c = lsode(@f, x0, t); - -fn = fopen("eulerpath.tex", "w"); - -fprintf(fn, "\\def\\fresnela{ (0,0)"); -for i = (2:n) - fprintf(fn, "\n\t-- (%.4f,%.4f)", c(i,1), c(i,2)); -end -fprintf(fn, "\n}\n\n"); - -fprintf(fn, "\\def\\fresnelb{ (0,0)"); -for i = (2:n) - fprintf(fn, "\n\t-- (%.4f,%.4f)", -c(i,1), -c(i,2)); -end -fprintf(fn, "\n}\n\n"); - -fprintf(fn, "\\def\\Cplotright{ (0,0)"); -for i = (2:N) - fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", t(i), c(i,1)); -end -fprintf(fn, "\n}\n\n"); - -fprintf(fn, "\\def\\Cplotleft{ (0,0)"); -for i = (2:N) - fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", -t(i), -c(i,1)); -end -fprintf(fn, "\n}\n\n"); - -fprintf(fn, "\\def\\Splotright{ (0,0)"); -for i = (2:N) - fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", t(i), c(i,2)); -end -fprintf(fn, "\n}\n\n"); - -fprintf(fn, "\\def\\Splotleft{ (0,0)"); -for i = (2:N) - fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", -t(i), -c(i,2)); -end -fprintf(fn, "\n}\n\n"); - -fclose(fn); diff --git a/buch/papers/fresnel/eulerspirale.pdf b/buch/papers/fresnel/eulerspirale.pdf deleted file mode 100644 index 4a85a50..0000000 Binary files a/buch/papers/fresnel/eulerspirale.pdf and /dev/null differ diff --git a/buch/papers/fresnel/eulerspirale.tex b/buch/papers/fresnel/eulerspirale.tex deleted file mode 100644 index 38ef756..0000000 --- a/buch/papers/fresnel/eulerspirale.tex +++ /dev/null @@ -1,41 +0,0 @@ -% -% eulerspirale.tex -- Darstellung der Eulerspirale -% -% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\documentclass[tikz]{standalone} -\usepackage{amsmath} -\usepackage{times} -\usepackage{txfonts} -\usepackage{pgfplots} -\usepackage{csvsimple} -\usetikzlibrary{arrows,intersections,math} -\begin{document} -\def\skala{1} -\definecolor{darkgreen}{rgb}{0,0.6,0} -\begin{tikzpicture}[>=latex,thick,scale=\skala] - -\input{eulerpath.tex} - -\def\s{8} - -\begin{scope}[scale=\s] -\draw[color=blue] (-0.5,-0.5) rectangle (0.5,0.5); -\draw[color=darkgreen,line width=1.4pt] \fresnela; -\draw[color=darkgreen,line width=1.4pt] \fresnelb; -\fill[color=blue] (0.5,0.5) circle[radius={0.1/\s}]; -\fill[color=blue] (-0.5,-0.5) circle[radius={0.1/\s}]; -\draw (-0.5,{-0.05/\s}) -- (-0.5,{0.05/\s}); -\draw (0.5,{-0.05/\s}) -- (0.5,{-0.05/\s}); -\node at (-0.5,0) [above left] {$\frac12$}; -\node at (0.5,0) [below right] {$\frac12$}; -\node at (0,-0.5) [below right] {$\frac12$}; -\node at (0,0.5) [above left] {$\frac12$}; -\end{scope} - -\draw[->] (-6.7,0) -- (6.9,0) coordinate[label={$C(x)$}];; -\draw[->] (0,-5.8) -- (0,6.1) coordinate[label={left:$S(x)$}];; - -\end{tikzpicture} -\end{document} - diff --git a/buch/papers/fresnel/fresnelgraph.pdf b/buch/papers/fresnel/fresnelgraph.pdf deleted file mode 100644 index 9ccad56..0000000 Binary files a/buch/papers/fresnel/fresnelgraph.pdf and /dev/null differ diff --git a/buch/papers/fresnel/fresnelgraph.tex b/buch/papers/fresnel/fresnelgraph.tex deleted file mode 100644 index 20df951..0000000 --- a/buch/papers/fresnel/fresnelgraph.tex +++ /dev/null @@ -1,46 +0,0 @@ -% -% fresnelgraph.tex -- Graphs of the fresnel functions -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\documentclass[tikz]{standalone} -\usepackage{amsmath} -\usepackage{times} -\usepackage{txfonts} -\usepackage{pgfplots} -\usepackage{csvsimple} -\usetikzlibrary{arrows,intersections,math} -\begin{document} -\def\skala{1} -\begin{tikzpicture}[>=latex,thick,scale=\skala] - -\input{eulerpath.tex} -\def\dx{1.3} -\def\dy{2.6} - -\draw[color=gray] (0,{0.5*\dy}) -- ({5*\dx},{0.5*\dy}); -\draw[color=gray] (0,{-0.5*\dy}) -- ({-5*\dx},{-0.5*\dy}); - -\draw[color=blue,line width=1.4pt] \Splotright; -\draw[color=blue,line width=1.4pt] \Splotleft; - -\draw[color=red,line width=1.4pt] \Cplotright; -\draw[color=red,line width=1.4pt] \Cplotleft; - -\draw[->] (-6.7,0) -- (6.9,0) coordinate[label={$x$}]; -\draw[->] (0,-2.3) -- (0,2.3) coordinate[label={$y$}]; - -\foreach \x in {1,2,3,4,5}{ - \draw ({\x*\dx},-0.05) -- ({\x*\dx},0.05); - \draw ({-\x*\dx},-0.05) -- ({-\x*\dx},0.05); - \node at ({\x*\dx},-0.05) [below] {$\x$}; - \node at ({-\x*\dx},0.05) [above] {$-\x$}; -} -\draw (-0.05,{0.5*\dy}) -- (0.05,{0.5*\dy}); -\node at (-0.05,{0.5*\dy}) [left] {$\frac12$}; -\draw (-0.05,{-0.5*\dy}) -- (0.05,{-0.5*\dy}); -\node at (0.05,{-0.5*\dy}) [right] {$-\frac12$}; - -\end{tikzpicture} -\end{document} - diff --git a/buch/papers/fresnel/images/Makefile b/buch/papers/fresnel/images/Makefile new file mode 100644 index 0000000..eb7dc57 --- /dev/null +++ b/buch/papers/fresnel/images/Makefile @@ -0,0 +1,38 @@ +# +# Makefile +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: schale.pdf \ + fresnelgraph.pdf \ + eulerspirale.pdf \ + pfad.pdf \ + apfel.pdf \ + kruemmung.pdf + +schale.png: schale.pov + povray +A0.1 -W1920 -H1080 -Oschale.png schale.pov + +schale.jpg: schale.png Makefile + convert -extract 1240x1080+340 schale.png -density 300 -units PixelsPerInch schale.jpg + +schale.pdf: schale.tex schale.jpg + pdflatex schale.tex + +eulerpath.tex: eulerspirale.m + octave eulerspirale.m + +fresnelgraph.pdf: fresnelgraph.tex eulerpath.tex + pdflatex fresnelgraph.tex + +eulerspirale.pdf: eulerspirale.tex eulerpath.tex + pdflatex eulerspirale.tex + +pfad.pdf: pfad.tex + pdflatex pfad.tex + +apfel.pdf: apfel.tex apfel.jpg eulerpath.tex + pdflatex apfel.tex + +kruemmung.pdf: kruemmung.tex + pdflatex kruemmung.tex diff --git a/buch/papers/fresnel/images/apfel.jpg b/buch/papers/fresnel/images/apfel.jpg new file mode 100644 index 0000000..76e48e7 Binary files /dev/null and b/buch/papers/fresnel/images/apfel.jpg differ diff --git a/buch/papers/fresnel/images/apfel.pdf b/buch/papers/fresnel/images/apfel.pdf new file mode 100644 index 0000000..69e5092 Binary files /dev/null and b/buch/papers/fresnel/images/apfel.pdf differ diff --git a/buch/papers/fresnel/images/apfel.tex b/buch/papers/fresnel/images/apfel.tex new file mode 100644 index 0000000..754886b --- /dev/null +++ b/buch/papers/fresnel/images/apfel.tex @@ -0,0 +1,49 @@ +% +% apfel.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{7} +\def\hoehe{4} + +\input{eulerpath.tex} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\begin{scope} +\clip(-0.6,-0.6) rectangle (7,6); +\node at (3.1,2.2) [rotate=-3] {\includegraphics[width=9.4cm]{apfel.jpg}}; +\end{scope} + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\draw[color=gray!50] (0,0) rectangle (4,4); +\draw[->] (-0.5,0) -- (7.5,0) coordinate[label={$C(t)$}]; +\draw[->] (0,-0.5) -- (0,6.0) coordinate[label={left:$S(t)$}]; +\begin{scope}[scale=8] +\draw[color=red,opacity=0.5,line width=1.4pt] \fresnela; +\end{scope} + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/fresnel/images/eulerspirale.m b/buch/papers/fresnel/images/eulerspirale.m new file mode 100644 index 0000000..84e3696 --- /dev/null +++ b/buch/papers/fresnel/images/eulerspirale.m @@ -0,0 +1,61 @@ +# +# eulerspirale.m +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue +# +global n; +n = 1000; +global tmax; +tmax = 10; +global N; +N = round(n*5/tmax); + +function retval = f(x, t) + x = pi * t^2 / 2; + retval = [ cos(x); sin(x) ]; +endfunction + +x0 = [ 0; 0 ]; +t = tmax * (0:n) / n; + +c = lsode(@f, x0, t); + +fn = fopen("eulerpath.tex", "w"); + +fprintf(fn, "\\def\\fresnela{ (0,0)"); +for i = (2:n) + fprintf(fn, "\n\t-- (%.4f,%.4f)", c(i,1), c(i,2)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\fresnelb{ (0,0)"); +for i = (2:n) + fprintf(fn, "\n\t-- (%.4f,%.4f)", -c(i,1), -c(i,2)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\Cplotright{ (0,0)"); +for i = (2:N) + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", t(i), c(i,1)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\Cplotleft{ (0,0)"); +for i = (2:N) + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", -t(i), -c(i,1)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\Splotright{ (0,0)"); +for i = (2:N) + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", t(i), c(i,2)); +end +fprintf(fn, "\n}\n\n"); + +fprintf(fn, "\\def\\Splotleft{ (0,0)"); +for i = (2:N) + fprintf(fn, "\n\t-- ({%.4f*\\dx},{%.4f*\\dy})", -t(i), -c(i,2)); +end +fprintf(fn, "\n}\n\n"); + +fclose(fn); diff --git a/buch/papers/fresnel/images/eulerspirale.pdf b/buch/papers/fresnel/images/eulerspirale.pdf new file mode 100644 index 0000000..db74e4b Binary files /dev/null and b/buch/papers/fresnel/images/eulerspirale.pdf differ diff --git a/buch/papers/fresnel/images/eulerspirale.tex b/buch/papers/fresnel/images/eulerspirale.tex new file mode 100644 index 0000000..38ef756 --- /dev/null +++ b/buch/papers/fresnel/images/eulerspirale.tex @@ -0,0 +1,41 @@ +% +% eulerspirale.tex -- Darstellung der Eulerspirale +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{eulerpath.tex} + +\def\s{8} + +\begin{scope}[scale=\s] +\draw[color=blue] (-0.5,-0.5) rectangle (0.5,0.5); +\draw[color=darkgreen,line width=1.4pt] \fresnela; +\draw[color=darkgreen,line width=1.4pt] \fresnelb; +\fill[color=blue] (0.5,0.5) circle[radius={0.1/\s}]; +\fill[color=blue] (-0.5,-0.5) circle[radius={0.1/\s}]; +\draw (-0.5,{-0.05/\s}) -- (-0.5,{0.05/\s}); +\draw (0.5,{-0.05/\s}) -- (0.5,{-0.05/\s}); +\node at (-0.5,0) [above left] {$\frac12$}; +\node at (0.5,0) [below right] {$\frac12$}; +\node at (0,-0.5) [below right] {$\frac12$}; +\node at (0,0.5) [above left] {$\frac12$}; +\end{scope} + +\draw[->] (-6.7,0) -- (6.9,0) coordinate[label={$C(x)$}];; +\draw[->] (0,-5.8) -- (0,6.1) coordinate[label={left:$S(x)$}];; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/images/fresnelgraph.pdf b/buch/papers/fresnel/images/fresnelgraph.pdf new file mode 100644 index 0000000..c658901 Binary files /dev/null and b/buch/papers/fresnel/images/fresnelgraph.pdf differ diff --git a/buch/papers/fresnel/images/fresnelgraph.tex b/buch/papers/fresnel/images/fresnelgraph.tex new file mode 100644 index 0000000..20df951 --- /dev/null +++ b/buch/papers/fresnel/images/fresnelgraph.tex @@ -0,0 +1,46 @@ +% +% fresnelgraph.tex -- Graphs of the fresnel functions +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\input{eulerpath.tex} +\def\dx{1.3} +\def\dy{2.6} + +\draw[color=gray] (0,{0.5*\dy}) -- ({5*\dx},{0.5*\dy}); +\draw[color=gray] (0,{-0.5*\dy}) -- ({-5*\dx},{-0.5*\dy}); + +\draw[color=blue,line width=1.4pt] \Splotright; +\draw[color=blue,line width=1.4pt] \Splotleft; + +\draw[color=red,line width=1.4pt] \Cplotright; +\draw[color=red,line width=1.4pt] \Cplotleft; + +\draw[->] (-6.7,0) -- (6.9,0) coordinate[label={$x$}]; +\draw[->] (0,-2.3) -- (0,2.3) coordinate[label={$y$}]; + +\foreach \x in {1,2,3,4,5}{ + \draw ({\x*\dx},-0.05) -- ({\x*\dx},0.05); + \draw ({-\x*\dx},-0.05) -- ({-\x*\dx},0.05); + \node at ({\x*\dx},-0.05) [below] {$\x$}; + \node at ({-\x*\dx},0.05) [above] {$-\x$}; +} +\draw (-0.05,{0.5*\dy}) -- (0.05,{0.5*\dy}); +\node at (-0.05,{0.5*\dy}) [left] {$\frac12$}; +\draw (-0.05,{-0.5*\dy}) -- (0.05,{-0.5*\dy}); +\node at (0.05,{-0.5*\dy}) [right] {$-\frac12$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/images/kruemmung.pdf b/buch/papers/fresnel/images/kruemmung.pdf new file mode 100644 index 0000000..1180116 Binary files /dev/null and b/buch/papers/fresnel/images/kruemmung.pdf differ diff --git a/buch/papers/fresnel/images/kruemmung.tex b/buch/papers/fresnel/images/kruemmung.tex new file mode 100644 index 0000000..af0a1a9 --- /dev/null +++ b/buch/papers/fresnel/images/kruemmung.tex @@ -0,0 +1,51 @@ +% +% kruemmung.tex -- Krümmung einer ebenen Kurve +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\begin{scope} +\clip (-1,-1) rectangle (4,4); + +\def\r{3} +\def\winkel{30} + +\fill[color=blue!20] (0,0) -- (0:{0.6*\r}) arc (0:\winkel:{0.6*\r}) -- cycle; +\fill[color=blue!20] (\winkel:\r) + -- ($(\winkel:\r)+(0,{0.6*\r})$) arc (90:{90+\winkel}:{0.6*\r}) -- cycle; +\node[color=blue] at ({0.5*\winkel}:{0.45*\r}) {$\Delta\varphi$}; + +\node[color=blue] at ($(\winkel:\r)+({90+0.5*\winkel}:{0.45*\r})$) + {$\Delta\varphi$}; + +\draw[line width=0.3pt] (0,0) circle[radius=\r]; + +\draw[->] (0,0) -- (0:\r); +\draw[->] (0,0) -- (\winkel:\r); + +\draw[->] (0:\r) -- ($(0:\r)+(90:0.7*\r)$); +\draw[->] (\winkel:\r) -- ($(\winkel:\r)+({90+\winkel}:0.7*\r)$); +\draw[->,color=gray] (\winkel:\r) -- ($(\winkel:\r)+(0,0.7*\r)$); + +\draw[color=red,line width=1.4pt] (0:\r) arc (0:\winkel:\r); +\node[color=red] at ({0.5*\winkel}:\r) [left] {$\Delta s$}; +\fill[color=red] (0:\r) circle[radius=0.05]; +\fill[color=red] (\winkel:\r) circle[radius=0.05]; + +\node at (\winkel:{0.5*\r}) [above] {$r$}; +\node at (0:{0.5*\r}) [below] {$r$}; +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/images/pfad.pdf b/buch/papers/fresnel/images/pfad.pdf new file mode 100644 index 0000000..df3c7af Binary files /dev/null and b/buch/papers/fresnel/images/pfad.pdf differ diff --git a/buch/papers/fresnel/images/pfad.tex b/buch/papers/fresnel/images/pfad.tex new file mode 100644 index 0000000..680cd78 --- /dev/null +++ b/buch/papers/fresnel/images/pfad.tex @@ -0,0 +1,37 @@ +% +% pfad.tex -- template for standalon tikz images +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\fill[color=gray!40] (0,0) -- (2,0) arc (0:45:2) -- cycle; +\node at (22.5:1.4) {$\displaystyle\frac{\pi}4$}; + +\draw[->] (-1,0) -- (9,0) coordinate[label={$\operatorname{Re}$}]; +\draw[->] (0,-1) -- (0,6) coordinate[label={left:$\operatorname{Im}$}]; + +\draw[->,color=red,line width=1.4pt] (0,0) -- (7,0); +\draw[->,color=blue,line width=1.4pt] (7,0) arc (0:45:7); +\draw[->,color=darkgreen,line width=1.4pt] (45:7) -- (0,0); + +\node[color=red] at (3.5,0) [below] {$\gamma_1(t) = tR$}; +\node[color=blue] at (25:7) [right] {$\gamma_2(t) = Re^{it}$}; +\node[color=darkgreen] at (45:3.5) [above left] {$\gamma_3(t) = te^{i\pi/4}$}; + +\node at (7,0) [below] {$R$}; +\node at (45:7) [above] {$Re^{i\pi/4}$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/fresnel/images/schale.pdf b/buch/papers/fresnel/images/schale.pdf new file mode 100644 index 0000000..9c21951 Binary files /dev/null and b/buch/papers/fresnel/images/schale.pdf differ diff --git a/buch/papers/fresnel/images/schale.pov b/buch/papers/fresnel/images/schale.pov new file mode 100644 index 0000000..085a6a4 --- /dev/null +++ b/buch/papers/fresnel/images/schale.pov @@ -0,0 +1,191 @@ +// +// schale.pov -- +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +#declare O = <0,0,0>; + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.036; + +camera { + location <40, 20, -20> + look_at <0, 0.5, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <10, 10, -40> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +sphere { + <0, 0, 0>, 1 + pigment { + color rgb<0.8,0.8,0.8> + } + finish { + specular 0.95 + metallic + } +} + +#declare stripcolor = rgb<0.2,0.2,0.8>; + +#declare R = 1.002; + +#macro punkt(phi,theta) +R * < cos(phi) * cos(theta), sin(theta), sin(phi) * cos(theta) > +#end + +#declare N = 24; +#declare thetaphi = 0.01; +#declare thetawidth = pi * 0.008; +#declare theta = function(phi) { phi * thetaphi } + +#declare axisdiameter = 0.007; + +cylinder { + < 0, -2, 0>, < 0, 2, 0>, axisdiameter + pigment { + color White + } + finish { + specular 0.95 + metallic + } +} + +#declare curvaturecircle = 0.008; +#declare curvaturecirclecolor = rgb<0.4,0.8,0.4>; + +#declare phit = 12.8 * 2 * pi; +#declare P = punkt(phit, theta(phit)); +#declare Q = <0, R / sin(theta(phit)), 0>; + +#declare e1 = vnormalize(P - Q) / tan(theta(phit)); +#declare e2 = vnormalize(vcross(e1, <0,1,0>)) / tan(theta(phit)); +#declare psimin = -0.1 * pi; +#declare psimax = 0.1 * pi; +#declare psistep = (psimax - psimin) / 30; + +union { + #declare psi = psimin; + #declare K = Q + cos(psi) * e1 + sin(psi) * e2; + #while (psi < psimax - psistep/2) + sphere { K, curvaturecircle } + #declare psi = psi + psistep; + #declare K2 = Q + cos(psi) * e1 + sin(psi) * e2; + cylinder { K, K2, curvaturecircle } + #declare K = K2; + #end + sphere { K, curvaturecircle } + pigment { + color curvaturecirclecolor + } + finish { + specular 0.95 + metallic + } +} + +object { + mesh { + #declare psi = psimin; + #declare K = Q + cos(psi) * e1 + sin(psi) * e2; + #while (psi < psimax - psistep/2) + #declare psi = psi + psistep; + #declare K2 = Q + cos(psi) * e1 + sin(psi) * e2; + triangle { K, K2, Q } + #declare K = K2; + #end + } + pigment { + color rgbt<0.4,0.8,0.4,0.5> + } + finish { + specular 0.95 + metallic + } +} + +union { + sphere { P, 0.02 } + sphere { Q, 0.02 } + cylinder { P, Q, 0.01 } + pigment { + color Red + } + finish { + specular 0.95 + metallic + } +} + +#declare phisteps = 300; +#declare phistep = 2 * pi / phisteps; +#declare phimin = 0; +#declare phimax = N * 2 * pi; + +object { + mesh { + #declare phi = phimin; + #declare Poben = punkt(phi, theta(phi) + thetawidth); + #declare Punten = punkt(phi, theta(phi) - thetawidth); + triangle { O, Punten, Poben } + #while (phi < phimax - phistep/2) + #declare phi = phi + phistep; + #declare Poben2 = punkt(phi, theta(phi) + thetawidth); + #declare Punten2 = punkt(phi, theta(phi) - thetawidth); + triangle { O, Punten, Punten2 } + triangle { O, Poben, Poben2 } + triangle { Punten, Punten2, Poben } + triangle { Punten2, Poben2, Poben } + #declare Poben = Poben2; + #declare Punten = Punten2; + #end + triangle { O, Punten, Poben } + } + pigment { + color stripcolor + } + finish { + specular 0.8 + metallic + } +} + +union { + #declare phi = phimin; + #declare P = punkt(phi, theta(phi)); + #while (phi < phimax - phistep/2) + sphere { P, 0.003 } + #declare phi = phi + phistep; + #declare P2 = punkt(phi, theta(phi)); + cylinder { P, P2, 0.003 } + #declare P = P2; + #end + sphere { P, 0.003 } + pigment { + color stripcolor + } + finish { + specular 0.8 + metallic + } +} diff --git a/buch/papers/fresnel/images/schale.tex b/buch/papers/fresnel/images/schale.tex new file mode 100644 index 0000000..577ede4 --- /dev/null +++ b/buch/papers/fresnel/images/schale.tex @@ -0,0 +1,77 @@ +% +% schlange.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math,calc} +\usepackage{ifthen} +\begin{document} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} +\def\a{47} +\def\r{3.3} +\def\skala{0.95} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\begin{scope}[xshift=-7.4cm,yshift=-1.2cm] + \clip (-3.6,-2.2) rectangle (3.6,5.1); + + \fill[color=blue!20] (0,0) + -- ({180-\a}:{0.4*\r}) arc ({180-\a}:180:{0.4*\r}) + -- cycle; + \node[color=blue] at ({180-\a/2}:{0.3*\r}) {$\vartheta$}; + + \fill[color=blue!20] (0,{\r/sin(\a)}) + -- ($(0,{\r/sin(\a)})+({270-\a}:{0.3*\r})$) + arc ({270-\a}:270:{0.3*\r}) + -- cycle; + \node[color=blue] at ($(0,{\r/sin(\a)})+({270-\a/2}:{0.2*\r})$) + {$\vartheta$}; + + + \draw (0,0) circle[radius=\r]; + \draw[->] (0,-3.0) -- (0,5); + \draw ({-\r-0.2},0) -- ({\r+0.2},0); + \fill (0,0) circle[radius=0.06]; + + \draw (0,0) -- ({180-\a}:\r); + \node at ({180-\a+3}:{0.65*\r}) [above right] {$1$}; + + \draw[color=red,line width=1.4pt] + ({180-\a}:\r) -- (0,{\r/cos(90-\a)}); + \fill[color=red] ({180-\a}:\r) circle[radius=0.08]; + \fill[color=red] (0,{\r/cos(90-\a)}) circle[radius=0.08]; + \node[color=red] at (-1.0,3.7) [left] {$r=\cot\vartheta$}; + \node[color=red] at ({180-\a}:\r) [above left] {$P$}; + \node[color=red] at (0,{\r/sin(\a)}) [right] {$Q$}; +\end{scope} + +% Povray Bild +\node at (0,0) {\includegraphics[width=7.6cm]{schale.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node[color=red] at (-1.4,1.4) {$r$}; +\node[color=red] at (-2.2,-0.2) {$P$}; +\node[color=red] at (0,3.3) [right] {$Q$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/fresnel/main.tex b/buch/papers/fresnel/main.tex index e6ee3b5..2050fd4 100644 --- a/buch/papers/fresnel/main.tex +++ b/buch/papers/fresnel/main.tex @@ -8,6 +8,11 @@ \begin{refsection} \chapterauthor{Andreas Müller} +{\parindent0pt Die} Fresnel-Integrale tauchen in der Untersuchung der Beugung +in paraxialer Näherung auf, auch bekannt als die Fresnel-Approximation. +In diesem Kapitel betrachen wir jedoch nur die geometrische +Anwendung der Fresnel-Integrale als Parametrisierung der Euler-Spirale +und zeigen, dass letztere eine Klothoide ist. \input{papers/fresnel/teil0.tex} \input{papers/fresnel/teil1.tex} diff --git a/buch/papers/fresnel/pfad.pdf b/buch/papers/fresnel/pfad.pdf deleted file mode 100644 index ff514cc..0000000 Binary files a/buch/papers/fresnel/pfad.pdf and /dev/null differ diff --git a/buch/papers/fresnel/pfad.tex b/buch/papers/fresnel/pfad.tex deleted file mode 100644 index 5439a71..0000000 --- a/buch/papers/fresnel/pfad.tex +++ /dev/null @@ -1,34 +0,0 @@ -% -% pfad.tex -- template for standalon tikz images -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\documentclass[tikz]{standalone} -\usepackage{amsmath} -\usepackage{times} -\usepackage{txfonts} -\usepackage{pgfplots} -\usepackage{csvsimple} -\usetikzlibrary{arrows,intersections,math} -\begin{document} -\def\skala{1} -\definecolor{darkgreen}{rgb}{0,0.6,0} -\begin{tikzpicture}[>=latex,thick,scale=\skala] - -\draw[->] (-1,0) -- (9,0) coordinate[label={$\operatorname{Re}$}]; -\draw[->] (0,-1) -- (0,6) coordinate[label={left:$\operatorname{Im}$}]; - -\draw[->,color=red,line width=1.4pt] (0,0) -- (7,0); -\draw[->,color=blue,line width=1.4pt] (7,0) arc (0:45:7); -\draw[->,color=darkgreen,line width=1.4pt] (45:7) -- (0,0); - -\node[color=red] at (3.5,0) [below] {$\gamma_1(t) = tR$}; -\node[color=blue] at (25:7) [right] {$\gamma_2(t) = Re^{it}$}; -\node[color=darkgreen] at (45:3.5) [above left] {$\gamma_3(t) = te^{i\pi/4}$}; - -\node at (7,0) [below] {$R$}; -\node at (45:7) [above] {$Re^{i\pi/4}$}; - -\end{tikzpicture} -\end{document} - diff --git a/buch/papers/fresnel/references.bib b/buch/papers/fresnel/references.bib index 58e9242..cf8fb21 100644 --- a/buch/papers/fresnel/references.bib +++ b/buch/papers/fresnel/references.bib @@ -44,3 +44,9 @@ title = { Fresnel Integral }, date = { 2022-05-13 } } + +@online{fresnel:schale, + url = { https://www.youtube.com/watch?v=D3tdW9l1690 }, + title = { A Strange Map Projection (Euler Spiral) - Numberphile }, + date = { 2022-05-14 } +} diff --git a/buch/papers/fresnel/teil0.tex b/buch/papers/fresnel/teil0.tex index 253e2f3..85b8bf7 100644 --- a/buch/papers/fresnel/teil0.tex +++ b/buch/papers/fresnel/teil0.tex @@ -20,7 +20,7 @@ C(x) &= \int_0^x \cos\biggl(\frac{\pi}2 t^2\biggr)\,dt \\ S(x) &= \int_0^x \sin\biggl(\frac{\pi}2 t^2\biggr)\,dt \end{align*} -heissen die Fesnel-Integrale. +heissen die Fresnel-Integrale. \end{definition} Der Faktor $\frac{\pi}2$ ist einigermassen willkürlich, man könnte @@ -39,7 +39,7 @@ C(x) &= C_{\frac{\pi}2}(x), S(x) &= S_{\frac{\pi}2}(x). \end{aligned} \] -Durch eine Substution $t=bs$ erhält man +Durch eine Substitution $t=bs$ erhält man \begin{align*} C_a(x) &= @@ -91,7 +91,7 @@ $C_1(x)$ und $S_1(x)$ betrachten, da in diesem Fall die Formeln einfacher werden. \begin{figure} \centering -\includegraphics{papers/fresnel/fresnelgraph.pdf} +\includegraphics{papers/fresnel/images/fresnelgraph.pdf} \caption{Graph der Funktionen $C(x)$ ({\color{red}rot}) und $S(x)$ ({\color{blue}blau}) \label{fresnel:figure:plot}} diff --git a/buch/papers/fresnel/teil1.tex b/buch/papers/fresnel/teil1.tex index a41ddb7..c716cd7 100644 --- a/buch/papers/fresnel/teil1.tex +++ b/buch/papers/fresnel/teil1.tex @@ -8,7 +8,7 @@ \rhead{Euler-Spirale} \begin{figure} \centering -\includegraphics{papers/fresnel/eulerspirale.pdf} +\includegraphics{papers/fresnel/images/eulerspirale.pdf} \caption{Die Eulerspirale ist die Kurve mit der Parameterdarstellung $x\mapsto (C(x),S(x))$, sie ist rot dargestellt. Sie windet sich unendlich oft um die beiden Punkte $(\pm\frac12,\pm\frac12)$. @@ -25,7 +25,7 @@ $(\pm\frac12,\pm\frac12)$ zu winden. \begin{figure} \centering -\includegraphics{papers/fresnel/pfad.pdf} +\includegraphics{papers/fresnel/images/pfad.pdf} \caption{Pfad zur Berechnung der Grenzwerte $C_1(\infty)$ und $S_1(\infty)$ mit Hilfe des Cauchy-Integralsatzes \label{fresnel:figure:pfad}} @@ -182,7 +182,7 @@ muss, folgt $C_1(\infty)=S_1(\infty)$. Nach Multlikation mit $\sqrt{2}$ folgt aus der Tatsache, dass auch der Realteil verschwinden muss \[ -\frac{\sqrt{\pi}}{\sqrt{2}} = C_1(\infty)+S_1(\infty) +\sqrt{\frac{\pi}{2}} = C_1(\infty)+S_1(\infty) \qquad \Rightarrow \qquad @@ -190,7 +190,10 @@ C_1(\infty) = S_1(\infty) = -\frac{\sqrt{\pi}}{2\sqrt{2}}. +\frac12 +\sqrt{ +\frac{\pi}{2} +}. \] Aus \eqref{fresnel:equation:arg} diff --git a/buch/papers/fresnel/teil2.tex b/buch/papers/fresnel/teil2.tex index 22d2a89..ec8c896 100644 --- a/buch/papers/fresnel/teil2.tex +++ b/buch/papers/fresnel/teil2.tex @@ -15,10 +15,165 @@ Eine ebene Kurve, deren Krümmung proportionale zur Kurvenlänge ist, heisst {\em Klothoide}. \end{definition} -Die Klothoide wird zum Beispiel im Strassenbau bei Autobahnkurven -angewendet. -Fährt man mit konstanter Geschwindigkeit mit entlang einer Klothoide, +Die Klothoide wird zum Beispiel im Strassenbau für Autobahnkurven +verwendet. +Fährt man mit konstanter Geschwindigkeit entlang einer Klothoide, muss man die Krümmung mit konstaner Geschwindigkeit ändern, also das Lenkrad mit konstanter Geschwindigkeit drehen. Dies ermöglicht eine ruhige Fahrweise. +\subsection{Krümmung einer ebenen Kurve} +\begin{figure} +\centering +\includegraphics{papers/fresnel/images/kruemmung.pdf} +\caption{Berechnung der Krümmung einer ebenen Kurve. +\label{fresnel:figure:kruemmung}} +\end{figure} +Abbildung~\ref{fresnel:figure:kruemmung} erinnert daran, dass der +Bogen eines Kreises vom Radius $r$, entlang dem sich die Richtung +der Tangente um $\Delta\varphi$ ändert, die Länge +$\Delta s = r\Delta\varphi$. +Die Krümmung ist der Kehrwert des Krümmungsradius, daraus kann +man ablesen, dass +\[ +\kappa = \frac{1}{r} = \frac{\Delta \varphi}{\Delta s}. +\] +Für eine beliebige ebene Kurve ist daher die Krümmung +\[ +\kappa = \frac{d\varphi}{ds}. +\] + +\subsection{Krümmung der Euler-Spirale} +Wir betrachten jetzt die Euler-Spirale mit der Parametrisierung +$\gamma(s) = (C_1(s),S_1(s))$. +Zunächst stellen wir fest, dass die Länge der Tangente +\[ +\dot{\gamma}(s) += +\frac{d\gamma}{ds} += +\begin{pmatrix} +\dot{C}_1(s)\\ +\dot{S}_1(s) +\end{pmatrix} += +\begin{pmatrix} +\cos s^2\\ +\sin s^2 +\end{pmatrix} +\qquad\Rightarrow\qquad +|\dot{\gamma}(s)| += +\sqrt{\cos^2s^2+\sin^2s^2} += +1. +\] +Insbesondere ist der Parameter $s$ der Kurve $\gamma(s)$ die +Bogenlänge. + +Der zu $\dot{\gamma}(s)$ gehörige Polarwinkel kann aus dem Vergleich +mit einem Vektor mit bekanntem Polarwinkel $\varphi$ abgelesen werden: +\[ +\begin{pmatrix} +\cos \varphi\\ +\sin \varphi +\end{pmatrix} += +\dot{\gamma}(s) += +\begin{pmatrix} +\cos s^2\\\sin s^2 +\end{pmatrix}, +\] +der Polarwinkel +ist daher $\varphi = s^2$. +Die Krümmung ist die Ableitung des Polarwinkels nach $s$, also +\[ +\kappa += +\frac{d\varphi}{ds} += +\frac{ds^2}{ds} += +2s, +\] +sie ist somit proportional zur Bogenlänge $s$. +Damit folgt, dass die Euler-Spirale eine Klothoide ist. + +\subsection{Eine Kugel schälen} +\begin{figure} +\centering +\includegraphics[width=\textwidth]{papers/fresnel/images/schale.pdf} +\caption{Schält man eine einen Streifen konstanter Breite beginnend am +Äquator von einer Kugel ab und breitet ihn in der Ebene aus, entsteht +eine Klothoide. +\label{fresnel:figure:schale}} +\end{figure} +\begin{figure} +\centering +\includegraphics{papers/fresnel/images/apfel.pdf} +\caption{Klothoide erhalten durch Abschälen eines Streifens von einem +Apfel (vgl.~Abbildung~\ref{fresnel:figure:schale}) +\label{fresnel:figure:apfel}} +\end{figure} +Schält man einen Streifen konstanter Breite beginnend parallel zum Äquator +von einer Kugel ab und breitet ihn in die Ebene aus, entsteht eine +Approximation einer Klothoide. +Abbildung~\ref{fresnel:figure:schale} zeigt blau den abgeschälten Streifen, +Abbildung~\ref{fresnel:figure:apfel} zeigt das Resultat dieses Versuches +an einem Apfel, das Youtube-Video \cite{fresnel:schale} des +Numberphile-Kanals illustriert das Problem anhand eines aufblasbaren +Globus. + +Windet sich die Kurve in Abbildung~\ref{fresnel:figure:schale} $n$ +mal um die vertikale Achse, bevor sie den Nordpol erreicht, dann kann +die Kurve mit der Funktion +\[ +\gamma(t) += +\begin{pmatrix} +\cos(t) \cos(t/n) \\ +\sin(t) \cos(t/n) \\ +\sin(t/n) +\end{pmatrix} +\] +parametrisiert werden. +Der Tangentialvektor +\[ +\dot{\gamma}(t) += +\begin{pmatrix} +-\sin(t)\cos(t/n) - \cos(t)\sin(t/n)/n \\ +\cos(t)\cos(t/n) - \sin(t)\sin(t/n)/n \\ +\cos(t/n)/n +\end{pmatrix} +\] +hat die Länge +\[ +| \dot{\gamma}(t) |^2 += +\frac{1}{n^2} ++ +\cos^2\frac{t}{n}. +\] +Die Ableitung der Bogenlänge ist daher +\[ +\dot{s}(t) += +\sqrt{ +\frac{1}{n^2} ++ +\cos^2\frac{t}{n} +}. +\] + + +Der Krümmungsradius des blauen Streifens, der die Kugel im Punkt $P$ bei +geographischer $\vartheta$ berührt, hat die Länge der Tangente, die +die Kugel im Punkt $P$ berührt und im Punkt $Q$ durch die Achse der +Kugel geht (Abbildung~\ref{fresnel:figure:schale}). +Die Krümmung in Abhängigkeit von $\vartheta$ ist daher $\tan\vartheta$. + + + + diff --git a/buch/papers/fresnel/teil3.tex b/buch/papers/fresnel/teil3.tex index 37e6bee..ceddbe0 100644 --- a/buch/papers/fresnel/teil3.tex +++ b/buch/papers/fresnel/teil3.tex @@ -42,8 +42,8 @@ C'(x) = \cos \biggl(\frac{\pi}2 x^2\biggr) \qquad\text{und}\qquad S'(x) = \sin \biggl(\frac{\pi}2 x^2\biggr) \] -erfüllen, kann man eine Methode zur Lösung von Differentialgleichung -verwenden. +erfüllen, kann man eine Methode zur numerischen Lösung von +Differentialgleichung verwenden. Die Abbildungen~\ref{fresnel:figure:plot} und \ref{fresnel:figure:eulerspirale} wurden auf diese Weise erzeugt. -- cgit v1.2.1 From 0fe9bb56da147bf7986852e6f657149206d967a4 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 19 May 2022 17:31:23 +0200 Subject: fixes --- buch/papers/nav/Makefile.inc | 1 - buch/papers/nav/nautischesdreieck.tex | 2 +- 2 files changed, 1 insertion(+), 2 deletions(-) (limited to 'buch') diff --git a/buch/papers/nav/Makefile.inc b/buch/papers/nav/Makefile.inc index 24ab4ee..5e86543 100644 --- a/buch/papers/nav/Makefile.inc +++ b/buch/papers/nav/Makefile.inc @@ -8,7 +8,6 @@ dependencies-nav = \ papers/nav/main.tex \ papers/nav/einleitung.tex \ papers/nav/flatearth.tex \ - papers/nav/geschichte.tex \ papers/nav/nautischesdreieck.tex \ papers/nav/sincos.tex \ papers/nav/trigo.tex \ diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex index 0a498f0..c1ad38a 100644 --- a/buch/papers/nav/nautischesdreieck.tex +++ b/buch/papers/nav/nautischesdreieck.tex @@ -195,4 +195,4 @@ Für die Geographische Länge $\lambda$ des eigenen Standortes muss man den Wink Somit ist \[ \omega=\sin^{-1}[\sin(pc) \cdot \frac{\sin(\gamma)}{\sin(l)}] \]und unsere gesuchte geographische Länge schlussendlich \[\lambda=\lambda_1 - \omega\] -mit $\lambda_1$=Längengrad Bildpunkt $X +mit $\lambda_1$=Längengrad Bildpunkt $X$ -- cgit v1.2.1 From f0a6f930187eb0226ddd4735feba1d93667b8a58 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 19 May 2022 22:12:27 +0200 Subject: add dreieck3d9.pov --- buch/papers/nav/images/Makefile | 7 ++++ buch/papers/nav/images/common.inc | 60 +++++++++++++++++++------------ buch/papers/nav/images/dreieck3d9.pov | 66 +++++++++++++++++++++++++++++++++++ 3 files changed, 111 insertions(+), 22 deletions(-) create mode 100644 buch/papers/nav/images/dreieck3d9.pov (limited to 'buch') diff --git a/buch/papers/nav/images/Makefile b/buch/papers/nav/images/Makefile index bbdea2f..da4defa 100644 --- a/buch/papers/nav/images/Makefile +++ b/buch/papers/nav/images/Makefile @@ -114,3 +114,10 @@ dreieck3d8.jpg: dreieck3d8.png dreieck3d8.pdf: dreieck3d8.tex dreieck3d8.jpg pdflatex dreieck3d8.tex +dreieck3d9.png: dreieck3d9.pov common.inc + povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d9.png dreieck3d9.pov +dreieck3d9.jpg: dreieck3d9.png + convert dreieck3d9.png -density 300 -units PixelsPerInch dreieck3d9.jpg +dreieck3d9.pdf: dreieck3d9.tex dreieck3d9.jpg + pdflatex dreieck3d9.tex + diff --git a/buch/papers/nav/images/common.inc b/buch/papers/nav/images/common.inc index e2a1ed0..2c0ae6e 100644 --- a/buch/papers/nav/images/common.inc +++ b/buch/papers/nav/images/common.inc @@ -12,6 +12,7 @@ global_settings { #declare imagescale = 0.034; +#declare O = <0, 0, 0>; #declare A = vnormalize(< 0, 1, 0>); #declare B = vnormalize(< 1, 2, -8>); #declare C = vnormalize(< 5, 1, 0>); @@ -102,8 +103,8 @@ union { #declare pp = vnormalize(p - vdot(n, p) * n); #declare qq = vnormalize(q - vdot(n, q) * n); intersection { - sphere { <0, 0, 0>, 1 + staerke } - cone { <0, 0, 0>, 0, 1.2 * vnormalize(w), r } + sphere { O, 1 + staerke } + cone { O, 0, 1.2 * vnormalize(w), r } plane { -vcross(n, qq) * vdot(vcross(n, qq), pp), 0 } plane { -vcross(n, pp) * vdot(vcross(n, pp), qq), 0 } } @@ -132,6 +133,35 @@ union { } #end +#macro ebenerwinkel(a, p, q, s, r, farbe) + #declare n = vnormalize(-vcross(p, q)); + #declare np = vnormalize(-vcross(p, n)); + #declare nq = -vnormalize(-vcross(q, n)); +// arrow(a, a + n, 0.02, White) +// arrow(a, a + np, 0.01, Red) +// arrow(a, a + nq, 0.01, Blue) + intersection { + cylinder { a - (s/2) * n, a + (s/2) * n, r } + plane { np, vdot(np, a) } + plane { nq, vdot(nq, a) } + pigment { + farbe + } + finish { + metallic + specular 0.5 + } + } +#end + +#macro komplement(a, p, q, s, r, farbe) + #declare n = vnormalize(-vcross(p, q)); +// arrow(a, a + n, 0.015, Orange) + #declare m = vnormalize(-vcross(q, n)); +// arrow(a, a + m, 0.015, Pink) + ebenerwinkel(a, p, m, s, r, farbe) +#end + #declare fett = 0.015; #declare fein = 0.010; @@ -143,29 +173,15 @@ union { #declare gruen = rgb<0,0.6,0>; #declare blau = rgb<0.2,0.2,0.8>; +#declare kugelfarbe = rgb<0.8,0.8,0.8>; +#declare kugeltransparent = rgbt<0.8,0.8,0.8,0.5>; + +#macro kugel(farbe) sphere { <0, 0, 0>, 1 pigment { - color rgb<0.8,0.8,0.8> + color farbe } } +#end -//union { -// sphere { A, 0.02 } -// sphere { B, 0.02 } -// sphere { C, 0.02 } -// sphere { P, 0.02 } -// pigment { -// color Red -// } -//} - -//union { -// winkel(A, B, C) -// winkel(B, P, C) -// seite(B, C, 0.01) -// seite(B, P, 0.01) -// pigment { -// color rgb<0,0.6,0> -// } -//} diff --git a/buch/papers/nav/images/dreieck3d9.pov b/buch/papers/nav/images/dreieck3d9.pov new file mode 100644 index 0000000..24d3843 --- /dev/null +++ b/buch/papers/nav/images/dreieck3d9.pov @@ -0,0 +1,66 @@ +// +// dreiecke3d8.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +//union { +// seite(A, B, fein) +// seite(B, C, fein) +// seite(A, C, fein) +// seite(A, P, fein) +// seite(B, P, fett) +// seite(C, P, fett) +// punkt(A, fein) +// punkt(B, fett) +// punkt(C, fett) +// punkt(P, fett) +// pigment { +// color dreieckfarbe +// } +// finish { +// specular 0.95 +// metallic +// } +//} + +//dreieck(A, B, C, White) + +kugel(kugeltransparent) + +ebenerwinkel(O, C, P, 0.01, 1.001, rot) +ebenerwinkel(P, C, P, 0.01, 0.3, rot) +komplement(P, C, P, 0.01, 0.3, Yellow) + +ebenerwinkel(O, B, P, 0.01, 1.001, blau) +ebenerwinkel(P, B, P, 0.01, 0.3, blau) +komplement(P, B, P, 0.01, 0.3, Green) + +arrow(B, 1.5 * B, 0.015, White) +arrow(C, 1.5 * C, 0.015, White) +arrow(P, 1.5 * P, 0.015, White) + +union { + cylinder { O, P, 0.7 * fein } + + cylinder { P, P + 3 * B, 0.7 * fein } + cylinder { O, B + 3 * B, 0.7 * fein } + + cylinder { P, P + 3 * C, 0.7 * fein } + cylinder { O, C + 3 * C, 0.7 * fein } + + pigment { + color White + } +} + +#declare imagescale = 0.044; + +camera { + location <40, 20, -20> + look_at <0, 0.24, -0.20> + right x * imagescale + up y * imagescale +} + -- cgit v1.2.1 From 5d1cd4306966a5433bcc8375d627989aade53a3c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 21 May 2022 07:21:47 +0200 Subject: add new script for risch part --- buch/chapters/060-integral/experiments/rxy.maxima | 9 +++++++++ 1 file changed, 9 insertions(+) create mode 100644 buch/chapters/060-integral/experiments/rxy.maxima (limited to 'buch') diff --git a/buch/chapters/060-integral/experiments/rxy.maxima b/buch/chapters/060-integral/experiments/rxy.maxima new file mode 100644 index 0000000..0d5a56d --- /dev/null +++ b/buch/chapters/060-integral/experiments/rxy.maxima @@ -0,0 +1,9 @@ +y: sqrt(a*x^2+b*x+c); + +F: log(x + b/(2 * a) + y/sqrt(a))/sqrt(a); + +f: diff(F, x); + +ratsimp(f); + +ratsimp(y*f); -- cgit v1.2.1 From eceae67b3a13bc28acc446288429a90be2efa99d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 21 May 2022 12:45:42 +0200 Subject: curvature graph --- buch/papers/kugel/images/Makefile | 13 ++++++ buch/papers/kugel/images/curvature.pov | 72 +++++++++++++++++++++++++++++ buch/papers/kugel/images/curvgraph.m | 83 ++++++++++++++++++++++++++++++++++ 3 files changed, 168 insertions(+) create mode 100644 buch/papers/kugel/images/Makefile create mode 100644 buch/papers/kugel/images/curvature.pov create mode 100644 buch/papers/kugel/images/curvgraph.m (limited to 'buch') diff --git a/buch/papers/kugel/images/Makefile b/buch/papers/kugel/images/Makefile new file mode 100644 index 0000000..8efa228 --- /dev/null +++ b/buch/papers/kugel/images/Makefile @@ -0,0 +1,13 @@ +# +# Makefile -- build images +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: curvature.png + +curvature.inc: curvgraph.m + octave curvgraph.m + +curvature.png: curvature.pov curvature.inc + povray +A0.1 +W1920 +H1080 +Ocurvature.png curvature.pov + diff --git a/buch/papers/kugel/images/curvature.pov b/buch/papers/kugel/images/curvature.pov new file mode 100644 index 0000000..3535488 --- /dev/null +++ b/buch/papers/kugel/images/curvature.pov @@ -0,0 +1,72 @@ +// +// curvature.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.1; + +camera { + location <40, 10, -20> + look_at <0, 0, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <10, 10, -40> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +// +// draw an arrow from to with thickness with +// color +// +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +arrow(<-3.1,0,0>, <3.1,0,0>, 0.01, White) +arrow(<0,-1,0>, <0,1,0>, 0.01, White) +arrow(<0,0,-2.1>, <0,0,2.1>, 0.01, White) + +#include "curvature.inc" diff --git a/buch/papers/kugel/images/curvgraph.m b/buch/papers/kugel/images/curvgraph.m new file mode 100644 index 0000000..96ca4b1 --- /dev/null +++ b/buch/papers/kugel/images/curvgraph.m @@ -0,0 +1,83 @@ +# +# curvature.m +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +global N; +N = 10; + +global sigma2; +sigma2 = 1; + +global s; +s = 1; + +xmin = -3; +xmax = 3; +xsteps = 1000; +hx = (xmax - xmin) / xsteps; + +ymin = -2; +ymax = 2; +ysteps = 1000; +hy = (ymax - ymin) / ysteps; + +function retval = f0(r) + global sigma2; + retval = exp(-r^2/sigma2)/sigma2 - exp(-r^2/(2*sigma2))/(sqrt(2)*sigma2); +end + +global N0; +N0 = f0(0); + +function retval = f1(x,y) + global N0; + retval = f0(hypot(x, y)) / N0; +endfunction + +function retval = f(x, y) + global s; + retval = f1(x+s, y) - f1(x-s, y); +endfunction + +function retval = curvature0(r) + global sigma2; + retval = ( + (2*sigma2-r^2)*exp(-r^2/(2*sigma2)) + + + 4*(r^2-sigma2)*exp(-r^2/sigma2) + ) / (sigma2^2); +endfunction + +function retval = curvature1(x, y) + retval = curvature0(hypot(x, y)); +endfunction + +function retval = curvature(x, y) + global s; + retval = curvature1(x+s, y) + curvature1(x-s, y); +endfunction + +function retval = farbe(x, y) + c = curvature(x, y); + retval = c * ones(1,3); +endfunction + +fn = fopen("curvature.inc", "w"); + +for ix = (0:xsteps) + x = xmin + ix * hx; + for iy = (0:ysteps) + y = ymin + iy * hy; + fprintf(fn, "sphere { <%.4f, %.4f, %.4f>, 0.01\n", + x, f(x, y), y); + color = farbe(x, y); + fprintf(fn, "pigment { color rgb<%.4f,%.4f,%.4f> }\n", + color(1,1), color(1,2), color(1,3)); + fprintf(fn, "finish { metallic specular 0.5 }\n"); + fprintf(fn, "}\n"); + end +end + +fclose(fn); -- cgit v1.2.1 From 411fb410f790fcc1bb3da381c17119ebb5130032 Mon Sep 17 00:00:00 2001 From: "ENEZ-PC\\erdem" Date: Sat, 21 May 2022 18:56:21 +0200 Subject: Korrektur 21.05 --- buch/papers/nav/bilder/ephe.png | Bin 184799 -> 543515 bytes buch/papers/nav/bilder/recht.jpg | Bin 0 -> 42889 bytes buch/papers/nav/bilder/sextant.jpg | Bin 0 -> 8280 bytes buch/papers/nav/einleitung.tex | 2 +- buch/papers/nav/flatearth.tex | 15 +++-- buch/papers/nav/main.tex | 2 +- buch/papers/nav/nautischesdreieck.tex | 123 +++++++++++++++++----------------- buch/papers/nav/sincos.tex | 6 +- buch/papers/nav/trigo.tex | 66 ++++++++++-------- 9 files changed, 114 insertions(+), 100 deletions(-) create mode 100644 buch/papers/nav/bilder/recht.jpg create mode 100644 buch/papers/nav/bilder/sextant.jpg (limited to 'buch') diff --git a/buch/papers/nav/bilder/ephe.png b/buch/papers/nav/bilder/ephe.png index 0aeef6f..3f99a36 100644 Binary files a/buch/papers/nav/bilder/ephe.png and b/buch/papers/nav/bilder/ephe.png differ diff --git a/buch/papers/nav/bilder/recht.jpg b/buch/papers/nav/bilder/recht.jpg new file mode 100644 index 0000000..3f60370 Binary files /dev/null and b/buch/papers/nav/bilder/recht.jpg differ diff --git a/buch/papers/nav/bilder/sextant.jpg b/buch/papers/nav/bilder/sextant.jpg new file mode 100644 index 0000000..53dd784 Binary files /dev/null and b/buch/papers/nav/bilder/sextant.jpg differ diff --git a/buch/papers/nav/einleitung.tex b/buch/papers/nav/einleitung.tex index 8d8c5c1..aafa107 100644 --- a/buch/papers/nav/einleitung.tex +++ b/buch/papers/nav/einleitung.tex @@ -4,6 +4,6 @@ Heutzutage ist die Navigation ein Teil des Lebens. Man sendet dem Kollegen seinen eigenen Standort, um sich das ewige Erklären zu sparen oder gibt die Adresse des Ziels ein, damit man seinen Aufenthaltsort zum Beispiel auf einer riesigen Wiese am See findet. Dies wird durch Technologien wie Funknavigation, welches ein auf Langzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt. -Vor all diesen technologischen Fortschritten gab es lediglich die Astronavigation, welche heute noch auf kleineren Schiffen benötigt wird im Falle eines Stromausfalls. +Vor all diesen technologischen Fortschritten gab es lediglich die Astronavigation, welche heute noch auf Schiffen verwendet wird im Falle eines Stromausfalls. Aber wie funktioniert die Navigation mit den Sternen? Welche Hilfsmittel benötigt man, welche Rolle spielt die Mathematik und weshalb kann die Erde nicht flach sein? In diesem Kapitel werden genau diese Fragen mithilfe des nautischen Dreiecks, der sphärischen Trigonometrie und einigen Hilfsmitteln und Messgeräten beantwortet. \ No newline at end of file diff --git a/buch/papers/nav/flatearth.tex b/buch/papers/nav/flatearth.tex index bec242e..5bfc1b7 100644 --- a/buch/papers/nav/flatearth.tex +++ b/buch/papers/nav/flatearth.tex @@ -9,19 +9,20 @@ \end{center} \end{figure} -Es gibt heut zu Tage viele Beweise dafür, dass die Erde eine Kugel ist. +Es gibt heutzutage viele Beweise dafür, dass die Erde eine Kugel ist. Die Fotos von unserem Planeten oder die Berichte der Astronauten. -Aber schon vor ca. 2300 Jahren hat Aristotoles bemerkt, dass Schiffe im Horizont verschwinden und die einzige Erklärung dafür die Kugelgestalt der Erde ist oder der Erdschatten bei einer Mondfinsternis immer rund ist. +Aber schon vor ca. 2300 Jahren hat Aristoteles bemerkt, dass Schiffe im Horizont verschwinden und die einzige Erklärung dafür die Kugelgestalt der Erde ist. +Auch der Erdschatten bei einer Mondfinsternis ist immer rund. Eratosthenes konnte etwa 100 Jahre später den Erdumfang berechnen. Er beobachtete, dass die Sonne in Syene mittags im Zenit steht und gleichzeitig in Alexandria unter einem Winkel einfällt. Mithilfe der Trigonometrie konnte er mit dem Abstand der Städte und dem Einfallswinkel den Umfang berechnen. -Auch in der Navigation würden grobe Fehler passieren, wenn man davon ausgeht, dass die Erde eine Scheibe ist. -Man sieht es zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. +Der Kartograph Gerhard Mercator projizierte die Erdkugel auf ein Papier und erstellte so eine winkeltreue Karte. +Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. +Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen. + +Dies sieht man zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. Grönland ist auf der Weltkarte so gross wie Afrika. In der Anwendung Google Earth jedoch ist Grönland etwa so gross wie Algerien. Das liegt daran, das man die 3D – Weltkarte nicht einfach auslegen kann. -Der Kartograph Gerhard Mercator projizierte die Erdkugel auf ein Papier und erstellte so eine winkeltreue Karte. -Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. -Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen. \ No newline at end of file diff --git a/buch/papers/nav/main.tex b/buch/papers/nav/main.tex index 47764e8..e16dc2a 100644 --- a/buch/papers/nav/main.tex +++ b/buch/papers/nav/main.tex @@ -3,7 +3,7 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Spährische Navigation\label{chapter:nav}} +\chapter{Sphärische Navigation\label{chapter:nav}} \lhead{Sphärische Navigation} \begin{refsection} \chapterauthor{Enez Erdem und Marc Kühne} diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex index c1ad38a..c239d64 100644 --- a/buch/papers/nav/nautischesdreieck.tex +++ b/buch/papers/nav/nautischesdreieck.tex @@ -1,22 +1,14 @@ \section{Das Nautische Dreieck} \subsection{Definition des Nautischen Dreiecks} -Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der Himmelskugel. Die Himmelskugel ist eine gedachte Kugel, welche die Erde und dessen Beobachter umgibt und als Rechenfläche für Koordinaten in der Astronomie und Geodäsie dient. -Das nautische Dreieck definiert sich durch folgende Ecken: Zenit, Gestirn und Himmelspol. - Der Zenit ist jener Punkt, der vom Erdmittelpunkt durch denn eigenen Standort an die Himmelskugel verlängert wird. Ein Gestirn ist ein Planet oder ein Fixstern, zu welchen es diverse Jahrbücher mit allen astronomischen Eigenschaften gibt. Der Himmelspol ist der Nordpol an die Himmelskugel projiziert. +Das nautische Dreieck definiert sich durch folgende Ecken: Zenit, Gestirn und Himmelspol. -Zur Anwendung der Formeln der sphärischen Trigonometrie gelten folgende einfache Zusammenhänge: -\begin{itemize} - \item Seitenlänge Zenit zu Himmelspol $= \frac{\pi}{2} - \phi $ - \item Seitenlänge Himmelspol zu Gestirn $= \frac{\pi}{2} - \delta$ - \item Seitenlänge Zenit zu Gestirn $= \frac{\pi}{2} - h$ - \item Winkel von Zenit zu Himmelsnordpol zu Gestirn$=\pi - \alpha$ - \item Winkel von Himmelsnordpol zu Zenit und Gestirn$= \tau$ -\end{itemize} -Um mit diesen Zusammenhängen zu rechnen benötigt man folgende Legende: +Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der Himmelskugel zu bestimmen. + +Für die Definition gilt: \begin{center} \begin{tabular}{ c c c } Winkel && Name / Beziehung \\ @@ -31,6 +23,15 @@ Um mit diesen Zusammenhängen zu rechnen benötigt man folgende Legende: \end{tabular} \end{center} +\begin{itemize} + \item Seitenlänge Zenit zu Himmelspol $= \frac{\pi}{2} - \phi $ + \item Seitenlänge Himmelspol zu Gestirn $= \frac{\pi}{2} - \delta$ + \item Seitenlänge Zenit zu Gestirn $= \frac{\pi}{2} - h$ + \item Winkel von Zenit zu Himmelsnordpol zu Gestirn$=\pi - \alpha$ + \item Winkel von Himmelsnordpol zu Zenit und Gestirn$= \tau$ +\end{itemize} + + \subsection{Zusammenhang des nautischen Dreiecks und des Kugeldreiecks auf der Erdkugel} \begin{figure} \begin{center} @@ -39,15 +40,13 @@ Um mit diesen Zusammenhängen zu rechnen benötigt man folgende Legende: \end{center} \end{figure} -Wie man im oberen Bild sieht, liegt das nautische Dreieck auf der Himmelskugel mit den Ecken Zenit, Gestirn und Himmelsnordpol. -Das selbe Dreieck kann man aber auch auf die Erdkugel projizieren und es hat dann die Ecken Standort, Bildpunkt und Nordpol. -Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. - +Wie man in der Abbildung 21.4 sieht, liegt das nautische Dreieck auf der Himmelskugel mit den Ecken Zenit, Gestirn und Himmelsnordpol. +Das selbe Dreieck kann man aber auch auf die Erdkugel projizieren. Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. Die Projektion auf der Erdkugel hat die Ecken Nordpol, Standort und Bildpunkt. \section{Standortbestimmung ohne elektronische Hilfsmittel} -Um den eigenen Standort herauszufinden, wird in diesem Kapitel die Projektion Nautische Dreieck auf der Erdkugel zur Hilfe genommen. -Mithilfe einiger Hilfsmittel und der Sphärischen Trigonometrie kann man dann die Längen- und Breitengrade des eigenen Standortes bestimmen. - +Um den eigenen Standort herauszufinden, wird in diesem Kapitel die Projektion des nautische Dreiecks auf die Erdkugel zur Hilfe genommen. +Mithilfe eines Sextanten, einem Jahrbuch und der sphärischen Trigonometrie kann man dann die Längen- und Breitengrade des eigenen Standortes bestimmen. +Was ein Sextant und ein Jahrbuch ist, wird im Kapitel 21.6 erklärt. \begin{figure} \begin{center} \includegraphics[width=10cm]{papers/nav/bilder/dreieck.pdf} @@ -60,31 +59,30 @@ Mithilfe einiger Hilfsmittel und der Sphärischen Trigonometrie kann man dann di \subsection{Ecke $P$ und $A$} Unser eigener Standort ist der gesuchte Ecke $P$ und die Ecke $A$ ist in unserem Fall der Nordpol. -Der Vorteil ander Idee des Nautischen Dreiecks ist, dass eine Ecke immer der Nordpol ist. +Der Vorteil ander Idee des nautischen Dreiecks ist, dass eine Ecke immer der Nordpol ist. Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so simpel. -\subsection{Ecke $B$ und $C$ - Bildpunkt X und Y} +\subsection{Ecke $B$ und $C$ - Bildpunkt $X$ und $Y$} Für die Standortermittlung benötigt man als weiteren Punkt ein Gestirn bzw. seinen Bildpunkt auf der Erdkugel. Damit das trigonometrische Rechnen einfacher wird, werden hier zwei Gestirne zur Hilfe genommen. Es gibt diverse Gestirne, die man nutzen kann wie zum Beispiel die Sonne, der Mond oder die vier Navigationsplaneten Venus, Mars, Jupiter und Saturn. +Die Bildpunkte von den beiden Gestirnen $X$ und $Y$ bilden die beiden Ecken $B$ und $C$ im Dreieck der Abbildung 21.5. \subsection{Ephemeriden} Zu all diesen Gestirnen gibt es Ephemeriden, die man auch Jahrbücher nennt. In diesen findet man Begriffe wie Rektaszension, Deklination und Sternzeit. -Da diese Angaben in Stundenabständen gegeben sind, muss man für die minutengenaue Bestimmung zwischen den Stunden interpolieren. -Was diese Begriffe bedeuten, wird in den kommenden beiden Abschnitten erklärt. \begin{figure} \begin{center} - \includegraphics[width=18cm]{papers/nav/bilder/ephe.png} - \caption[Astrodienst - Ephemeriden Januar 2022]{Astrodienst - Ephemeriden Januar 2022} + \includegraphics[width=\textwidth]{papers/nav/bilder/ephe.png} + \caption[Nautical Almanac Mai 2002]{Nautical Almanac Mai 2002} \end{center} \end{figure} \subsubsection{Deklination} -Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und ergibt schlussendlich den Breitengrad. +Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und entspricht dem Breitengrad des Gestirns. -\subsubsection{Sternzeit und Rektaszension} +\subsubsection{Rektaszension und Sternzeit} Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt steht und geht vom Koordinatensystem der Himmelskugel aus. Der Frühlungspunkt ist der Nullpunkt auf dem Himmelsäquator. Die Tatsache, dass sich die Himmelskugel ca. vier Minuten schneller um die eigene Achse dreht als die Erdkugel, stellt hier ein kleines Problem dar. @@ -98,19 +96,28 @@ Für die Standortermittlung auf der Erdkugel ist es am einfachsten, wenn man die Für die Sternzeit von Greenwich $\theta $braucht man als erstes das Julianische Datum $T$ vom aktuellen Tag, welches sich leicht recherchieren lässt. Im Anschluss berechnet man die Sternzeit von Greenwich -$\theta = 6^h 41^m 50^s,54841 + 8640184^s,812866 \cdot T + 0^s,093104 \cdot T^2 - 0^s,0000062 \cdot T^3$. +\[\theta = 6^h 41^m 50^s,54841 + 8640184^s,812866 \cdot T + 0^s,093104 \cdot T^2 - 0^s,0000062 \cdot T^3.\] -Wenn mann die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ mithilfe der Rektaszension und Sternzeit von Greenwich bestimmen. +Wenn man die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ mithilfe der Rektaszension und Sternzeit von Greenwich bestimmen. Dies gilt analog auch für das zweite Gestirn. +\subsubsection{Sextant} +Ein Sextant ist ein nautisches Messinstrument, mit dem man den Winkel zwischen der Blickrichtung zu weit entfernten Objekten bestimmen kann, insbesondere den Winkelabstand zu einem Gestirn vom Horizont. Man nutze ihn vor allem für die astronomische Navigation auf See. -\subsection{Bestimmung des eigenen Standortes P} +\begin{figure} + \begin{center} + \includegraphics[width=10cm]{papers/nav/bilder/sextant.jpg} + \caption[Sextant]{Sextant} + \end{center} +\end{figure} + +\subsection{Bestimmung des eigenen Standortes $P$} Nun hat man die Koordinaten der beiden Gestirne und man weiss die Koordinaten des Nordpols. Damit wir unseren Standort bestimmen können, bilden wir zuerst das Dreieck $ABC$, dann das Dreieck $BPC$ und zum Schluss noch das Dreieck $ABP$. Mithilfe dieser Dreiecken können wir die einfachen Sätze der sphärischen Trigonometrie anwenden und benötigen lediglich ein Ephemeride zu den Gestirnen und einen Sextant. \begin{figure} \begin{center} - \includegraphics[width=10cm]{papers/nav/bilder/dreieck.pdf} + \includegraphics[width=8cm]{papers/nav/bilder/dreieck.pdf} \caption[Dreieck für die Standortbestimmung]{Dreieck für die Standortbestimmung} \end{center} \end{figure} @@ -128,15 +135,15 @@ Mithilfe dieser Dreiecken können wir die einfachen Sätze der sphärischen Trig \end{tabular} \end{center} -Mithilfe des sphärischen Trigonometrie und den darausfolgenden Zusammenhängen des Nautischen Dreiecks können wir nun alle Seiten des Dreiecks $ABC$ berechnen. +Mit unserem erlangten Wissen können wir nun alle Seiten des Dreiecks $ABC$ berechnen. -Die Seitenlänge der Seite "Nordpol zum Bildpunkt X" sei $c$. +Die Seite vom Nordpol zum Bildpunkt $X$ sei $c$. Dann ist $c = \frac{\pi}{2} - \delta_1$. -Die Seitenlänge der Seite "Nordpol zum Bildpunkt Y" sei $b$. +Die Seite vom Nordpol zum Bildpunkt $Y$ sei $b$. Dann ist $b = \frac{\pi}{2} - \delta_2$. -Der Innenwinkel beim der Ecke "Nordpol" sei $\alpha$. +Der Innenwinkel bei der Ecke, wo der Nordpol ist sei $\alpha$. Dann ist $ \alpha = |\lambda_1 - \lambda_2|$. mit @@ -144,55 +151,49 @@ mit \begin{tabular}{ c c c } Ecke && Name \\ \hline - $\delta_1$ && Deklination Bildpunkt $X$ \\ - $\delta_2$ && Deklination Bildpunk $Y$ \\ - $\lambda_1 $&& Längengrad Bildpunkt $X$\\ - $\lambda_2$ && Längengrad Bildpunkt $Y$ + $\delta_1$ && Deklination vom Bildpunkt $X$ \\ + $\delta_2$ && Deklination vom Bildpunk $Y$ \\ + $\lambda_1 $&& Längengrad vom Bildpunkt $X$\\ + $\lambda_2$ && Längengrad vom Bildpunkt $Y$ \end{tabular} \end{center} -Wichtig ist: Die Differenz der Längengrade ist gleich der Innenwinkel Alpha, deswegen der Betrag! - -Nun haben wir die beiden Seiten $c\ und\ b$ und den Winkel $\alpha$, der sich zwischen diesen Seiten befindet. +Nun haben wir die beiden Seiten $c$ und $b$ und den Winkel $\alpha$, der sich zwischen diesen Seiten befindet. Mithilfe des Seiten-Kosinussatzes $\cos(a) = \cos(b)\cdot \cos(c) + \sin(b) \cdot \sin(c)\cdot \cos(\alpha)$ können wir nun die dritte Seitenlänge bestimmen. Es ist darauf zu achten, dass hier natürlich die Seitenlängen in Bogenmass sind und dementsprechend der Kosinus und Sinus verwendet wird. -Jetzt fehlen noch die beiden anderen Innenwinkel $\beta \ und\ \gamma$. -Diese bestimmen wir mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$. +Jetzt fehlen noch die beiden anderen Innenwinkel $\beta$ und\ $\gamma$. +Diese bestimmen wir mithilfe des Sinussatzes \[\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}.\] Hier muss man aufpassen, dass man Seite von Winkel unterscheiden kann. Im Zähler sind die Seiten, im Nenner die Winkel. -Somit ist $\beta =\sin^{-1} [\sin(b) \cdot \frac{\sin(\alpha)}{\sin(a)}] $. +Somit ist \[\beta =\sin^{-1} [\sin(b) \cdot \frac{\sin(\alpha)}{\sin(a)}].\] -Schlussendlich haben wir die Seiten $a,b\ und \ c$, die Ecken A,B und C und die Winkel $\alpha, \beta \ und \ \gamma$ bestimmt und somit das ganze erste Kugeldreieck berechnet. +Schlussendlich haben wir die Seiten $a,b\ und \ c$, die Ecken A,B und C und die Winkel $\alpha$, $\beta$ und $\gamma$ bestimmt und somit das ganze Kugeldreieck $ABC$ berechnet. \subsubsection{Dreieck $BPC$} -Wir bilden nun ein zweites Dreieck, welches die Ecken B und C des ersten Dreiecks besitzt. -Die dritte Ecke ist der eigene Standort P. +Wir bilden nun ein zweites Dreieck, welches die Ecken $B$ und $C$ des ersten Dreiecks besitzt. +Die dritte Ecke ist der eigene Standort $P$. Unser Standort definiere sich aus einer geographischen Breite $\delta$ und einer geographischen Länge $\lambda$. -Die Seite von P zu B sei $pb$ und die Seite von P zu C sei $pc$. +Die Seite von $P$ zu $B$ sei $pb$ und die Seite von $P$ zu $C$ sei $pc$. Die beiden Seitenlängen kann man mit dem Sextant messen und durch eine einfache Formel bestimmen, nämlich $pb=\frac{\pi}{2} - h_{B}$ und $pc=\frac{\pi}{2} - h_{C}$ -mit $h_B=$ Höhe von Gestirn in B und $h_C=$ Höhe von Gestirn in C mit Sextant gemessen. +mit $h_B=$ Höhe von Gestirn in $B$ und $h_C=$ Höhe von Gestirn in $C$ mit Sextant gemessen. -Zum Schluss müssen wir noch den Winkel $\beta1$ mithilfe des Seiten-Kosinussatzes mit den bekannten Seiten $pc$, $pb$ und $a$ bestimmen. +Zum Schluss müssen wir noch den Winkel $\beta_1$ mithilfe des Seiten-Kosinussatzes \[\cos(pb)=\cos(pc)\cdot\cos(a)+\sin(pc)\cdot\sin(a)\cdot\cos(\beta_1)\] mit den bekannten Seiten $pc$, $pb$ und $a$ bestimmen. \subsubsection{Dreieck $ABP$} -Nun muss man eine Verbindungslinie ziehen zwischen P und A. Die Länge $l$ dieser Linie entspricht der gesuchten geographischen Breite $\delta$. Diese lässt sich mithilfe des Dreiecks $ABP$, den bekannten Seiten $c\ und \ pb$ und des Seiten-Kosinussatzes berechnen. - -Für den Seiten-Kosinussatz benötigt es noch $\kappa=\beta + \beta1$. - -Somit ist $\cos(l) = \cos(c)\cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)$ - +Nun muss man eine Verbindungslinie ziehen zwischen $P$ und $A$. Die Länge $l$ dieser Linie entspricht der gesuchten geographischen Breite $\delta$. Diese lässt sich mithilfe des Dreiecks $ABP$, den bekannten Seiten $c$ und $pb$ und des Seiten-Kosinussatzes berechnen. +Für den Seiten-Kosinussatz benötigt es noch $\kappa=\beta + \beta_1$. +Somit ist \[\cos(l) = \cos(c)\cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)\] und - \[ \delta =\cos^{-1} [\cos(c) \cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)]. \] -Für die Geographische Länge $\lambda$ des eigenen Standortes muss man den Winkel $\omega$, welcher sich im Dreieck $ACP$ in der Ecke bei $A$ befindet mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$ bestimmen. - +Für die Geographische Länge $\lambda$ des eigenen Standortes muss man den Winkel $\omega$, welcher sich im Dreieck $ACP$ in der Ecke bei $A$ befindet. +Mithilfe des Sinussatzes \[\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}\] können wir das bestimmen. Somit ist \[ \omega=\sin^{-1}[\sin(pc) \cdot \frac{\sin(\gamma)}{\sin(l)}] \]und unsere gesuchte geographische Länge schlussendlich \[\lambda=\lambda_1 - \omega\] -mit $\lambda_1$=Längengrad Bildpunkt $X$ +wobei $\lambda_1$ die Länge des Bildpunktes $X$ von $C$ ist. diff --git a/buch/papers/nav/sincos.tex b/buch/papers/nav/sincos.tex index bb7f1e4..d56d482 100644 --- a/buch/papers/nav/sincos.tex +++ b/buch/papers/nav/sincos.tex @@ -6,14 +6,16 @@ Es gibt Hinweise, dass sich schon die Babylonier und Ägypter vor 4000 Jahren si Jedoch konnten sie dieses Problem nicht lösen. Die Geschichte der sphärischen Trigonometrie ist daher eng mit der Astronomie verknüpft. Ca. 350 vor Christus dachten die Griechen über Kugelgeometrie nach und sie wurde zu einer Hilfswissenschaft der Astronomen. -In Folge werden auch die ersten Sätze aufgestellt und wenige Jahrhunderte später wurden Berechnungen mithilfe des Sternkataloges von Hipparchos angestellt und darauffolgend Kartenmaterial erstellt. +Zwischen 190 v. Chr. und 120 v. Chr. lebte ein griechischer Astronom names Hipparchos. +Dieser entwickelte unter anderem die Chordentafeln, welche die Chord - Funktionen, auch Chord genannt, beinhalten. +Chord ist der Vorgänger der Sinusfunktion und galt damals als wichtigste Grundlage der Trigonometrie. In dieser Zeit wurden auch die ersten Sternenkarten angefertigt, jedoch kannte man damals die Sinusfunktion noch nicht. + Aus Indien stammten die ersten Ansätze zu den Kosinussätzen. Aufbauend auf den indischen und griechischen Forschungen entwickeln die Araber um das 9. Jahrhundert den Sinussatz. Doch ein paar weitere Jahrhunderte vergingen bis zu diesem Thema wieder verstärkt Forschung betrieben wurde, da im 15. Jahrhundert grosse Entdeckungsreisen, hauptsächlich per Schiff, erfolgten und die Orientierung mit Sternen vermehrt an Wichtigkeit gewann. Man nutzte für die Kartographie nun die Kugelgeometrie, um die Genauigkeit zu erhöhen. Der Sinussatz, die Tangensfunktion und der neu entwickelte Seitenkosinussatz wurden in dieser Zeit bereits verwendet und im darauffolgenden Jahrhundert folgte der Winkelkosinussatz. - Durch weitere mathematische Entwicklungen wie den Logarithmus wurden im Laufe des nächsten Jahrhunderts viele neue Methoden und kartographische Anwendungen der Kugelgeometrie entdeckt. Im 19. und 20. Jahrhundert wurden weitere nicht-euklidische Geometrien entwickelt und die sphärische Trigonometrie fand auch ihre Anwendung in der Relativitätstheorie. \ No newline at end of file diff --git a/buch/papers/nav/trigo.tex b/buch/papers/nav/trigo.tex index cf2f242..ce367f6 100644 --- a/buch/papers/nav/trigo.tex +++ b/buch/papers/nav/trigo.tex @@ -2,33 +2,35 @@ \section{Sphärische Trigonometrie} In der sphärischen Trigonometrie gibt es eine Symetrie zwischen Seiten und Winkel, also zu jedem Satz über Seiten und Winkel gibt es einen entsprechenden Satz, mit dem man Winkel durch Seiten und Seiten durch Winkel ersetzt hat. Dabei gibt es folgenden Zusammenhang zwischen der ebenen- und sphärischen Trigonometrie: -\begin{center} - - -\begin{tabular}{ccc} - Eben & $\leftrightarrow$ & sphärisch \\ - \hline - $a$ & $\leftrightarrow$ & $\sin \ a$ \\ - - $a^2$ & $\leftrightarrow$ & $-\cos \ a$ \\ -\end{tabular} -\end{center} \subsection{Das Kugeldreieck} +Damit man die Definition des Kugeldreiecks versteht, müssen wir zuerst Begriffe wie "Grosskreisebene" und "Grosskreisbögen" verstehen. +Ein Grosskreis ist ein größtmöglicher Kreis auf einer Kugeloberfläche. +Sein Mittelpunkt fällt immer mit dem Mittelpunkt der Kugel zusammen und ein Schnitt auf dem Großkreis teilt die Kugel in jedem Fall in zwei gleich grosse Hälften. +Da es unendlich viele Möglichkeiten gibt, eine Kugel so zu zerschneiden, dass die Schnittebene den Kugelmittelpunkt trifft, gibt es auch unendlich viele Grosskreise. +Grosskreisbögen sind die Verbindungslinien zwischen zwei Punkten auf der Kugel, welche auch "Seiten" eines Kugeldreiecks gennant werden. Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden, so entsteht ein Kugeldreieck $ABC$. Für ein Kugeldreieck gilt, dass die Summe der drei Seiten kleiner als $2\pi$ aber grösser als 0 ist. -$A$, $B$ und $C$ sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten. -Ein Grosskreis ist ein größtmöglicher Kreis auf einer Kugeloberfläche. -Sein Mittelpunkt fällt immer mit dem Mittelpunkt der Kugel zusammen und ein Schnitt auf dem Großkreis teilt die Kugel in jedem Fall in zwei gleich grosse Hälften. +$A$, $B$ und $C$ sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten (siehe Abbildung 21.2). -Da es unendlich viele Möglichkeiten gibt, eine Kugel so zu zerschneiden, dass die Schnittebene den Kugelmittelpunkt trifft, gibt es auch unendlich viele Grosskreise. Da die Länge der Grosskreisbögen wegen der Abhängigkeit vom Kugelradius ungeeignet ist, wird die Grösse einer Seite mit dem zugehörigen Mittelpunktwinkel des Grosskreisbogens angegeben. -Laut dieser Definition ist die Seite $c$ der Winkel $AMB$. +Laut dieser Definition ist die Seite $c$ der Winkel $AMB$, wobei der Punkt $M$ die Erdmitte ist. Man kann bei Kugeldreiecken nicht so einfach unterscheiden, was Innen oder Aussen ist. Wenn man drei Eckpunkte miteinander verbindet, ergeben sich immer 16 Kugeldreiecke. -Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha, \beta, \gamma < \pi$ nennt man Eulersche Dreiecke. +Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha, \beta, \gamma < \pi$ nennt man Eulersches Dreieck. + +Es gibt einen Zusammenhang zwischen der ebenen- und sphärischen Trigonometrie, wobei folgend $a$ eine Seite beschreibt: +\begin{center} + \begin{tabular}{ccc} + Eben & $\leftrightarrow$ & sphärisch \\ + \hline + $a$ & $\leftrightarrow$ & $\sin \ a$ \\ + + $a^2$ & $\leftrightarrow$ & $-\cos \ a$ \\ + \end{tabular} +\end{center} \begin{figure} \begin{center} @@ -38,9 +40,16 @@ Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha \end{figure} -\subsection{Rechtwinkliges Dreieck und Rechtseitiges Dreieck} -Wie auch im uns bekannten Dreieck gibt es beim Kugeldreieck auch ein Rechtwinkliges Kugeldreieck, bei dem ein Winkel $\frac{\pi}{2}$ ist. +\subsection{Rechtwinkliges Dreieck und rechtseitiges Dreieck} +Wie auch im ebenen Dreieck gibt es beim Kugeldreieck auch ein rechtwinkliges Kugeldreieck, bei dem ein Winkel $\frac{\pi}{2}$ ist. Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss. +\begin{figure} + + \begin{center} + \includegraphics[width=10cm]{papers/nav/bilder/recht.jpg} + \caption[Rechtseitiges Kugeldreieck]{Rechtseitiges Kugeldreieck} + \end{center} +\end{figure} \subsection{Winkelsumme} \begin{figure} @@ -55,8 +64,9 @@ Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine S Die Winkel eines Kugeldreiecks sind die, welche die Halbtangenten in den Eckpunkten einschliessen. Für die Summe der Innenwinkel gilt \begin{align} - \alpha+\beta+\gamma &= \frac{A}{r^2} + \pi \ \text{und} \ \alpha+\beta+\gamma > \pi. \nonumber + \alpha+\beta+\gamma &= \frac{F}{r^2} + \pi \ \text{und} \ \alpha+\beta+\gamma > \pi, \nonumber \end{align} +wobei F der Flächeninhalt des Kugeldreiecks ist. \subsubsection{Sphärischer Exzess} Der sphärische Exzess \begin{align} @@ -65,31 +75,31 @@ Der sphärische Exzess beschreibt die Abweichung der Innenwinkelsumme von $\pi$ und ist proportional zum Flächeninhalt des Kugeldreiecks. \subsubsection{Flächeninnhalt} -Der Flächeninhalt $A$ lässt sich aus den Winkeln $\alpha,\ \beta, \ \gamma$ und dem Kugelradius $r$ berechnen. +Mithilfe des Radius $r$ und dem sphärischen Exzess $\epsilon$ gilt für den Flächeninhalt +\[ F=\frac{\pi \cdot r^2}{\frac{\pi}{2}} \cdot \epsilon\]. \subsection{Sphärischer Sinussatz} In jedem Dreieck ist das Verhältnis des Sinus einer Seite zum Sinus des Gegenwinkels konstant. - Das bedeutet, dass \begin{align} - \frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} \nonumber \ \text{auch beim Kugeldreieck gilt.} + \frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} \nonumber \end{align} +auch beim Kugeldreieck gilt. -\subsection{Sphärischer Kosinussätze} +\subsection{Sphärische Kosinussätze} Auch in der sphärischen Trigonometrie gibt es den Seitenkosinussatz \begin{align} - cos \ a = \cos b \cdot \cos c + \sin b \cdot \sin c \cdot \cos \alpha \nonumber + \cos \ a = \cos b \cdot \cos c + \sin b \cdot \sin c \cdot \cos \alpha \nonumber \end{align} %Seitenkosinussatz und den Winkelkosinussatz \begin{align} - \cos \gamma = -\cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \cdot \cos c \nonumber + \cos \gamma = -\cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \cdot \cos c. \nonumber \end{align} \subsection{Sphärischer Satz des Pythagoras für das rechtwinklige Kugeldreieck} Es gibt in der sphärischen Trigonometrie eigentlich garkeinen "Satz des Pythagoras", wie man ihn aus der zweidimensionalen Geometrie kennt. -In der sphärischen Trigonometrie gibt es aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks in eine Beziehung bringt. - +In der sphärischen Trigonometrie gibt es aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks, nicht aber für das rechtseitige Kugeldreieck, in eine Beziehung bringt. Es gilt nämlich: \begin{align} \cos c = \cos a \cdot \cos b \ \text{wenn} \nonumber & -- cgit v1.2.1 From ab62c3937cc111ce1d61d76f0bdf396a4a5a9297 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 21 May 2022 20:36:01 +0200 Subject: add image code --- buch/papers/kugel/images/Makefile | 4 +- buch/papers/kugel/images/curvature.maxima | 6 ++ buch/papers/kugel/images/curvature.pov | 72 +++++++++++++++++++++++- buch/papers/kugel/images/curvgraph.m | 93 +++++++++++++++++++++++++------ 4 files changed, 153 insertions(+), 22 deletions(-) create mode 100644 buch/papers/kugel/images/curvature.maxima (limited to 'buch') diff --git a/buch/papers/kugel/images/Makefile b/buch/papers/kugel/images/Makefile index 8efa228..e8bf919 100644 --- a/buch/papers/kugel/images/Makefile +++ b/buch/papers/kugel/images/Makefile @@ -3,7 +3,7 @@ # # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -all: curvature.png +all: curvature.jpg curvature.inc: curvgraph.m octave curvgraph.m @@ -11,3 +11,5 @@ curvature.inc: curvgraph.m curvature.png: curvature.pov curvature.inc povray +A0.1 +W1920 +H1080 +Ocurvature.png curvature.pov +curvature.jpg: curvature.png + convert curvature.png -density 300 -units PixelsPerInch curvature.jpg diff --git a/buch/papers/kugel/images/curvature.maxima b/buch/papers/kugel/images/curvature.maxima new file mode 100644 index 0000000..6313642 --- /dev/null +++ b/buch/papers/kugel/images/curvature.maxima @@ -0,0 +1,6 @@ + +f: exp(-r^2/sigma^2)/sigma; +laplacef: ratsimp(diff(r * diff(f,r), r) / r); +f: exp(-r^2/(2*sigma^2))/(sqrt(2)*sigma); +laplacef: ratsimp(diff(r * diff(f,r), r) / r); + diff --git a/buch/papers/kugel/images/curvature.pov b/buch/papers/kugel/images/curvature.pov index 3535488..9dbaa86 100644 --- a/buch/papers/kugel/images/curvature.pov +++ b/buch/papers/kugel/images/curvature.pov @@ -11,17 +11,17 @@ global_settings { assumed_gamma 1 } -#declare imagescale = 0.1; +#declare imagescale = 0.09; camera { - location <40, 10, -20> + location <10, 10, -40> look_at <0, 0, 0> right 16/9 * x * imagescale up y * imagescale } light_source { - <10, 10, -40> color White + <-10, 10, -40> color White area_light <1,0,0> <0,0,1>, 10, 10 adaptive 1 jitter @@ -70,3 +70,69 @@ arrow(<0,-1,0>, <0,1,0>, 0.01, White) arrow(<0,0,-2.1>, <0,0,2.1>, 0.01, White) #include "curvature.inc" + +#declare sigma = 1; +#declare N0 = 0.5; +#declare funktion = function(r) { + (exp(-r*r/(sigma*sigma)) / sigma + - + exp(-r*r/(2*sigma*sigma)) / (sqrt(2)*sigma)) / N0 +}; +#declare hypot = function(xx, yy) { sqrt(xx*xx+yy*yy) }; + +#declare Funktion = function(x,y) { funktion(hypot(x+1,y)) - funktion(hypot(x-1,y)) }; +#macro punkt(xx,yy) + +#end + +#declare griddiameter = 0.006; +union { + #declare xmin = -3; + #declare xmax = 3; + #declare ymin = -2; + #declare ymax = 2; + + + #declare xstep = 0.2; + #declare ystep = 0.02; + #declare xx = xmin; + #while (xx < xmax + xstep/2) + #declare yy = ymin; + #declare P = punkt(xx, yy); + #while (yy < ymax - ystep/2) + #declare yy = yy + ystep; + #declare Q = punkt(xx, yy); + sphere { P, griddiameter } + cylinder { P, Q, griddiameter } + #declare P = Q; + #end + sphere { P, griddiameter } + #declare xx = xx + xstep; + #end + + #declare xstep = 0.02; + #declare ystep = 0.2; + #declare yy = ymin; + #while (yy < ymax + ystep/2) + #declare xx = xmin; + #declare P = punkt(xx, yy); + #while (xx < xmax - xstep/2) + #declare xx = xx + xstep; + #declare Q = punkt(xx, yy); + sphere { P, griddiameter } + cylinder { P, Q, griddiameter } + #declare P = Q; + #end + sphere { P, griddiameter } + #declare yy = yy + ystep; + #end + + pigment { + color rgb<0.8,0.8,0.8> + } + finish { + metallic + specular 0.8 + } +} + diff --git a/buch/papers/kugel/images/curvgraph.m b/buch/papers/kugel/images/curvgraph.m index 96ca4b1..b83c877 100644 --- a/buch/papers/kugel/images/curvgraph.m +++ b/buch/papers/kugel/images/curvgraph.m @@ -13,23 +13,34 @@ sigma2 = 1; global s; s = 1; +global cmax; +cmax = 0.9; +global cmin; +cmin = -0.9; + +global Cmax; +global Cmin; +Cmax = 0; +Cmin = 0; + xmin = -3; xmax = 3; -xsteps = 1000; +xsteps = 200; hx = (xmax - xmin) / xsteps; ymin = -2; ymax = 2; -ysteps = 1000; +ysteps = 200; hy = (ymax - ymin) / ysteps; function retval = f0(r) global sigma2; - retval = exp(-r^2/sigma2)/sigma2 - exp(-r^2/(2*sigma2))/(sqrt(2)*sigma2); + retval = exp(-r^2/sigma2)/sqrt(sigma2) - exp(-r^2/(2*sigma2))/(sqrt(2*sigma2)); end global N0; -N0 = f0(0); +N0 = f0(0) +N0 = 0.5; function retval = f1(x,y) global N0; @@ -44,10 +55,10 @@ endfunction function retval = curvature0(r) global sigma2; retval = ( - (2*sigma2-r^2)*exp(-r^2/(2*sigma2)) + -4*(sigma2-r^2)*exp(-r^2/sigma2) + - 4*(r^2-sigma2)*exp(-r^2/sigma2) - ) / (sigma2^2); + (2*sigma2-r^2)*exp(-r^2/(2*sigma2)) + ) / (sigma2^(5/2)); endfunction function retval = curvature1(x, y) @@ -56,28 +67,74 @@ endfunction function retval = curvature(x, y) global s; - retval = curvature1(x+s, y) + curvature1(x-s, y); + retval = curvature1(x+s, y) - curvature1(x-s, y); endfunction function retval = farbe(x, y) + global Cmax; + global Cmin; + global cmax; + global cmin; c = curvature(x, y); - retval = c * ones(1,3); + if (c < Cmin) + Cmin = c + endif + if (c > Cmax) + Cmax = c + endif + u = (c - cmin) / (cmax - cmin); + if (u > 1) + u = 1; + endif + if (u < 0) + u = 0; + endif + color = [ u, 0.5, 1-u ]; + color = color/max(color); + color(1,4) = c/2; + retval = color; endfunction -fn = fopen("curvature.inc", "w"); +function dreieck(fn, A, B, C) + fprintf(fn, "\ttriangle {\n"); + fprintf(fn, "\t <%.4f,%.4f,%.4f>,\n", A(1,1), A(1,3), A(1,2)); + fprintf(fn, "\t <%.4f,%.4f,%.4f>,\n", B(1,1), B(1,3), B(1,2)); + fprintf(fn, "\t <%.4f,%.4f,%.4f>\n", C(1,1), C(1,3), C(1,2)); + fprintf(fn, "\t}\n"); +endfunction +function viereck(fn, punkte) + color = farbe(mean(punkte(:,1)), mean(punkte(:,2))); + fprintf(fn, " mesh {\n"); + dreieck(fn, punkte(1,:), punkte(2,:), punkte(3,:)); + dreieck(fn, punkte(2,:), punkte(3,:), punkte(4,:)); + fprintf(fn, "\tpigment { color rgb<%.4f,%.4f,%.4f> } // %.4f\n", + color(1,1), color(1,2), color(1,3), color(1,4)); + fprintf(fn, " }\n"); +endfunction + +fn = fopen("curvature.inc", "w"); +punkte = zeros(4,3); for ix = (0:xsteps) x = xmin + ix * hx; + punkte(1,1) = x; + punkte(2,1) = x; + punkte(3,1) = x + hx; + punkte(4,1) = x + hx; for iy = (0:ysteps) y = ymin + iy * hy; - fprintf(fn, "sphere { <%.4f, %.4f, %.4f>, 0.01\n", - x, f(x, y), y); - color = farbe(x, y); - fprintf(fn, "pigment { color rgb<%.4f,%.4f,%.4f> }\n", - color(1,1), color(1,2), color(1,3)); - fprintf(fn, "finish { metallic specular 0.5 }\n"); - fprintf(fn, "}\n"); + punkte(1,2) = y; + punkte(2,2) = y + hy; + punkte(3,2) = y; + punkte(4,2) = y + hy; + for i = (1:4) + punkte(i,3) = f(punkte(i,1), punkte(i,2)); + endfor + viereck(fn, punkte); end end - +#fprintf(fn, " finish { metallic specular 0.5 }\n"); fclose(fn); + +printf("Cmax = %.4f\n", Cmax); +printf("Cmin = %.4f\n", Cmin); -- cgit v1.2.1 From d8d6a61a2ab45d9171a93e4a72d254a3ed5ef87f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 21 May 2022 20:42:47 +0200 Subject: fix some bugs --- buch/papers/kugel/images/curvature.pov | 5 +++-- buch/papers/kugel/images/curvgraph.m | 8 ++++---- 2 files changed, 7 insertions(+), 6 deletions(-) (limited to 'buch') diff --git a/buch/papers/kugel/images/curvature.pov b/buch/papers/kugel/images/curvature.pov index 9dbaa86..3b15d77 100644 --- a/buch/papers/kugel/images/curvature.pov +++ b/buch/papers/kugel/images/curvature.pov @@ -72,7 +72,8 @@ arrow(<0,0,-2.1>, <0,0,2.1>, 0.01, White) #include "curvature.inc" #declare sigma = 1; -#declare N0 = 0.5; +#declare s = 1.4; +#declare N0 = 0.4; #declare funktion = function(r) { (exp(-r*r/(sigma*sigma)) / sigma - @@ -80,7 +81,7 @@ arrow(<0,0,-2.1>, <0,0,2.1>, 0.01, White) }; #declare hypot = function(xx, yy) { sqrt(xx*xx+yy*yy) }; -#declare Funktion = function(x,y) { funktion(hypot(x+1,y)) - funktion(hypot(x-1,y)) }; +#declare Funktion = function(x,y) { funktion(hypot(x+s,y)) - funktion(hypot(x-s,y)) }; #macro punkt(xx,yy) #end diff --git a/buch/papers/kugel/images/curvgraph.m b/buch/papers/kugel/images/curvgraph.m index b83c877..75effd6 100644 --- a/buch/papers/kugel/images/curvgraph.m +++ b/buch/papers/kugel/images/curvgraph.m @@ -11,7 +11,7 @@ global sigma2; sigma2 = 1; global s; -s = 1; +s = 1.4; global cmax; cmax = 0.9; @@ -40,7 +40,7 @@ end global N0; N0 = f0(0) -N0 = 0.5; +N0 = 0.4; function retval = f1(x,y) global N0; @@ -115,13 +115,13 @@ endfunction fn = fopen("curvature.inc", "w"); punkte = zeros(4,3); -for ix = (0:xsteps) +for ix = (0:xsteps-1) x = xmin + ix * hx; punkte(1,1) = x; punkte(2,1) = x; punkte(3,1) = x + hx; punkte(4,1) = x + hx; - for iy = (0:ysteps) + for iy = (0:ysteps-1) y = ymin + iy * hy; punkte(1,2) = y; punkte(2,2) = y + hy; -- cgit v1.2.1 From 53aea87685ea9f37f982f1ec90a82ce168d6d7cb Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 23 May 2022 11:34:57 +0200 Subject: rewriting the risch algorithm stuff --- buch/chapters/060-integral/Makefile.inc | 7 + buch/chapters/060-integral/differentialkoerper.tex | 1957 +------------------- .../chapters/060-integral/differentialkoerper2.tex | 1953 +++++++++++++++++++ buch/chapters/060-integral/diffke.tex | 20 + buch/chapters/060-integral/elementar.tex | 7 + buch/chapters/060-integral/erweiterungen.tex | 12 + buch/chapters/060-integral/iproblem.tex | 93 + buch/chapters/060-integral/irat.tex | 140 ++ buch/chapters/060-integral/logexp.tex | 27 + buch/chapters/060-integral/rational.tex | 8 + buch/chapters/060-integral/risch.tex | 3 +- buch/chapters/060-integral/sqrat.tex | 8 + 12 files changed, 2286 insertions(+), 1949 deletions(-) create mode 100644 buch/chapters/060-integral/differentialkoerper2.tex create mode 100644 buch/chapters/060-integral/diffke.tex create mode 100644 buch/chapters/060-integral/elementar.tex create mode 100644 buch/chapters/060-integral/erweiterungen.tex create mode 100644 buch/chapters/060-integral/iproblem.tex create mode 100644 buch/chapters/060-integral/irat.tex create mode 100644 buch/chapters/060-integral/logexp.tex create mode 100644 buch/chapters/060-integral/rational.tex create mode 100644 buch/chapters/060-integral/sqrat.tex (limited to 'buch') diff --git a/buch/chapters/060-integral/Makefile.inc b/buch/chapters/060-integral/Makefile.inc index d85caad..e0dfc21 100644 --- a/buch/chapters/060-integral/Makefile.inc +++ b/buch/chapters/060-integral/Makefile.inc @@ -8,5 +8,12 @@ CHAPTERFILES += \ chapters/060-integral/fehlerfunktion.tex \ chapters/060-integral/eulertransformation.tex \ chapters/060-integral/differentialkoerper.tex \ + chapters/060-integral/rational.tex \ + chapters/060-integral/erweiterungen.tex \ + chapters/060-integral/diffke.tex \ + chapters/060-integral/irat.tex \ + chapters/060-integral/sqratrat.tex \ chapters/060-integral/risch.tex \ + chapters/060-integral/logexp.tex \ + chapters/060-integral/elementar.tex \ chapters/060-integral/chapter.tex diff --git a/buch/chapters/060-integral/differentialkoerper.tex b/buch/chapters/060-integral/differentialkoerper.tex index f41d3ba..66bb0c1 100644 --- a/buch/chapters/060-integral/differentialkoerper.tex +++ b/buch/chapters/060-integral/differentialkoerper.tex @@ -1,1953 +1,14 @@ % -% differentialalgebren.tex +% differentialkoerper.tex % % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % -\section{Differentialkörper und der Satz von Liouville +\section{Differentialkörper \label{buch:integrale:section:dkoerper}} -\rhead{Differentialkörper und der Satz von Liouville} -Das Problem der Darstellbarkeit eines Integrals in geschlossener -Form verlangt zunächst einmal nach einer Definition dessen, was man -als ``geschlossene Form'' akzeptieren will. -Die sogenannten {\em elementaren Funktionen} von -Abschnitt~\ref{buch:integrale:section:elementar} -bilden dafür den theoretischen Rahmen. -Das Problem ist dann die Frage zu beantworten, ob ein Integral eine -Stammfunktion hat, die eine elementare Funktion ist. -Der Satz von Liouville von Abschnitt~\ref{buch:integrale:section:liouville} -löst das Problem. - -\subsection{Eine Analogie -\label{buch:integrale:section:analogie}} -% XXX Analogie: Formel für Polynom-Nullstellen -% XXX Stammfunktion als elementare Funktion -Das Analysis-Problem, eine Stammfunktion zu finden, ist analog zum -wohlbekannten algebraischen Problem, Nullstellen von Polynomen zu finden. -Wir entwickeln diese Analogie in etwas mehr Detail, um zu sehen, ob man -aus dem algebraischen Problem etwas über das Problem der Analysis -lernen kann. - -Für ein Polynom $p(X) = a_nX^n+a_{n-1}X^{n-1}+\dots+a_1X+a_0\in\mathbb{C}[X]$ -mit Koeffizienten $a_k\in\mathbb{C}$ ist es sehr einfach, für jede beliebige -komplexe Zahl $z\in\mathbb{C}$ den Wert $p(z)$ des Polynoms auszurechnen. -Ein paar wenige Rechenregeln genügen dazu, man kann leicht einem Kind -beibringen, mit einem Taschenrechner so einen Wert auszurechnen. - -Ähnlich sieht es mit der Ableitungsoperation aus. -Einige wenige Ableitungsregeln, die man in der Analysis~I lernt, -erlauben, auf mehr oder weniger mechanische Art und Weise, jede -beliebige Funktion abzuleiten. -Man kann auch leicht einen Computer dazu programmieren, solche Ableitungen -symbolisch zu berechnen. - -Aus dem Fundamentalsatz der Algebra, der von Gauss vollständig bewiesen -wurde, ist bekannt, dass jedes Polynom mit Koeffizienten in $\mathbb{C}$ -genau so viele Lösungen in $\mathbb{C}$, wie der Grad des Polynoms angibt. -Dies ist aber ein Existenzsatz, er sagt nichts darüber aus, wie man diese -Lösungen finden kann. -In Spezialfällen, wie zum Beispiel für quadratische Polynome, gibt -es spezialsierte Lösungsverfahren, mit denen man Lösungen angeben kann. -Natürlich existieren numerische Methoden wie zum Beispiel das -Newton-Verfahren, mit dem man Nullstellen von Polynomen beliebig genau -bestimmen kann. - -Der Fundamentalsatz der Integralrechnung besagt, dass jede stetige -Funktion eine Stammfunktion hat, die bis auf eine Konstante eindeutig -bestimmt ist. -Auch dieser Existenzsatz gibt keinerlei Hinweise darauf, wie man die -Stammfunktion finden kann. -In der Analysis-Vorlesung lernt man viele Tricks, die in einer -beindruckenden Zahl von Spezialfällen ermöglichen, ein passende -Funktion anzugeben. -Man lernt auch numerische Verfahren kennen, mit denen sich Werte der -Stammfunktion, also bestimmte Integrale, mit beliebiger Genauigkeit -finden kann. - -Die numerische Lösung des Nullstellenproblems ist insofern unbefriedigend, -als sie nur schwer eine Diskussion der Abhängigkeit der Nullstellen von -den Koeffizienten des Polynoms ermöglichen. -Eine Formel wie die Lösungsformel für die quadratische Gleichung -stellt genau für solche Fälle ein ideales Werkzeug bereit. -Was man sich also wünscht ist nicht nur einfach eine Lösung, sondern eine -einfache Formel zur Bestimmung aller Lösungen. -Im Zusammenhang mit algebraischen Gleichungen erwartet man eine Formel, -in der nur arithmetische Operationen und Wurzeln vorkommen. -Für quadratische Gleichungen ist so eine Formel seit dem Altertum bekannt, -Formeln für die kubische Gleichung und die Gleichung vierten Grades wurden -im 16.~Jahrhundert von Cardano bzw.~Ferrari gefunden. -Erst viel später haben Abel und Ruffini gezeigt, dass so eine allgemeine -Formel für Polynome höheren Grades als 4 nicht existiert. -Die Galois-Theorie, die auf den Ideen von Évariste Galois beruht, -stellt eine vollständige Theorie unter anderem für die Lösbarkeit -von Gleichungen durch Wurzelausdrücke dar. - -Numerische Integralwerte haben ebenfalls den Nachteil, dass damit -Diskussionen wie die Abhängigkeit von Parametern eines Integranden -nur schwer möglich sind. -Was man sich daher wünscht ist eine Formel für die Stammfunktion, -die Werte als Zusammensetzung gut bekannter Funktionen wie der Exponential- -und Logarithmus-Funktionen oder der trigonometrischen Funktionen -sowie Wurzeln, Potenzen und den arithmetischen Operationen. -Man sagt, man möchte die Stammfunktion in ``geschlossener Form'' -dargestellt haben. -Tatsächlich ist dieses Problem auch zu Beginn des 19.~Jahrhunderts -von Joseph Liouville genauer untersucht worden. -Er hat zunächst eine Klasse von ``elementaren Funktionen'' definiert, -die als Darstellungen einer Stammfunktion in Frage kommen. -Der Satz von Liouville besagt dann, dass nur Funktionen mit einer -ganz speziellen Form eine elementare Stammfunktion haben. -Damit wird es möglich, zu entscheiden, ob ein Integrand wie $e^{-x^2}$ -eine elementare Stammfunktion hat. -Seit dieser Zeit weiss man zum Beispiel, dass die Fehlerfunktion nicht -mit den bekannten Funktionen dargestellt werden kann. - -Mit dem Aufkommen der Computer und vor allem der Computer-Algebra-System (CAS) -wurde die Frage nach der Bestimmung einer Stammfunktion erneut aktuell. -Die ebenfalls weiter entwickelte abstrakte Algebra hat ermöglicht, die -Ideen von Liouville in eine erweiterte, sogenannte differentielle -Galois-Theorie zu verpacken, die eine vollständige Lösung des Problems -darstellt. -Robert Henry Risch hat in den Sechzigerjahren auf dieser Basis -einen Algorithmus entwickelt, mit dem es möglich wird, zu entscheiden, -ob eine Funktion eine elementare Stammfunktion hat und diese -gegebenenfalls auch zu finden. -Moderne CAS implementieren diesen Algorithmus -in Teilen, besonders weit zu gehen scheint das quelloffene System -Axiom. - -Der Risch-Algorithmus hat allerdings eine Achillesferse: er benötigt -eine Method zu entscheiden, ob zwei Ausdrücke übereinstimmen. -Dies ist jedoch ein im Allgemeinen nicht entscheidbares Problem. -Moderne CAS treiben einigen Aufwand, um die -Gleichheit von Ausdrücken zu entscheiden, sie können das Problem -aber grundsätzlich nicht vollständig lösen. -Damit kann der Risch-Algorithmus in praktischen Anwendungen das -Stammfunktionsproblem ebenfalls nur mit Einschränkungen lösen, -die durch die Fähigkeiten des Ausdrucksvergleichs in einem CAS -gesetzt werden. - -Im Folgenden sollen elementare Funktionen definiert werden, es sollen -die Grundideen der differentiellen Galois-Theorie zusammengetragen werden -und der Satz von Liouvill vorgestellt werden. -An Hand der Fehler-Funktion soll dann gezeigt werden, wie man jetzt -einsehen kann, dass die Fehlerfunktion nicht elementar darstellbar ist. -Im nächsten Abschnitt dann soll der Risch-Algorithmus skizziert werden. - -\subsection{Elementare Funktionen -\label{buch:integrale:section:elementar}} -Es soll die Frage beantwortet werden, welche Stammfunktionen sich -in ``geschlossener Form'' oder durch ``wohlbekannte Funktionen'' -ausdrücken lassen. -Welche Funktionen dabei als ``wohlbekannt'' gelten dürfen ist -ziemlich willkürlich. -Sicher möchte man Potenzen und Wurzeln, Logarithmus und Exponentialfunktion, -aber auch die trigonometrischen Funktionen dazu zählen dürfen. -Ausserdem will man beliebig mit den arithmetischen Operationen -rechnen. -So entsteht die Menge der Funktionen, die man ``elementar'' nennen -will. - -In der Menge der elementaren Funktionen möchte man jetzt -Stammfunktionen ausgewählter Funktionen suchen. -Dazu muss man von jeder Funktion ihre Ableitung kennen. -Die Ableitungsoperation macht aus der Funktionenmenge eine -differentielle Algebra. -Der Satz von Liouville (Satz~\ref{buch:integrale:satz:liouville1}) -liefert Bedingungen, die erfüllt sein müssen, wenn eine Funktion -eine elementare Stammfunktion hat. -Sind diese Bedingungen nicht erfüllbar, ist auch keine -elementare Stammfunktion möglich. - -In den folgenden Abschnitten soll die differentielle Algebra -der elementaren Funktionen konstruiert werden. - -\subsubsection{Körper} -Die einfachsten Funktionen sind die die Konstanten, für die wir -für die nachfolgenden Betrachtungen fast immer die komplexen Zahlen -$\mathbb{C}$ -zu Grunde legen wollen. -Dabei ist vor allem wichtig, dass sich darin alle arithmetischen -Operationen durchführen lassen mit der einzigen Ausnahme, dass -nicht durch $0$ dividiert werden darf. -Man nennt $\mathbb{C}$ daher ein {\em Körper}. -\index{Körper}% -\label{buch:integrale:def:koerper} - -\subsubsection{Polynome und rationale Funktionen} -Die Polynome einer Variablen beschreiben eine Menge von -Funktionen, in der Addition, Subtraktion, Multiplikation -von Funktionen und Multiplikation mit komplexen Zahlen -uneingeschränkt möglich ist. -Wir bezeichen wie früher die Menge der Polynome in $z$ mit -$\mathbb{C}[z]$. - -Die Division ist erst möglich, wenn man beliebige Brüche -zulässt, deren Zähler und Nenner Polynome sind. -Die Menge -\[ -\mathbb{C}(z) -= -\biggl\{ -\frac{p(z)}{q(z)} -\;\bigg|\; -p,q\in \mathbb{C}[z] -\biggr\} -\] -heisst die Menge der {\em rationalen Funktionen}. -\label{buch:integrale:def:rationalefunktion} -\index{Funktion, rationale}% -\index{rationale Funktion}% -In ihr sind jetzt alle arithmetischen Operationen ausführbar -ausser natürlich die Division durch die Nullfunktion. -Die rationalen Funktionen bilden also wieder eine Körper. - -Die Tatsache, dass die rationalen Funktionen einen Körper -bilden bedeutet auch, dass die Konstruktion erneut durchgeführt -werden kann. -Ausgehend von einem beliebigen Körper $K$ können wieder zunächst -die Polynome $K[X]$ und anschliesen die rationalen Funktionen $K[X]$ -in der neuen Variablen, jetzt aber mit Koeffizienten in $K$ -gebildet werden. -So entstehen Funktionen von mehreren Variablen und, indem -wir für die neue Variable $X$ zum Beispiel die im übernächsten -Abschnitt betrachtete Wurzel $X=\sqrt{z}$ -einsetzen, rationale Funktionen in $z$ und $\sqrt{z}$. - -Solche Funktionenkörper werden im folgenden mit geschweiften -Buchstaben $\mathscr{D}$ bezeichnet. -\index{Funktionenkörper}% - -\subsubsection{Ableitungsoperation} -In allen Untersuchungen soll immer die Ableitungsoperation -mit berücksichtigt werden. -In unserer Betrachtungsweise spielt es keine Rolle, dass die -Ableitung aus einem Grenzwert entsteht, es sind nur die algebraischen -Eigenschaften wichtig. -Diese sind in der folgenden Definition zusammengefasst. - -\begin{definition} -\label{buch:integrale:def:derivation} -Ein {\em Ableitungsoperator} oder eine {\em Derivation} einer Algebra -$\mathscr{D}$ von Funktionen ist eine lineare Abbildung -\[ -\frac{d}{dz} -\colon \mathscr{D} \to \mathscr{D} -: -f \mapsto \frac{df}{dz} = f', -\] -die zusätzlich die Produktregel -\begin{equation} -\frac{d}{dz} (fg) -= -\frac{df}{dz} \cdot g + f \cdot \frac{dg}{dz} -\qquad\Leftrightarrow\qquad -(fg)' = f' g + fg' -\label{buch:integrale:eqn:produktregel} -\end{equation} -\index{Produktregel}% -erfüllt. -Die Funktion $f'\in \mathscr{D}$ heisst auch die {\em Ableitung} -von $f\in\mathscr{D}$. -\index{Derivation}% -\index{Ableitungsoperator}% -\index{Ableitung}% -\end{definition} - -Die Produktregel hat zum Beispiel auch die bekannten Quotientenregel -zur Folge. -Dazu betrachten wir das Produkt $f= (f/g)\cdot g$ und leiten es mit -Hilfe der Produktregel ab: -\[ -\frac{d}{dz}f -= -\frac{d}{dz} -\biggl( -\frac{f}{g}\cdot g -\biggr) -= -{\color{darkred} -\frac{d}{dz} -\biggl( -\frac{f}{g} -\biggr)} -\cdot g -+ -\frac{f}{g}\cdot \frac{d}{dz}g. -\] -Jetzt lösen wir nach der {\color{darkred}roten} Ableitung des Quotienten -auf und erhalten -\begin{equation} -\biggl(\frac{f}{g}\biggr)' -= -\frac{d}{dz}\biggl(\frac{f}{g}\biggr) -= -\frac1g\biggl( -\frac{d}{dz}f - \frac{f}{g}\cdot \frac{d}{dz}g -\biggr) -= -\frac{1}{g} -\biggl( -f'-\frac{fg'}{g} -\biggr) -= -\frac{f'g-fg'}{g^2}. -\label{buch:integrale:eqn:quotientenregel} -\end{equation} -Dies ist die Quotientenregel. - -Aus der Produktregel folgt natürlich sofort auch die Potenzregel -für die Ableitung der $n$ten Potenz einer Funktion $f\in\mathscr{D}$, -sie lautet: -\begin{equation} -\frac{d}{dz} f^n -= -\underbrace{ -f'f^{n-1} + ff'f^{n-2} + f^2f'f^{n-3}+\dots f^{n-1}f' -}_{\displaystyle \text{$n$ Terme}} -= -nf^{n-1}f'. -\label{buch:integrale:eqn:potenzregel} -\end{equation} -In dieser Form versteckt sich natürlich auch die Kettenregel, die -Potenzfunktion ist die äussere Funktion, $f$ die innere, $f'$ ist also -die Ableitung er inneren Funktion, wie in der Kettenregel verlangt. -Falls $f$ ein Element von $\mathscr{D}$ ist mit der Eigenschaft -$df/dz=1$, dann entsteht die übliche Produktregel. - -\begin{definition} -Eine Algebra $\mathscr{D}$ von Funktionen mit einem Ableitungsoperator -$d/dz$ heisst eine {\em differentielle Algebra}. -\index{differentielle Algebra}% -\index{Algebra, differentielle}% -In einer differentiellen Algebra gelten die üblichen -Ableitungsregeln. -\end{definition} - -Die Potenzregel war in der Form~\eqref{buch:integrale:eqn:potenzregel} -geschrieben worden, nicht als die Ableitung von $z$. -Der Grund dafür ist, dass wir gar nicht voraussetzen wollen, dass in -unserer differentiellen Algebra eine Funktion existiert, die die -Rolle von $z$ hat. -Dies ist gar nicht nötig, wie das folgende Beispiel zeigt. - -\begin{beispiel} -Als Funktionenmenge $\mathscr{D}$ nehmen wir rationale Funktionen -in zwei Variablen, die wir $\cos x $ und $\sin x$ nennen. -Diese Menge bezeichnen wir mit -$\mathscr{D}=\mathbb{Q}(\cos x,\sin x)$ -Der Ableitungsoperator ist -\begin{align*} -\frac{d}{dx} \cos x &= -\sin x -\\ -\frac{d}{dx} \sin x &= \phantom{-}\cos x. -\end{align*} -Die Funktionen von $\mathbb{Q}(\cos x,\sin x)$ sind also Brüche, -deren Zähler und Nenner Polynome in $\cos x$ und $\sin x$ sind. -Aus den Produkt- und Quotientenregeln und den Ableitungsregeln für -$\cos x$ und $\sin x$ folgt, dass die Ableitung einer Funktion in -$\mathscr{D}$ wieder in $\mathscr{D}$ ist, $\mathscr{D}$ ist eine -differentielle Algebra. -\end{beispiel} - -Die konstanten Funktionen spielen eine besondere Rolle. -Da wir bei der Ableitung nicht von der Vorstellung einer -Funktion mit einem variablen Argument ausgehen wollten und -die Ableitung nicht als Grenzwert definieren wollten, müssen -wir auch bei der Definition der ``Konstanten'' einen neuen -Weg gehen. -In der Analysis sind die Konstanten genau die Funktionen, -deren Ableitung $0$ ist. - -\begin{definition} -\label{buch:integrale:def:konstante} -Ein Element $f\in \mathscr{D}$ mit $df/dz=f'=0$ heissen -{\em Konstante} in $\mathscr{D}$. -\index{Konstante}% -\end{definition} - -Die in der Potenzregel~\eqref{buch:integrale:eqn:potenzregel} -vermisste Funktion $z$ kann man ähnlich zu den Konstanten -zu definieren versuchen. -$z$ müsste ein Element von $\mathscr{D}$ mit $z' = 1$ sein. -Allerdings gibt es viele solche Elemente, ist $c$ eine Konstanten -und $z'=1$, dann ist auch $(z+c)'=1$, $(z+c)$ hat also für -die Zwecke unserer Untersuchung die gleichen Eigenschaften wie -$z$. -Dies deckt sich natürlich auch mit der Erwartung, dass Stammfunktionen -nur bis auf eine Konstante bestimmt sind. -Eine differentielle Algebra muss allerdings kein Element $z$ mit der -Eigenschaft $z'=1$ enthalten. - -\begin{beispiel} -In $\mathscr{D}=\mathbb{Q}(\cos x,\sin x)$ gibt es kein Element $x$. -Ein solches wäre von der Form -\[ -x = \frac{p(\cos x,\sin x)}{q(\cos x,\sin x)}. -\] -Eine solche goniometrische Beziehung würde für $x=\frac{\pi}4$ bedeuten, -dass -\[ -\frac{\pi}4 -= -\frac{p(\sqrt{2}/2,\sqrt{2}/2)}{q(\sqrt{2}/2,\sqrt{2}/2)}. -\] -Auf der rechten Seite steht ein Quotient von Polynome, in dessen -Argument nur rationale Zahlen und $\sqrt{2}$ steht. -So ein Ausdruck kann immer in die Form -\[ -\pi -= -4\frac{a\sqrt{2}+b}{c\sqrt{2}+d} -= -\frac{4(a\sqrt{2}+b)(c\sqrt{2}-d)}{2c^2+d^2} -= -r\sqrt{2}+s -\] -gebracht werden. -Die Zahl auf der rechten Seite ist zwar irrational, aber sie ist Nullstelle -des quadratischen Polynoms -\[ -p(x) -= -(x-r\sqrt{2}-s)(x+r\sqrt{2}-s) -= -x^2 --2sx --2r^2+s^2 -\] -mit rationalen Koeffizienten, wie man mit der Lösungsformel für die -quadratische Gleichung nachprüfen kann. -Es ist bekannt, dass $\pi$ als transzendente Zahl nicht Nullstelle -eines Polynoms mit rationalen Koeffizienten ist. -Dieser Widerspruch zeigt, dass $x$ nicht in $\mathbb{Q}(\cos x, \sin x)$ -vorkommen kann. -\end{beispiel} - -In einer differentiellen Algebra kann jetzt die Frage nach der -Existenz einer Stammfunktion gestellt werden. - -\begin{aufgabe} -\label{buch:integrale:aufgabe:existenz-stammfunktion} -Gegeben eine differentielle Algebra $\mathscr{D}$ und ein Element -$f\in\mathscr{D}$, entscheide, ob es ein Element $F\in\mathscr{D}$ -gibt mit der Eigenschaft $F'=f$. -Ein solches $F\in\mathscr{D}$ heisst {\em Stammfunktion} von $f$. -\end{aufgabe} - -\begin{satz} -In einer differentiellen Algebra $\mathscr{D}$ mit $z\in\mathscr{D}$ -hat die Potenzfunktion $f=z^n$ für $n\in\mathbb{N}\setminus\{-1\}$ -ein Stammfunktion, nämlich -\[ -F = \frac{1}{n+1} z^{n+1}. -\] -\label{buch:integrale:satz:potenzstammfunktion} -\end{satz} - -\begin{proof}[Beweis] -Tatsächlich kann man dies sofort nachrechnen, muss allerdings die -Fälle $n+1 >0$ und $n+1<0$ unterscheiden, da die Potenzregel -\eqref{buch:integrale:eqn:potenzregel} nur für natürliche Exponenten -gilt. -Man erhält -\begin{align*} -n+1&>0\colon -& -\frac{d}{dz}\frac{1}{n+1}z^{n+1} -&= -\frac{1}{n+1}(n+1)z^{n+1-1} -= -z^n, -\\ -n+1&<0\colon -& -\frac{d}{dz}\frac{1}{n+1}\frac{1}{z^{-(n+1)}} -&= -\frac{1}{n+1}\frac{1'z^{-(n+1)}-1(-(n+1))z^{-n-1-1}}{z^{-2n-2}} -\\ -&& -&= -\frac{1}{n+1} -\frac{(n+1)z^n{-n-2}}{z^{-2n-2}} -\\ -&& -&= -\frac{1}{z^{-n}}=z^n. -\end{align*} -Man beachte, dass in dieser Rechnung nichts anderes als die -algebraischen Eigenschaften der Produkt- und Quotientenregel -verwendet wurden. -\end{proof} - -\subsubsection{Wurzeln} -Die Wurzelfunktionen sollen natürlich als elementare Funktionen -erlaubt sein. -Es ist bekannt, dass $\sqrt{z}\not\in \mathscr{D}=\mathbb{C}(z)$ -ist, ein solches Element müsste also erst noch hinzugefügt werden. -Dabei muss auch seine Ableitung definiert werden. -Auch dabei dürfen wir nicht auf eine Grenzwertüberlegung zurückgreifen, -vielmehr müssen wir die Ableitung auf vollständig algebraische -Weise bestimmen. - -Wir schreiben $f=\sqrt{z}$ und leiten die Gleichung $f^2=z$ nach $z$ ab. -Dabei ergibt sich nach der Potenzregel -\[ -\frac{d}{dz}f^2 = 2f'f = \frac{d}{dz}z=1 -\qquad\Rightarrow\qquad f' = \frac{1}{2f}. -\] -Diese Rechnung lässt sich auch auf $n$-Wurzeln $g=\root{n}\of{z}$ mit -der Gleichung $g^n = z$ verallgemeinern. -Die Ableitung der $n$-ten Wurzel ist -\begin{equation} -\frac{d}{dz}g^n -= -ng^{n-1} = \frac{d}{dz}z=1 -\qquad\Rightarrow\qquad -\frac{d}{dz}g = \frac{1}{ng^{n-1}}. -\end{equation} -Es ist also möglich, eine differentielle Algebra $\mathscr{D}$ mit einer -$n$-ten Wurzel $g$ zu einer grösseren differentiellen Algebra $\mathscr{D}(g)$ -zu erweitern, in der wieder alle Regeln für das Rechnen mit Ableitungen -erfüllt sind. - -\subsubsection{Algebraische Elemente} -Die Charakterisierung der Wurzelfunktionen passt zwar zum verlangten -algebraischen Vorgehen, ist aber zu spezielle und nicht gut für die -nachfolgenden Untersuchengen geeignet. -Etwas allgemeiner ist der Begriff der algebraischen Elemente. - -\begin{definition} -\label{buch:integrale:def:algebraisches-element} -Seien $K\subset L$ zwei Körper. -Ein Element $\alpha \in L$ heisst {\em algebraisch} über $K$, -wenn $\alpha$ Nullstelle eines Polynoms $p\in K[X]$ mit Koeffizienten -in $K$ ist. -\index{algebraisch}% -\end{definition} - -Jedes Element $\alpha\in K$ ist algebraisch, da $\alpha$ Nullstelle -von $X-\alpha\in K[X]$ ist. -Die $n$tem Wurzeln eines Elemente $\alpha\in K$ sind ebenfalls algebraisch, -da sie Nullstellen des Polynoms $p(X) = X^n - \alpha$ sind. -Allerdings ist nicht klar, dass diese Wurzeln überhaupt existieren. -Nach dem Satz von Abel~\ref{buch:potenzen:satz:abel} gibt es aber -Nullstellen von Polynomen, die sich nicht als Wurzelausdrücke schreiben -lassen. -Der Begriff der algebraischen Elemente ist also allgemeiner als der -Begriff der Wurzel. - -\begin{definition} -\label{buch:integrale:def:algebraisch-abgeschlossen} -Ein Körper $K$ heisst {\em algebraisch abgeschlossen}, wenn jedes Polynom mit -Koeffizienten in $K$ eine Nullstelle in $K$ hat. -\end{definition} - -Der Körper $\mathbb{C}$ ist nach dem -Fundamentalsatz~\label{buch:potenzen:satz:fundamentalsatz} -der Algebra algebraisch abgeschlossen. -Da wir aber mit Funktionen arbeiten, müssen wir auch Wurzeln -von Funktionen finden können. -Dies ist nicht selbstverständlich, wie das folgende Beispiel zeigt. - -\begin{beispiel} -Es gibt keine stetige Funktion $f\colon \mathbb{C}\to\mathbb{C}$, die -die Gleichung $f(z)^2 = z$ und $f(1)=1$ erfüllt. -Für die Argumente $z(t)= e^{it}$ folgt, dass $f(z(t)) = e^{it/2}$ sein -muss. -Setzt man aber $t=\pm \pi$ ein, ergeben sich die Werte -$f(z(\pm\pi))=e^{\pm i\pi/2}=\pm 1$, die beiden Grenzwerte -für $t\to\pm\pi$ sind also verschieden. -\end{beispiel} - -Die Mathematik hat verschiedene ``Tricks'' entwickelt, wie mit diesem -Problem umgegangen werden kann: Funktionskeime, Garben, Riemannsche -Flächen. -Sie sind alle gleichermassen gut geeignet, das Problem zu lösen. -Für die vorliegende Aufgabe genügt es aber, dass es tatsächlich -immer ein wie auch immer geartetes Element gibt, welches Nullstelle -des Polynoms ist. - -Ist $f$ eine Nullstelle des Polynoms $p(X)$ mit Koeffizienten in -$\mathscr{D}$, dann kann man die Ableitung wie folgt berechnen. -Zunächst leitet man $p(f)$ ab: -\begin{align} -0&= -\frac{d}{dz}(a_nf^n + a_{n-1}f^{n-1}+\ldots+a_1f+a_0) -\notag -\\ -&= -a_n'f^n + a_{n-1}'f^{n-1}+\ldots+a_1'f+a_0' -+ -na_nf^{n-1}f' -+ -(n-1)a_nf^{n-2}f' -+ -\ldots -+ -a_2ff' -+ -a_1f' -\notag -\\ -&= -a_n'f^n + a_{n-1}'f^{n-1}+\ldots+a_1'f+a_0' -+ -( -na_nf^{n-1} -+ -(n-1)a_nf^{n-2} -+ -\ldots -+ -a_2f -+ -a_1 -)f' -\notag -\\ -\Rightarrow -\qquad -f'&=\frac{ -a_n'f^n + a_{n-1}'f^{n-1}+\dots+a_1'f+a_0' -}{ -na_nf^{n-1} -+ -(n-1)a_nf^{n-2} -+ -\dots -+ -a_1 -}. -\label{buch:integrale:eqn:algabl} -\end{align} -Das einzige, was dabei schief gehen könnte ist, dass der Nenner ebenfalls -verschwindet. -Dieses Problem kann man dadurch lösen, dass man als Polynom das -sogenannte Minimalpolynom verwendet. - -\begin{definition} -Das {\em Minimalpolynome} $m(X)$ eines algebraischen Elementes $\alpha$ ist -das Polynom kleinsten Grades, welches $m(\alpha)=0$ erfüllt. -\end{definition} - -Da das Minimalpolynom den kleinstmöglichen Grad hat, kann der Nenner -von~\eqref{buch:integrale:eqn:algabl}, -der noch kleineren Grad hat, unmöglich verschwinden. -Das Minimalpolynom ist auch im wesentlichen eindeutig. -Gäbe es nämlich zwei verschiedene Minimalpolynome $m_1$ und $m_2$, -dann müsste $\alpha$ auch eine Nullstelle des grössten gemeinsamen -Teilers $m_3=\operatorname{ggT}(m_1,m_2)$ sein. -Wären die beiden Polynome wesentlich verschieden, dann hätte $m_3$ -kleineren Grad, im Widerspruch zur Definition des Minimalpolynoms. -Also unterscheiden sich die beiden Polynome $m_1$ und $m_2$ nur um -einen skalaren Faktor. - -\subsubsection{Konjugation, Spur und Norm} -% Konjugation, Spur und Norm -Das Minimalpolynom eines algebraischen Elementes ist nicht -eindeutig bestimmt. -Zum Beispiel ist $\sqrt{2}$ algebraisch über $\mathbb{Q}$, das -Minimalpolynom ist $m(X)=X^2-2\in\mathbb{Q}[X]$. -Es hat aber noch eine zweite Nullstelle $-\sqrt{2}$. -Mit rein algebraischen Mitteln sind die beiden Nullstellen $\pm\sqrt{2}$ -nicht zu unterscheiden, erst die Verwendung der Vergleichsrelation -ermöglicht, sie zu unterscheiden. - -Dasselbe gilt für die imaginäre Einheit $i$, die das Minimalpolynom -$m(X)=X^2+1\in\mathbb{R}[X]$ hat. -Hier gibt es nicht einmal mehr eine Vergleichsrelation, mit der man -die beiden Nullstellen unterscheiden könnte. -In der Tat ändert sich aus algebraischer Sicht nichts, wenn man in -allen Formeln $i$ durch $-i$ ersetzt. - -Etwas komplizierter wird es bei $\root{3}\of{2}$. -Das Polynom $m=x^3-2\in\mathbb{Q}[X]$ hat $\root{3}\of{2}$ als -Nullstelle und dies ist auch tatsächlich das Minimalpolynom. -Das Polynom hat noch zwei weitere Nullstellen -\[ -\alpha_+ = \frac{-1+i\sqrt{3}}{2}\root{3}\of{2} -\qquad\text{und}\qquad -\alpha_- = \frac{-1-i\sqrt{3}}{2}\root{3}\of{2}. -\] -Die beiden Lösungen gehen durch die Vertauschung von $i$ und $-i$ -auseinander hervor. -Betrachtet man dasselbe Polynom aber als Polynom in $\mathbb{R}[X]$, -dann ist es nicht mehr das Minimalpolynom von $\root{3}\of{2}$, da -$X-\root{3}\of{2}\in\mathbb{R}[X]$ kleineren Grad und $\root{3}\of{2}$ -als Nullstelle hat. -Indem man -\[ -m(X)/(X-\root{3}\of{2})=X^2+\root{3}\of{2}X+\root{3}\of{2}^2=m_2(X) -\] -rechnet, bekommt man das Minimalpolynom der beiden Nullstellen $\alpha_+$ -und $\alpha_-$. -Wir lernen aus diesen Beispielen, dass das Minimalpolynom vom Grundkörper -abhängig ist (Die Faktorisierung $(X-\root{3}\of{2})\cdot m_2(X)$ von -$m(X)$ ist in $\mathbb{Q}[X]$ nicht möglich) und dass wir keine -algebraische Möglichkeit haben, die verschiedenen Nullstellen des -Minimalpolynoms zu unterscheiden. - -Die beiden Nullstellen $\alpha_+$ und $\alpha_-$ des Polynoms $m_2(X)$ -erlauben, $m_2(X)=(X-\alpha_+)(X-\alpha_-)$ zu faktorisieren. -Durch Ausmultiplizieren -\[ -(X-\alpha_+)(X-\alpha_-) -= -X^2 -(\alpha_++\alpha_-)X+\alpha_+\alpha_- -\] -und Koeffizientenvergleich mit $m_2(X)$ findet man die symmetrischen -Formeln -\[ -\alpha_+ + \alpha_- = \root{3}\of{2} -\qquad\text{und}\qquad -\alpha_+ \alpha_ = \root{3}\of{2}. -\] -Diese Ausdrücke sind nicht mehr abhängig von einer speziellen Wahl -der Nullstellen. - -Das Problem verschärft sich nocheinmal, wenn wir Funktionen betrachten. -Das Polynom $m(X)=X^3-z$ ist das Minimalpolynom der Funktion $\root{3}\of{z}$. -Die komplexe Zahl $z=re^{i\varphi}$ hat aber drei die algebraisch nicht -unterscheidbaren Nullstellen -\[ -\alpha_0(z)=\root{3}\of{r}e^{i\varphi/3}, -\quad -\alpha_1(z)=\root{3}\of{r}e^{i\varphi/3+2\pi/3} -\qquad\text{und}\qquad -\alpha_2(z)=\root{3}\of{r}e^{i\varphi/3+4\pi/3}. -\] -Aus der Faktorisierung $ (X-\alpha_0(z)) (X-\alpha_1(z)) (X-\alpha_2(z))$ -und dem Koeffizientenvergleich mit dem Minimalpolynom kann man wieder -schliessen, dass die Relationen -\[ -\alpha_0(z) + \alpha_1(z) + \alpha_2(z)=0 -\qquad\text{und}\qquad -\alpha_0(z) \alpha_1(z) \alpha_2(z) = z -\] -gelten. - -Wir können also oft keine Aussagen über individuelle Nullstellen -eines Minimalpolynoms machen, sondern nur über deren Summe oder -Produkt. - -\begin{definition} -\index{buch:integrale:def:spur-und-norm} -Sie $m(X)\in K[X]$ das Minimalpolynom eines über $K$ algebraischen -Elements und -\[ -m(X) = a_nX^n + a_{n-1}X^{n-1} + \ldots + a_1X + a_0. -\] -Dann heissen -\[ -\operatorname{Tr}(\alpha) = -a_{n-1} -\qquad\text{und}\qquad -\operatorname{Norm}(\alpha) = (-1)^n a_0 -\] -die {\em Spur} und die {\em Norm} des Elementes $\alpha$. -\index{Spur eines algebraischen Elementes}% -\index{Norm eines algebraischen Elementes}% -\end{definition} - -Die Spur und die Norm können als Spur und Determinante einer Matrix -verstanden werden, diese allgemeineren Definitionen, die man in der -Fachliteratur, z.~B.~in~\cite{buch:lang} nachlesen kann, führen aber -für unsere Zwecke zu weit. - -\begin{hilfssatz} -Die Ableitungen von Spur und Norm sind -\[ -\operatorname{Tr}(\alpha)' -= -\operatorname{Tr}(\alpha') -\qquad\text{und}\qquad -\operatorname{Norm}(\alpha)' -= -\operatorname{Tr}(\alpha)' -\] -XXX Wirklich? -\end{hilfssatz} - -\subsubsection{Logarithmen und Exponentialfunktionen} -Die Funktion $z^{-1}$ musste im -Satz~\ref{buch:integrale:satz:potenzstammfunktion} -ausgeschlossen werden, sie hat keine Stammfunktion in $\mathbb{C}(z)$. -Aus der Analysis ist bekannt, dass die Logarithmusfunktion $\log z$ -eine Stammfunktion ist. -Der Logarithmus von $z$ aber auch der Logarithmus $\log f(z)$ -einer beliebigen Funktion $f(z)$ oder die Exponentialfunktion $e^{f(z)}$ -sollen ebenfalls elementare Funktionen sein. -Da wir aber auch hier nicht auf die analytischen Eigenschaften zurückgreifen -wollen, brauchen wir ein rein algebraische Definition. - -\begin{definition} -\label{buch:integrale:def:logexp} -Sei $\mathscr{D}$ ein differentielle Algebra und $f\in\mathscr{D}$. -Ein Element $\vartheta\in\mathscr{D}$ heisst ein {\em Logarithmus} -von $f$, geschrieben $\vartheta = \log f$, wenn $f\vartheta' = f'$ gilt. -$\vartheta$ heisst eine Exponentialfunktion von $f$ wenn -$\vartheta'=\vartheta f'$ gilt. -\end{definition} - -Die Formel für die Exponentialfunktion ist etwas vertrauter, sie ist -die bekannte Kettenregel -\begin{equation} -\vartheta' -= -\frac{d}{dz} e^f -= -e^f \cdot \frac{d}{dz} f -= -\vartheta \cdot f'. -\label{buch:integrale:eqn:exponentialableitung} -\end{equation} -Da wir uns vorstellen, dass Logarithmen Umkehrfunktionen von -Exponentialfunktionen sein sollen, -muss die definierende Gleichung genau wie -\eqref{buch:integrale:eqn:exponentialableitung} -aussehen, allerdings mit vertauschten Plätzen von $f$ und $\vartheta$, -also -\begin{equation} -\vartheta' = \vartheta\cdot f' -\qquad -\rightarrow -\qquad -f' = f\cdot \vartheta' -\;\Leftrightarrow\; -\vartheta' = (\log f)' = \frac{f'}{f}. -\label{buch:integrale:eqn:logarithmischeableitung} -\end{equation} -Dies ist die aus der Analysis bekannte Formel für die logarithmische -Ableitung. - -Der Logarithmus von $f$ und die Exponentialfunktion von $f$ sollen -also ebenfalls als elementare Funktionen betrachtet werden. - -\subsubsection{Die trigonometrischen Funktionen} -Die bekannten trigonometrischen Funktionen und ihre Umkehrfunktionen -sollten natürlich auch elementare Funktionen sein. -Dabei kommt uns zur Hilfe, dass sie sich mit Hilfe der Exponentialfunktion -als -\[ -\cos f = \frac{e^{if}+e^{-if}}2 -\qquad\text{und}\qquad -\sin f = \frac{e^{if}-e^{-if}}{2i} -\] -schreiben lassen. -Eine differentielle Algebra, die die Exponentialfunktionen von $if$ und -$-if$ enthält, enthält also automatisch auch die trigonometrischen -Funktionen. -Im Folgenden ist es daher nicht mehr nötig, die trigonometrischen -Funktionen speziell zu untersuchen. - -\subsubsection{Elementare Funktionen} -Damit sind wir nun in der Lage, den Begriff der elementaren Funktion -genau zu fassen. - -\begin{definition} -\label{buch:integrale:def:einfache-elementare-funktion} -Sie $\mathscr{D}$ eine differentielle Algebra über $\mathbb{C}$ und -$\mathscr{D}(\vartheta)$ eine Erweiterung von $\mathscr{D}$ um eine -neue Funktion $\vartheta$, dann heissen $\vartheta$ und die Elemente -von $\mathscr{D}(\vartheta)$ einfach elementar, wenn eine der folgenden -Bedingungen erfüllt ist: -\begin{enumerate} -\item $\vartheta$ ist algebraisch über $\mathscr{D}$, d.~h.~$\vartheta$ -ist eine ``Wurzel''. -\item $\vartheta$ ist ein Logarithmus einer Funktion in $\mathscr{D}$, -d.~h.~es gibt $f\in \mathscr{D}$ mit $f'=f\vartheta'$ -(Definition~\ref{buch:integrale:def:logexp}). -\item $\vartheta$ ist eine Exponentialfunktion einer Funktion in $\mathscr{D}$, -d.~h.~es bit $f\in\mathscr{D}$ mit $\vartheta'=\vartheta f'$ -(Definition~\ref{buch:integrale:def:logexp}). -\end{enumerate} -\end{definition} - -Einfache elementare Funktionen entstehen also ausgehend von einer -differentiellen Algebra, indem man genau einmal eine Wurzel, einen -Logarithmus oder eine Exponentialfunktion hinzufügt. -So etwas wie die zusammengesetzte Funktion $e^{\sqrt{z}}$ ist -damit noch nicht möglich. -Daher erlauben wir, dass man die gesuchten Funktionen in mehreren -Schritten aufbauen kann. - -\begin{definition} -Sei $\mathscr{F}$ eine differentielle Algebra, die die differentielle -Algebra $\mathscr{D}$ enthält, also $\mathscr{D}\subset\mathscr{F}$. -$\mathscr{F}$ und die Elemente von $\mathscr{F}$ heissen einfach, -wenn es endlich viele Elemente $\vartheta_1,\dots,\vartheta_n$ gibt -derart, dass -\[ -\renewcommand{\arraycolsep}{2pt} -\begin{array}{ccccccccccccc} -\mathscr{D} -&\subset& -\mathscr{D}(\vartheta_1) -&\subset& -\mathscr{D}(\vartheta_1,\vartheta_2) -&\subset& -\; -\cdots -\; -&\subset& -\mathscr{D}(\vartheta_1,\vartheta_2,\dots,\vartheta_{n-1}) -&\subset& -\mathscr{D}(\vartheta_1,\vartheta_2,\dots,\vartheta_{n-1},\vartheta_n) -&=& -\mathscr{F} -\\ -\| -&& -\| -&& -\| -&& -&& -\| -&& -\| -&& -\\ -\mathscr{F}_0 -&\subset& -\mathscr{F}_1 -&\subset& -\mathscr{F}_2 -&\subset& -\cdots -&\subset& -\mathscr{F}_{n-1} -&\subset& -\mathscr{F}_{n\mathstrut} -&& -\end{array} -\] -gilt so, dass jedes $\vartheta_{i+1}$ einfach ist über -$\mathscr{F}_i=\mathscr{D}(\vartheta_1,\dots,\vartheta_i)$. -\end{definition} - -In Worten bedeutet dies, dass man den Funktionen von $\mathscr{D}$ -nacheinander Wurzeln, Logarithmen oder Exponentialfunktionen einzelner -Funktionen hinzufügt. -Die Aufgabe~\ref{buch:integrale:aufgabe:existenz-stammfunktion} kann -jetzt so formuliert werden. - -\begin{aufgabe} -\label{buch:integrale:aufgabe:existenz-stammfunktion-dalg} -Gegeben ist eine Differentielle Algebra $\mathscr{D}$ und eine -Funktion $f\in \mathscr{D}$. -Gibt es eine Folge $\vartheta_1,\dots,\vartheta_n$ und eine Funktion -$F\in\mathscr{D}(\vartheta_1,\dots,\vartheta_n)$ derart, dass -$F'=f$. -\end{aufgabe} - -Das folgende Beispiel zeigt, wie man möglicherweise mehrere -Erweiterungsschritte vornehmen muss, um zu einer Stammfunktion -zu kommen. -Es illustriert auch die zentrale Rolle, die der Partialbruchzerlegung -in der weiteren Entwicklung zukommen wird. - -\begin{beispiel} -\label{buch:integrale:beispiel:nichteinfacheelementarefunktion} -Es soll eine Stammfunktion der Funktion -\[ -f(z) -= -\frac{z}{(az+b)(cz+d)} -\in -\mathbb{C}(z) -\] -gefunden werden. -In der Analysis lernt man, dass solche Integrale mit der -Partialbruchzerlegung -\[ -\frac{z}{(az+b)(cz+d)} -= -\frac{A_1}{az+b}+\frac{A_2}{cz+d} -= -\frac{A_1cz+A_1d+A_2az+A_2b}{(az+b)(cz+d)} -\quad\Rightarrow\quad -\left\{ -\renewcommand{\arraycolsep}{2pt} -\begin{array}{rcrcr} -cA_1&+&aA_2&=&1\\ -dA_1&+&bA_2&=&0 -\end{array} -\right. -\] -bestimmt werden. -Die Lösung des Gleichungssystems ergibt -$A_1=b/(bc-ad)$ und $A_2=d/(ad-bc)$. -Die Stammfunktion kann dann aus -\begin{align*} -\int f(z)\,dz -&= -\int\frac{A_1}{az+b}\,dz -+ -\int\frac{A_2}{cz+d}\,dz -= -\frac{A_1}{a}\int\frac{a}{az+b}\,dz -+ -\frac{A_2}{c}\int\frac{c}{cz+d}\,dz -\end{align*} -bestimmt werden. -In den Integralen auf der rechten Seite ist der Zähler jeweils die -Ableitung des Nenners, der Integrand hat also die Form $g'/g$. -Genau diese Form tritt in der Definition eines Logarithmus auf. -Die Stammfunktion ist jetzt -\[ -F(z) -= -\int f(z)\,dz -= -\frac{A_1}{a}\log(az+b) -+ -\frac{A_2}{c}\log(cz+d) -= -\frac{b\log(az+b)}{a(bc-ad)} -+ -\frac{d\log(cz+d)}{c(ad-bc)}. -\] -Die beiden Logarithmen kann man nicht durch rein rationale Operationen -ineinander überführen. -Sie müssen daher beide der Algebra $\mathscr{D}$ hinzugefügt werden. -\[ -\left. -\begin{aligned} -\vartheta_1&=\log(az+b)\\ -\vartheta_2&=\log(cz+d) -\end{aligned} -\quad -\right\} -\qquad\Rightarrow\qquad -F(z) \in \mathscr{F}=\mathscr{D}(\vartheta_1,\vartheta_2). -\] -Die Stammfunktion $F(z)$ ist also keine einfache elementare Funktion, -aber $F$ ist immer noch eine elementare Funktion. -\end{beispiel} - -\subsection{Partialbruchzerlegung -\label{buch:integrale:section:partialbruchzerlegung}} -Die Konstruktionen des letzten Abschnitts haben gezeigt, -wie man die Funktionen, die man als Stammfunktionen einer Funktion -zulassen möchte, schrittweise konstruieren kann. -Die Aufgabe~\ref{buch:integrale:aufgabe:existenz-stammfunktion-dalg} -ist eine rein algebraische Formulierung der ursprünglichen -Aufgabe~\ref{buch:integrale:aufgabe:existenz-stammfunktion}. -Schliesslich hat das Beispiel auf -Seite~\pageref{buch:integrale:beispiel:nichteinfacheelementarefunktion} -gezeigt, dass es im allgemeinen mehrere Schritte braucht, um zu einer -elementaren Stammfunktion zu gelangen. -Die Lösung setzt sich aus den Termen der Partialbruchzerlegung. -In diesem Abschnitt soll diese genauer studiert werden. - -In diesem Abschnitt gehen wir immer von einer differentiellen -Algebra über den komplexen Zahlen aus und verlangen, dass die -Konstanten in allen betrachteten differentiellen Algebren -$\mathbb{C}$ sind. - -\subsubsection{Monome} -Die beiden Funktionen $\vartheta-1=\log(az+b)$ und $\vartheta_2=(cz+d)$, -die im Beispiel hinzugefügt werden mussten, verhalten sich ich algebraischer -Hinsicht wie ein Monom: man kann es nicht faktorisieren oder bereits -bekannte Summanden aufspalten. -Solchen Funktionen kommt eine besondere Bedeutung zu. - -\begin{definition} -\label{buch:integrale:def:monom} -Die Funktion $\vartheta$ heisst ein Monom, wenn $\vartheta$ nicht -algebraisch ist über $\mathscr{D}$ und $\mathscr{D}(\vartheta)$ die -gleichen Konstanten enthält wie $\mathscr{D}$. -\end{definition} - -\begin{beispiel} -Als Beispiel beginnen wir mit den komplexen Zahlen $\mathbb{C}$ -und fügen die Funktion $\vartheta_1=z$ hinzu und erhalten -$\mathscr{D}=\mathbb{C}(z)$. -Die Funktionen $z^k$ sind für alle $k$ linear unabhängig, d.~h.~es -gibt keinen Ausdruck -\[ -a_nz^n + a_{n-1}z^{n-1}+\cdots+a_1z+a_0=0. -\] -Dies ist gleichbedeutend damit, dass $z$ nicht algebraisch ist. -Das Monom $z$ ist also auch ein Monom im Sinne der -Definition~\ref{buch:integrale:def:monom}. -\end{beispiel} - -\begin{beispiel} -Wir beginnen wieder mit $\mathbb{C}$ und fügen die Funktion -$e^z$ hinzu. -Gäbe es eine Beziehung -\[ -b_m(e^z)^m + b_{m-1}(e^z)^{m-1}+\dots+b_1e^z + b_0=0 -\] -mit komplexen Koeffizienten $b_i\in\mathbb{C}$, -dann würde daraus durch Einsetzen von $z=1$ die Relation -\[ -b_me^m + b_{m-1}e^{m-1} + \dots + b_1e + b_0=0, -\] -die zeigen würde, dass $e$ eine algebraische Zahl ist. -Es ist aber bekannt, dass $e$ transzendent ist. -Dieser Widersprich zeigt, dass $e^z$ ein Monom ist. -\end{beispiel} - -\begin{beispiel} -Jetzt fügen wir die Exponentialfunktion $\vartheta_2=e^z$ -der differentiellen Algebra $\mathscr{D}=\mathbb{C}(z)$ hinzu -und erhalten $\mathscr{F}_1=\mathscr{D}(e^z) = \mathbb{C}(z,e^z)$. -Gäbe es das Minimalpolynom -\begin{equation} -b_m(z)(e^z)^m + b_{m-1}(z)(e^z)^{m-1}+\dots+b_1(z)e^z + b_0(z)=0 -\label{buch:integrale:beweis:exp-analytisch} -\end{equation} -mit Koeffizienten $b_i\in\mathbb{C}(z)$, dann könnte man mit dem -gemeinsamen Nenner der Koeffizienten durchmultiplizieren und erhielte -eine Relation~\eqref{buch:integrale:beweis:exp-analytisch} mit -Koeffizienten in $\mathbb{C}[z]$. -Dividiert man durch $e^{mz}$ erhält man -\[ -b_m(z) + b_{m-1}(z)\frac{1}{e^z} + \dots + b_1(z)\frac{1}{(e^z)^{m-1}} + b_0(z)\frac{1}{(e^z)^m}=0. -\] -Aus der Analysis weiss man, dass die Exponentialfunktion schneller -anwächst als jedes Polynom, alle Terme auf der rechten Seite -konvergieren daher gegen 0 für $z\to\infty$. -Das bedeutet, dass $b_m(z)\to0$ für $z\to \infty$. -Das Polynom~\eqref{buch:integrale:beweis:exp-analytisch} wäre also gar -nicht das Minimalpolynom. -Dieser Widerspruch zeigt, dass $e^z$ nicht algebraisch ist über -$\mathbb{C}(z)$ und damit ein Monom ist\footnote{Etwas unbefriedigend -an diesem Argument ist, dass man hier wieder rein analytische statt -algebraische Eigenschaften von $e^z$ verwendet. -Gäbe es aber eine minimale Relation wie -\eqref{buch:integrale:beweis:exp-analytisch} -mit Polynomkoeffizienten, dann wäre sie von der Form -\[ -P(z,e^z)=p(z)(e^z)^m + q(z,e^z)=0, -\] -wobei Grad von $e^z$ in $q$ höchstens $m-1$ ist. -Die Ableitung wäre dann -\[ -Q(z,e^z) -= -mp(z)(e^z)^m + p'(z)(e^z)^m + r(z,e^z) -= -(mp(z) + p'(z))(e^z)^m + r(z,e^z) -=0, -\] -wobei der Grad von $e^z$ in $r$ wieder höchstens $m-1$ ist. -Bildet man $mP(z,e^z) - Q(z,e^z) = 0$ ensteht eine Relation, -in der der Grad des Koeffizienten von $(e^z)^m$ um eins abgenommen hat. -Wiederholt man dies $m$ mal, verschwindet der Term $(e^z)^m$, die -Relation~\eqref{buch:integrale:beweis:exp-analytisch} -war also gar nicht minimal. -Dieser Widerspruch zeigt wieder, dass $e^z$ nicht algebraisch ist, -verwendet aber nur die algebraischen Eigenschaften der differentiellen -Algebra. -}. -\end{beispiel} - -\begin{beispiel} -Wir hätten auch in $\mathbb{Q}$ arbeiten können und $\mathbb{Q}$ -erst die Exponentialfunktion $e^z$ und dann den Logarithmus $z$ von $e^z$ -hinzufügen können. -Es gibt aber noch weitere Logarithmen von $e^z$ zum Beispiel $z+2\pi i$. -Offenbar ist $\psi=z+2\pi i\not\in \mathbb{Q}(z,e^z)$, wir könnten also -auch noch $\psi$ hinzufügen. -Zwar ist $\psi$ auch nicht algebraisch, aber wenn wir $\psi$ hinzufügen, -dann wird aber die Menge der Konstanten grösser, sie umfasst jetzt -$\mathbb{Q}(2\pi i)$. -Die Bedingung in der Definition~\ref{buch:integrale:def:monom}, -dass die Menge der Konstanten nicht grösser werden darf, ist also -verletzt. - -Hätte man mit $\mathbb{Q}(e^z, z+2\pi i)$ begonnen, wäre $z$ aus -dem gleichen Grund kein Monom, aber $z+2\pi i$ wäre eines im Sinne -der Definition~\ref{buch:integrale:def:monom}. -In allen Rechnungen könnte man $\psi=z+2\pi i$ nicht weiter aufteilen, -da $\pi$ oder seine Potenzen keine Elemente von $\mathbb{Q}(e^z)$ sind. -\end{beispiel} - -Da wir im Folgenden davon ausgehen, dass die Konstanten unserer -differentiellen Körper immer $\mathbb{C}$ sind, wird es jeweils -genügen zu untersuchen, ob eine neu hinzuzufügende Funktion algebraisch -ist oder nicht. - -\subsubsection{Ableitungen von Polynomen und rationalen Funktionen von Monomen} -Fügt man einer differentiellen Algebra ein Monom hinzu, dann lässt -sich etwas mehr über Ableitungen von Polynomen oder Brüchen in diesen -Monomen sagen. -Diese Eigenschaften werden später bei der Auflösung der Partialbruchzerlegung -nützlich sein. - -\begin{satz} -\label{buch:integrale:satz:polynom-ableitung-grad} -Sei -\[ -P -= -A_nX^n + A_{n-1}X^{n-1} + \dots A_1X+A_0 -\in\mathscr{D}[X] -\] -ein Polynom mit Koeffizienten in einer differentiellen Algebra $\mathscr{D}$ -und $\vartheta$ ein Monom über $\mathscr{D}$. -Dann gilt -\begin{enumerate} -\item -\label{buch:integrale:satz:polynom-ableitung-grad-log} -Falls $\vartheta=\log f$ ist, ist $P(\vartheta)'$ ein -Polynom vom Grad $n$ in $\vartheta$, wenn der Leitkoeffizient $A_n$ -nicht konstant ist, andernfalls ein Polynom vom Grad $n-1$. -\item -\label{buch:integrale:satz:polynom-ableitung-grad-exp} -Falls $\vartheta = \exp f$ ist, dann ist $P(\vartheta)'$ ein Polynom -in $\vartheta$ vom Grad $n$. -\end{enumerate} -\end{satz} - -Der Satz macht also genaue Aussagen darüber, wie sich der Grad eines -Polynoms in $\vartheta$ beim Ableiten ändert. - -\begin{proof}[Beweis] -Für Exponentialfunktion ist $\vartheta'=\vartheta f'$, die Ableitung -fügt also einfach einen Faktor $f'$ hinzu. -Terme der Form $A_k\vartheta^k$ haben die Ableitung -\[ -(A_k\vartheta^k) -= -A'_k\vartheta^k + A_kk\vartheta^{k-1}\vartheta' -= -A'_k\vartheta^k + A_kk\vartheta^{k-1}\vartheta f' -= -(A'_k + kA_k f)\vartheta^k. -\] -Damit wird die Ableitung des Polynoms -\begin{equation} -P(\vartheta)' -= -\underbrace{(A'_n+nA_nf')\vartheta^n}_{\displaystyle=(A_n\vartheta^n)'} -+ -(A'_{n-1}+(n-1)A_{n-1}f')\vartheta^{n-1} -+ \dots + -(A'_1+A_1f')\vartheta + A_0'. -\label{buch:integrale:ableitung:polynom} -\end{equation} -Der Grad der Ableitung kann sich also nur ändern, wenn $A_n'+nA_nf'=0$ ist. -Dies bedeutet aber wegen -\( -(A_n\vartheta^n)' -= -0 -\), dass $A_n\vartheta^n=c$ eine Konstante ist. -Da alle Konstanten bereits in $\mathscr{D}$ sind, folgt, dass -\[ -\vartheta^n=\frac{c}{A_n} -\qquad\Rightarrow\qquad -\vartheta^n - \frac{c}{A_n}=0, -\] -also wäre $\vartheta$ algebraisch über $\mathscr{D}$, also auch kein Monom. -Dieser Widerspruch zeigt, dass der Leitkoeffizient nicht verschwinden kann. - -Für die erste Aussage ist die Ableitung der einzelnen Terme des Polynoms -\[ -(A_k\vartheta^k)' -= -A_k'\vartheta^k + A_kk\vartheta^{k-1}\vartheta' -= -A_k'\vartheta^k + A_kk\vartheta^{k-1}\frac{f'}{f} -= -\biggl(A_k'\vartheta + kA_k\frac{f'}{f}\biggr)\vartheta^{k-1}. -\] -Die Ableitung des Polynoms ist daher -\[ -P(\vartheta)' -= -A_n'\vartheta^n + \biggl(nA_n\frac{f'}{f}+ A'_{n-1}\biggr)\vartheta^{n-1}+\dots -\] -Wenn $A_n$ keine Konstante ist, ist $A_n'\ne 0$ und der Grad von -$P(\vartheta)'$ ist $n$. -Wenn $A_n$ eine Konstante ist, müssen wir noch zeigen, dass der nächste -Koeffizient nicht verschwinden kann. -Wäre der zweite Koeffizient $=0$, dann wäre die Ableitung -\[ -(nA_n\vartheta+A_{n-1})' -= -nA_n\vartheta'+A'_{n-1} -= -nA_n\frac{f'}{f}+A'_{n-1} -= -0, -\] -d.h. $nA_n\vartheta+A_{n-1}=c$ wäre eine Konstante. -Da alle Konstanten schon in $\mathscr{D}$ sind, müsste auch -\[ -\vartheta = \frac{c-A_{n-1}}{nA_n} \in \mathscr{D} -\] -sein, wieder wäre $\vartheta$ kein Monom. -\end{proof} - -Der nächste Satz gibt Auskunft über den führenden Term in -$(\log P(\vartheta))' = P(\vartheta)'/P(\vartheta)$. - -\begin{satz} -\label{buch:integrale:satz:log-polynom-ableitung-grad} -Sei $P$ ein Polynom vom Grad $n$ wie in -\label{buch:integrale:satz:log-polynom-ableitung} -welches zusätzlich normiert ist, also $A_n=1$. -\begin{enumerate} -\item -\label{buch:integrale:satz:log-polynom-ableitung-log} -Ist $\vartheta=\log f$, dann ist -$(\log P(\vartheta))' = P(\vartheta)'/P(\vartheta)$ und $P(\vartheta)'$ -hat Grad $n-1$. -\item -\label{buch:integrale:satz:log-polynom-ableitung-exp} -Ist $\vartheta=\exp f$, dann gibt es ein Polynom $N(\vartheta)$ so, dass -$(\log P(\vartheta))' -= -P(\vartheta)'/P(\vartheta) -= -N(\vartheta)/P(\vartheta)+nf'$ -ist. -Falls $P(\vartheta)=\vartheta$ ist $N=0$, andernfalls ist $N(\vartheta)$ -ein Polynom vom Grad $0$ das kleinste $k$ so, dass $p<(k+1)q$. -Insbesondere ist dann $kq\le p$. -Nach dem euklidischen Satz für die Division von $P(X)$ durch $Q(X)^k$ -gibt es ein Polynom $P_k(X)$ vom Grad $\le p-qk$ derart, dass -\[ -P(X) = P_k(X)Q(X)^k + R_k(X) -\] -mit einem Rest $R_k(X)$ vom Grad $1$ können mit der Potenzregel -integriert werden, aber für eine Stammfunktion $1/(z-1)$ muss -der Logarithmus $\log(z-1)$ hinzugefügt werden. -Die Stammfunktion -\[ -\int f(z)\,dz -= -\int -\frac{1}{z-1} -\,dz -+ -\int -\frac{4}{(z-1)^2} -\,dz -+ -\int -\frac{4}{(z-1)^3} -\,dz -= -\log(z-1) -- -\underbrace{\frac{4z-2}{(z-1)^2}}_{\displaystyle\in\mathscr{D}} -\in \mathscr{D}(\log(z-1)) = \mathscr{F} -\] -hat eine sehr spezielle Form. -Sie besteht aus einem Term in $\mathscr{D}$ und einem Logarithmus -einer Funktion von $\mathscr{D}$, also einem Monom über $\mathscr{D}$. - -\subsubsection{Einfach elementare Stammfunktionen} -Der in diesem Abschnitt zu beweisende Satz von Liouville zeigt, -dass die im einführenden Beispiel konstruierte Form der Stammfunktion -eine allgemeine Eigenschaft elementar integrierbarer -Funktionen ist. -Zunächst aber soll dieses Bespiel etwas verallgemeinert werden. - -\begin{satz}[Liouville-Vorstufe für Monome] -\label{buch:integrale:satz:liouville-vorstufe-1} -Sei $\vartheta$ ein Monom über $\mathscr{D}$ und $g\in\mathscr{D}(\vartheta)$ -mit $g'\in\mathscr{D}$. -Dann hat $g$ die Form $v_0 + c_1\vartheta$ mit $v_0\in\mathscr{D}$ und -$c_1\in\mathbb{C}$. -\end{satz} - -\begin{proof}[Beweis] -In Anlehnung an das einführende Beispiel nehmen wir an, dass die -Stammfunktion $g\in\mathscr{D}[\vartheta]$ für ein Monom $\vartheta$ -über $\mathscr{D}$ ist. -Dann hat $g$ die Partialbruchzerlegung -\[ -g -= -H(\vartheta) -+ -\sum_{j\le r(i)} \frac{P_{ij}(\vartheta)}{Q_i(\vartheta)^j} -\] -mit irreduziblen normierten Polynomen $Q_i(\vartheta)$ und -Polynomen $P_{ij}(\vartheta)$ vom Grad kleiner als $\deg Q_i(\vartheta)$. -Ausserdem ist $H(\vartheta)$ ein Polynom. -Die Ableitung von $g$ muss jetzt aber wieder in $\mathscr{D}$ sein. -Zu ihrer Berechnung können die Sätze -\ref{buch:integrale:satz:polynom-ableitung-grad}, -\ref{buch:integrale:satz:log-polynom-ableitung-grad} -und -\ref{buch:integrale:satz:partialbruch-monom} -verwendet werden. -Diese besagen, dass in der Partialbruchzerlegung die Exponenten der -Nenner die Quotienten in der Summe nicht kleiner werden. -Die Ableitung $g'\in\mathscr{D}$ darf aber gar keine Nenner mit -$\vartheta$ enthalten, also dürfen die Quotienten gar nicht erst -vorkommen. -$g=H(\vartheta)$ muss also ein Polynom in $\vartheta$ sein. -Die Ableitung des Polynoms darf wegen $g'\in\mathscr{d}$ das Monom -$\vartheta$ ebenfalls nicht mehr enthalten, daher kann es höchstens vom -Grad $1$ sein. -Nach Satz~\ref{buch:integrale:satz:log-polynom-ableitung-grad} -muss ausserdem der Leitkoeffizient von $g$ eine Konstante sein, -das Polynom hat also genau die behauptete Form. -\end{proof} - -\begin{satz}[Liouville-Vorstufe für algebraische Elemente] -\label{buch:integrale:satz:liouville-vorstufe-2} -Sei $\vartheta$ algebraische über $\mathscr{D}$ und -$g\in\mathscr{D}(\vartheta)$ mit $g'\in\mathscr{D}$. -\end{satz} - -\subsubsection{Elementare Stammfunktionen} -Nach den Vorbereitungen über einfach elementare Stammfunktionen -in den Sätzen~\label{buch:integrale:satz:liouville-vorstufe-1} -und -\label{buch:integrale:satz:liouville-vorstufe-2} sind wir jetzt -in der Lage, den allgemeinen Satz von Liouville zu formulieren -und zu beweisen. - -\begin{satz}[Liouville] -Sei $\mathscr{D}$ ein Differentialkörper, $\mathscr{F}$ einfach über -$\mathscr{D}$ mit gleichem Konstantenkörper $\mathbb{C}$. -Wenn $g\in \mathscr{F}$ eine Stammfunktion von $f\in\mathscr{D}$ ist, -also $g'=f$, dann gibt es Zahlen $c_i\in\mathbb{C}$ und -$v_0,v_i\in\mathscr{D}$ derart, dass -\begin{equation} -g = v_0 + \sum_{i=1}^k c_i \log v_i -\qquad\Rightarrow\qquad -g' = v_0' + \sum_{i=1}^k c_i \frac{v_i'}{v_i} = f -\label{buch:integrale:satz:liouville-fform} -\end{equation} -gilt. -\end{satz} - -Der Satz hat zur Folge, dass eine elementare Stammfunktion für $f$ -nur dann existieren kann, wenn sich $f$ in der speziellen Form -\eqref{buch:integrale:satz:liouville-fform} -schreiben lässt. -Die Aufgabe~\ref{buch:integrale:aufgabe:existenz-stammfunktion-dalg} -lässt sich damit jetzt lösen. - - -\begin{proof}[Beweis] -Wenn die Stammfunktion $g\in\mathscr{D}$ ist, dann hat $g$ die Form -\eqref{buch:integrale:satz:liouville-fform} mit $v_0=g$, die Summe -wird nicht benötigt. - -Wir verwenden Induktion nach der Anzahl der Elemente, die zu $\mathscr{D}$ -hinzugefügt werden müssen, um einen Differentialkörper -$\mathscr{F}=\mathscr{D}(\vartheta_1,\dots,\vartheta_n)$ zu konstruieren, -der $g$ enthält. -Da $f\in\mathscr{D}\subset\mathscr{D}(\vartheta_1)$ ist, können wir die -Induktionsannahme auf die Erweiterung -\[ -\mathscr{D}(\vartheta_1)\subset\mathscr{D}(\vartheta_1,\vartheta_2) -\subset\cdots\subset \mathscr{D}(\vartheta_1,\cdots,\vartheta_n)=\mathscr{F} -\] -anwenden, die durch Hinzufügen von nur $n-1$ Elemente -$\vartheta_2,\dots,\vartheta_n$ aus $\mathscr{D}(\vartheta_1)$ den -Differentialkörper $\mathscr{F}$ erreicht, der $g$ enthält. -Sie besagt, dass sich $g$ schreiben lässt als -\[ -g = w_0 + \sum_{i=1}^{k_1} c_i\log w_i -\qquad\text{mit $c_i\in\mathbb{C}$ und $w_0,w_i\in\mathscr{D}(\vartheta_1)$.} -\] -Wir müssen jetzt zeigen, dass sich dieser Ausdruck umformen lässt -in den Ausdruck der Form~\eqref{buch:integrale:satz:liouville-fform}. - -Der Term $w_0\in\mathscr{D}(\vartheta_1)$ hat eine Partialbruchzerlegung -\[ -H(\vartheta_1) -+ -\sum_{j\le r(l)} \frac{P_{lj}(\vartheta_1)}{Q_l(\vartheta_1)^j} -\] -in der Variablen $\vartheta_1$. - -Da $w_i\in\mathscr{D}(\vartheta_1)$ ist, kann man Zähler und Nenner -von $w_i$ als Produkt irreduzibler normierter Polynome schreiben: -\[ -w_i -= -\frac{h_i Z_{i1}(\vartheta_1)^{s_{i1}}\cdots Z_{im(i)}^{s_{im(i)}} -}{ -N_{i1}(\vartheta_1)^{t_{i1}}\cdots N_{in(i)}(\vartheta_1)^{t_{in(i)}} -} -\] -Der Logarithmus hat die Form -\begin{align*} -\log w_i -&= \log h_i + -s_{i1} -\log Z_{i1}(\vartheta_1) -+ -\cdots -+ -s_{im(i)} -\log Z_{im(i)} -- -t_{i1} -\log -N_{i1}(\vartheta_1) -- -\cdots -- -t_{in(i)} -\log -N_{in(i)}(\vartheta_1). -\end{align*} -$g$ kann also geschrieben werden als eine Summe von Polynomen, Brüchen, -wie sie in der Partialbruchzerlegung vorkommen, Logarithmen von irreduziblen -normierten Polynomen und Logarithmen von Elementen von $\mathscr{D}$. - -Die Ableitung $g'$ muss jetzt aber wieder in $\mathscr{D}$ sein, beim -Ableiten müssen also alle Terme verschwinden, die $\vartheta_1$ enthalten. -Dabei spielt es eine Rolle, ob $\vartheta_1$ ein Monom oder algebraisch ist. -\begin{enumerate} -\item -Wenn $\vartheta_1$ ein Monom ist, dann kann man wie im Beweis des -Satzes~\ref{buch:integrale:satz:liouville-vorstufe-1} argumentieren, -dass die Brüchterme gar nicht vorkommen und -$H(\vartheta_1)=v_0+c_1\vartheta_1$ sein muss. -Die Ableitung Termen der Form $\log Z(\vartheta_1)$ ist ein Bruchterm -mit dem irreduziblen Nenner $Z(\vartheta_1)$, die ebenfalls verschwinden -müssen. -Ist $\vartheta_1$ eine Exponentialfunktion, dann ist -$\vartheta_1' \in \mathscr{D}(\vartheta_1)\setminus\mathscr{D}$, also muss -$c_1=0$ sein. -Ist $\vartheta_1$ ein Logarithmus, also $\vartheta_1=\log v_1$, dann -kommen nur noch Terme der in -\eqref{buch:integrale:satz:liouville-fform} -erlaubten Form vor. - -\item -Wenn $\vartheta_1$ algebraisch vom Grad $m$ ist, dann ist -\[ -g' = w_0' + \sum_{i=1}^{k_1} d_i\frac{w_i'}{w_i} = f. -\] -Weder $w_0$ noch $\log w_i$ sind in $\mathscr{D}(\vartheta_1)$. -Aber wenn man $\vartheta_1$ durch die $m$ konjugierten Elemente -ersetzt und alle summiert, dann ist -\[ -mf -= -\operatorname{Tr}(w_0) + \sum_{i=1}^{k_1} d_i \log\operatorname{Norm}(w_i). -\] -Da die Spur und die Norm in $\mathscr{D}$ sind, folgt, dass -\[ -f -= -\underbrace{\frac{1}{m} -\operatorname{Tr}(w_0)}_{\displaystyle= v_0} -+ -\sum_{i=1}^{k_1} \underbrace{\frac{d_i}{m}}_{\displaystyle=c_i} -\log -\underbrace{ \operatorname{Norm}(w_i)}_{\displaystyle=v_i} -= -v_0 + \sum_{i=1}^{k_1} c_i\log v_i -\] -die verlangte Form hat. -\qedhere -\end{enumerate} -\end{proof} - -\subsection{Die Fehlerfunktion ist keine elementare Funktion -\label{buch:integrale:section:fehlernichtelementar}} -% \url{https://youtu.be/bIdPQTVF5n4} -Mit Hilfe des Satzes von Liouville kann man jetzt beweisen, dass -die Fehlerfunktion keine elementare Funktion ist. -Dazu braucht man die folgende spezielle Form des Satzes. - -\begin{satz} -\label{buch:integrale:satz:elementarestammfunktion} -Wenn $f(x)$ und $g(x)$ rationale Funktionen von $x$ sind, dann -ist die Stammfunktion von $f(x)e^{g(x)}$ genau dann eine -elementare Funktion, wenn es eine rationale Funktion gibt, die -Lösung der Differentialgleichung -\[ -r'(x) + g'(x)r(x)=f(x) -\] -ist. -\end{satz} - -\begin{satz} -Die Funktion $x\mapsto e^{-x^2}$ hat keine elementare Stammfunktion. -\label{buch:iintegrale:satz:expx2} -\end{satz} - -\begin{proof}[Beweis] -Unter Anwendung des Satzes~\ref{buch:integrale:satz:elementarestammfunktion} -auf $f(x)=1$ und $g(x)=-x^2$ folgt, $e^{-x^2}$ genau dann eine rationale -Stammfunktion hat, wenn es eine rationale Funktion $r(x)$ gibt, die -Lösung der Differentialgleichung -\begin{equation} -r'(x) -2xr(x)=1 -\label{buch:integrale:expx2dgl} -\end{equation} -ist. - -Zunächst halten wir fest, dass $r(x)$ kein Polynom sein kann. -Wäre nämlich -\[ -r(x) -= -a_0 + a_1x + \dots + a_nx^n -= -\sum_{k=0}^n a_kx^k -\quad\Rightarrow\quad -r'(x) -= -a_1 + 2a_2x + \dots + na_nx^{n-1} -= -\sum_{k=1}^n -ka_kx^{k-1} -\] -ein Polynom, dann ergäbe sich beim Einsetzen in die Differentialgleichung -\begin{align*} -1 -&= -r'(x)-2xr(x) -\\ -&= -a_1 + 2a_2x + 3a_3x^2 + \dots + (n-1)a_{n-1}x^{n-2} + na_nx^{n-1} -\\ -&\qquad -- -2a_0x -2a_1x^2 -2a_2x^3 - \dots - 2a_{n-1}x^n - 2a_nx^{n+1} -\\ -& -\hspace{0.7pt} -\renewcommand{\arraycolsep}{1.8pt} -\begin{array}{crcrcrcrcrcrcrcr} -=&a_1&+&2a_2x&+&3a_3x^2&+&\dots&+&(n-1)a_{n-1}x^{n-2}&+&na_{n }x^{n-1}& & & & \\ - & &-&2a_0x&-&2a_1x^2&-&\dots&-& 2a_{n-3}x^{n-2}&-&2a_{n-2}x^{n-1}&-&2a_{n-1}x^n&-&2a_nx^{n+1} -\end{array} -\\ -&= -a_1 -+ -(2a_2-2a_0)x -+ -(3a_3-2a_1)x^2 -%+ -%(4a_4-2a_2)x^3 -+ -\dots -+ -(na_n-2a_{n-2})x^{n-1} -- -2a_{n-1}x^n -- -2a_nx^{n+1}. -\end{align*} -Koeffizientenvergleich zeigt, dass $a_1=1$ sein muss. -Aus den letzten zwei Termen liest man ebenfalls mittels Koeffizientenvergleich -ab, dass $a_n=0$ und $a_{n-1}=0$ sein müssen. -Aus den Koeffizienten $(ka_k-2a_{k-2})=0$ folgt, dass -$a_{k-2}=\frac{k}{2}a_k$ für alle $k>1$ sein muss, diese Koeffizienten -verschwinden also auch, inklusive $a_1=0$. -Dies ist allerdings im Widerspruch zu $a_1=1$. -Es folgt, dass $r(x)$ kein Polynom sein kann. - -Der Nenner der rationalen Funktion $r(x)$ hat also mindestens eine Nullstelle -$\alpha$, man kann daher $r(x)$ auch schreiben als -\[ -r(x) = \frac{s(x)}{(x-\alpha)^n}, -\] -wobei die rationale Funktion $s(x)$ keine Nullstellen und keine Pole hat. -Einsetzen in die Differentialgleichung ergibt: -\[ -1 -= -r'(x) -2xr(x) -= -\frac{s'(x)}{(x-\alpha)^n} --n -\frac{s(x)}{(x-\alpha)^{n+1}} -- -\frac{2xs(x)}{(x-\alpha)^n}. -\] -Multiplizieren mit $(x-\alpha)^{n+1}$ gibt -\[ -(x-\alpha)^{n+1} -= -s'(x)(x-\alpha) -- -ns(x) -- -2xs(x)(x-\alpha) -\] -Setzt man $x=\alpha$ ein, verschwinden alle Terme ausser dem mittleren -auf der rechten Seite, es bleibt -\[ -ns(\alpha) = 0. -\] -Dies widerspricht aber der Wahl der rationalen Funktion $s(x)$, für die -$\alpha$ keine Nullstelle ist. - -Somit kann es keine rationale Funktion $r(x)$ geben, die eine Lösung der -Differentialgleichung~\eqref{buch:integrale:expx2dgl} ist und -die Funktion $e^{-x^2}$ hat keine elementare Stammfunktion. -\end{proof} - -Der Satz~\ref{buch:iintegrale:satz:expx2} rechtfertigt die Einführung -der Fehlerfunktion $\operatorname{erf}(x)$ als neue spezielle Funktion, -mit deren Hilfe die Funktion $e^{-x^2}$ integriert werden kann. - - - +\rhead{Differentialkörper} +\input{chapters/060-integral/rational.tex} +\input{chapters/060-integral/erweiterungen.tex} +\input{chapters/060-integral/diffke.tex} +\input{chapters/060-integral/iproblem.tex} +\input{chapters/060-integral/irat.tex} +\input{chapters/060-integral/sqrat.tex} diff --git a/buch/chapters/060-integral/differentialkoerper2.tex b/buch/chapters/060-integral/differentialkoerper2.tex new file mode 100644 index 0000000..f41d3ba --- /dev/null +++ b/buch/chapters/060-integral/differentialkoerper2.tex @@ -0,0 +1,1953 @@ +% +% differentialalgebren.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\section{Differentialkörper und der Satz von Liouville +\label{buch:integrale:section:dkoerper}} +\rhead{Differentialkörper und der Satz von Liouville} +Das Problem der Darstellbarkeit eines Integrals in geschlossener +Form verlangt zunächst einmal nach einer Definition dessen, was man +als ``geschlossene Form'' akzeptieren will. +Die sogenannten {\em elementaren Funktionen} von +Abschnitt~\ref{buch:integrale:section:elementar} +bilden dafür den theoretischen Rahmen. +Das Problem ist dann die Frage zu beantworten, ob ein Integral eine +Stammfunktion hat, die eine elementare Funktion ist. +Der Satz von Liouville von Abschnitt~\ref{buch:integrale:section:liouville} +löst das Problem. + +\subsection{Eine Analogie +\label{buch:integrale:section:analogie}} +% XXX Analogie: Formel für Polynom-Nullstellen +% XXX Stammfunktion als elementare Funktion +Das Analysis-Problem, eine Stammfunktion zu finden, ist analog zum +wohlbekannten algebraischen Problem, Nullstellen von Polynomen zu finden. +Wir entwickeln diese Analogie in etwas mehr Detail, um zu sehen, ob man +aus dem algebraischen Problem etwas über das Problem der Analysis +lernen kann. + +Für ein Polynom $p(X) = a_nX^n+a_{n-1}X^{n-1}+\dots+a_1X+a_0\in\mathbb{C}[X]$ +mit Koeffizienten $a_k\in\mathbb{C}$ ist es sehr einfach, für jede beliebige +komplexe Zahl $z\in\mathbb{C}$ den Wert $p(z)$ des Polynoms auszurechnen. +Ein paar wenige Rechenregeln genügen dazu, man kann leicht einem Kind +beibringen, mit einem Taschenrechner so einen Wert auszurechnen. + +Ähnlich sieht es mit der Ableitungsoperation aus. +Einige wenige Ableitungsregeln, die man in der Analysis~I lernt, +erlauben, auf mehr oder weniger mechanische Art und Weise, jede +beliebige Funktion abzuleiten. +Man kann auch leicht einen Computer dazu programmieren, solche Ableitungen +symbolisch zu berechnen. + +Aus dem Fundamentalsatz der Algebra, der von Gauss vollständig bewiesen +wurde, ist bekannt, dass jedes Polynom mit Koeffizienten in $\mathbb{C}$ +genau so viele Lösungen in $\mathbb{C}$, wie der Grad des Polynoms angibt. +Dies ist aber ein Existenzsatz, er sagt nichts darüber aus, wie man diese +Lösungen finden kann. +In Spezialfällen, wie zum Beispiel für quadratische Polynome, gibt +es spezialsierte Lösungsverfahren, mit denen man Lösungen angeben kann. +Natürlich existieren numerische Methoden wie zum Beispiel das +Newton-Verfahren, mit dem man Nullstellen von Polynomen beliebig genau +bestimmen kann. + +Der Fundamentalsatz der Integralrechnung besagt, dass jede stetige +Funktion eine Stammfunktion hat, die bis auf eine Konstante eindeutig +bestimmt ist. +Auch dieser Existenzsatz gibt keinerlei Hinweise darauf, wie man die +Stammfunktion finden kann. +In der Analysis-Vorlesung lernt man viele Tricks, die in einer +beindruckenden Zahl von Spezialfällen ermöglichen, ein passende +Funktion anzugeben. +Man lernt auch numerische Verfahren kennen, mit denen sich Werte der +Stammfunktion, also bestimmte Integrale, mit beliebiger Genauigkeit +finden kann. + +Die numerische Lösung des Nullstellenproblems ist insofern unbefriedigend, +als sie nur schwer eine Diskussion der Abhängigkeit der Nullstellen von +den Koeffizienten des Polynoms ermöglichen. +Eine Formel wie die Lösungsformel für die quadratische Gleichung +stellt genau für solche Fälle ein ideales Werkzeug bereit. +Was man sich also wünscht ist nicht nur einfach eine Lösung, sondern eine +einfache Formel zur Bestimmung aller Lösungen. +Im Zusammenhang mit algebraischen Gleichungen erwartet man eine Formel, +in der nur arithmetische Operationen und Wurzeln vorkommen. +Für quadratische Gleichungen ist so eine Formel seit dem Altertum bekannt, +Formeln für die kubische Gleichung und die Gleichung vierten Grades wurden +im 16.~Jahrhundert von Cardano bzw.~Ferrari gefunden. +Erst viel später haben Abel und Ruffini gezeigt, dass so eine allgemeine +Formel für Polynome höheren Grades als 4 nicht existiert. +Die Galois-Theorie, die auf den Ideen von Évariste Galois beruht, +stellt eine vollständige Theorie unter anderem für die Lösbarkeit +von Gleichungen durch Wurzelausdrücke dar. + +Numerische Integralwerte haben ebenfalls den Nachteil, dass damit +Diskussionen wie die Abhängigkeit von Parametern eines Integranden +nur schwer möglich sind. +Was man sich daher wünscht ist eine Formel für die Stammfunktion, +die Werte als Zusammensetzung gut bekannter Funktionen wie der Exponential- +und Logarithmus-Funktionen oder der trigonometrischen Funktionen +sowie Wurzeln, Potenzen und den arithmetischen Operationen. +Man sagt, man möchte die Stammfunktion in ``geschlossener Form'' +dargestellt haben. +Tatsächlich ist dieses Problem auch zu Beginn des 19.~Jahrhunderts +von Joseph Liouville genauer untersucht worden. +Er hat zunächst eine Klasse von ``elementaren Funktionen'' definiert, +die als Darstellungen einer Stammfunktion in Frage kommen. +Der Satz von Liouville besagt dann, dass nur Funktionen mit einer +ganz speziellen Form eine elementare Stammfunktion haben. +Damit wird es möglich, zu entscheiden, ob ein Integrand wie $e^{-x^2}$ +eine elementare Stammfunktion hat. +Seit dieser Zeit weiss man zum Beispiel, dass die Fehlerfunktion nicht +mit den bekannten Funktionen dargestellt werden kann. + +Mit dem Aufkommen der Computer und vor allem der Computer-Algebra-System (CAS) +wurde die Frage nach der Bestimmung einer Stammfunktion erneut aktuell. +Die ebenfalls weiter entwickelte abstrakte Algebra hat ermöglicht, die +Ideen von Liouville in eine erweiterte, sogenannte differentielle +Galois-Theorie zu verpacken, die eine vollständige Lösung des Problems +darstellt. +Robert Henry Risch hat in den Sechzigerjahren auf dieser Basis +einen Algorithmus entwickelt, mit dem es möglich wird, zu entscheiden, +ob eine Funktion eine elementare Stammfunktion hat und diese +gegebenenfalls auch zu finden. +Moderne CAS implementieren diesen Algorithmus +in Teilen, besonders weit zu gehen scheint das quelloffene System +Axiom. + +Der Risch-Algorithmus hat allerdings eine Achillesferse: er benötigt +eine Method zu entscheiden, ob zwei Ausdrücke übereinstimmen. +Dies ist jedoch ein im Allgemeinen nicht entscheidbares Problem. +Moderne CAS treiben einigen Aufwand, um die +Gleichheit von Ausdrücken zu entscheiden, sie können das Problem +aber grundsätzlich nicht vollständig lösen. +Damit kann der Risch-Algorithmus in praktischen Anwendungen das +Stammfunktionsproblem ebenfalls nur mit Einschränkungen lösen, +die durch die Fähigkeiten des Ausdrucksvergleichs in einem CAS +gesetzt werden. + +Im Folgenden sollen elementare Funktionen definiert werden, es sollen +die Grundideen der differentiellen Galois-Theorie zusammengetragen werden +und der Satz von Liouvill vorgestellt werden. +An Hand der Fehler-Funktion soll dann gezeigt werden, wie man jetzt +einsehen kann, dass die Fehlerfunktion nicht elementar darstellbar ist. +Im nächsten Abschnitt dann soll der Risch-Algorithmus skizziert werden. + +\subsection{Elementare Funktionen +\label{buch:integrale:section:elementar}} +Es soll die Frage beantwortet werden, welche Stammfunktionen sich +in ``geschlossener Form'' oder durch ``wohlbekannte Funktionen'' +ausdrücken lassen. +Welche Funktionen dabei als ``wohlbekannt'' gelten dürfen ist +ziemlich willkürlich. +Sicher möchte man Potenzen und Wurzeln, Logarithmus und Exponentialfunktion, +aber auch die trigonometrischen Funktionen dazu zählen dürfen. +Ausserdem will man beliebig mit den arithmetischen Operationen +rechnen. +So entsteht die Menge der Funktionen, die man ``elementar'' nennen +will. + +In der Menge der elementaren Funktionen möchte man jetzt +Stammfunktionen ausgewählter Funktionen suchen. +Dazu muss man von jeder Funktion ihre Ableitung kennen. +Die Ableitungsoperation macht aus der Funktionenmenge eine +differentielle Algebra. +Der Satz von Liouville (Satz~\ref{buch:integrale:satz:liouville1}) +liefert Bedingungen, die erfüllt sein müssen, wenn eine Funktion +eine elementare Stammfunktion hat. +Sind diese Bedingungen nicht erfüllbar, ist auch keine +elementare Stammfunktion möglich. + +In den folgenden Abschnitten soll die differentielle Algebra +der elementaren Funktionen konstruiert werden. + +\subsubsection{Körper} +Die einfachsten Funktionen sind die die Konstanten, für die wir +für die nachfolgenden Betrachtungen fast immer die komplexen Zahlen +$\mathbb{C}$ +zu Grunde legen wollen. +Dabei ist vor allem wichtig, dass sich darin alle arithmetischen +Operationen durchführen lassen mit der einzigen Ausnahme, dass +nicht durch $0$ dividiert werden darf. +Man nennt $\mathbb{C}$ daher ein {\em Körper}. +\index{Körper}% +\label{buch:integrale:def:koerper} + +\subsubsection{Polynome und rationale Funktionen} +Die Polynome einer Variablen beschreiben eine Menge von +Funktionen, in der Addition, Subtraktion, Multiplikation +von Funktionen und Multiplikation mit komplexen Zahlen +uneingeschränkt möglich ist. +Wir bezeichen wie früher die Menge der Polynome in $z$ mit +$\mathbb{C}[z]$. + +Die Division ist erst möglich, wenn man beliebige Brüche +zulässt, deren Zähler und Nenner Polynome sind. +Die Menge +\[ +\mathbb{C}(z) += +\biggl\{ +\frac{p(z)}{q(z)} +\;\bigg|\; +p,q\in \mathbb{C}[z] +\biggr\} +\] +heisst die Menge der {\em rationalen Funktionen}. +\label{buch:integrale:def:rationalefunktion} +\index{Funktion, rationale}% +\index{rationale Funktion}% +In ihr sind jetzt alle arithmetischen Operationen ausführbar +ausser natürlich die Division durch die Nullfunktion. +Die rationalen Funktionen bilden also wieder eine Körper. + +Die Tatsache, dass die rationalen Funktionen einen Körper +bilden bedeutet auch, dass die Konstruktion erneut durchgeführt +werden kann. +Ausgehend von einem beliebigen Körper $K$ können wieder zunächst +die Polynome $K[X]$ und anschliesen die rationalen Funktionen $K[X]$ +in der neuen Variablen, jetzt aber mit Koeffizienten in $K$ +gebildet werden. +So entstehen Funktionen von mehreren Variablen und, indem +wir für die neue Variable $X$ zum Beispiel die im übernächsten +Abschnitt betrachtete Wurzel $X=\sqrt{z}$ +einsetzen, rationale Funktionen in $z$ und $\sqrt{z}$. + +Solche Funktionenkörper werden im folgenden mit geschweiften +Buchstaben $\mathscr{D}$ bezeichnet. +\index{Funktionenkörper}% + +\subsubsection{Ableitungsoperation} +In allen Untersuchungen soll immer die Ableitungsoperation +mit berücksichtigt werden. +In unserer Betrachtungsweise spielt es keine Rolle, dass die +Ableitung aus einem Grenzwert entsteht, es sind nur die algebraischen +Eigenschaften wichtig. +Diese sind in der folgenden Definition zusammengefasst. + +\begin{definition} +\label{buch:integrale:def:derivation} +Ein {\em Ableitungsoperator} oder eine {\em Derivation} einer Algebra +$\mathscr{D}$ von Funktionen ist eine lineare Abbildung +\[ +\frac{d}{dz} +\colon \mathscr{D} \to \mathscr{D} +: +f \mapsto \frac{df}{dz} = f', +\] +die zusätzlich die Produktregel +\begin{equation} +\frac{d}{dz} (fg) += +\frac{df}{dz} \cdot g + f \cdot \frac{dg}{dz} +\qquad\Leftrightarrow\qquad +(fg)' = f' g + fg' +\label{buch:integrale:eqn:produktregel} +\end{equation} +\index{Produktregel}% +erfüllt. +Die Funktion $f'\in \mathscr{D}$ heisst auch die {\em Ableitung} +von $f\in\mathscr{D}$. +\index{Derivation}% +\index{Ableitungsoperator}% +\index{Ableitung}% +\end{definition} + +Die Produktregel hat zum Beispiel auch die bekannten Quotientenregel +zur Folge. +Dazu betrachten wir das Produkt $f= (f/g)\cdot g$ und leiten es mit +Hilfe der Produktregel ab: +\[ +\frac{d}{dz}f += +\frac{d}{dz} +\biggl( +\frac{f}{g}\cdot g +\biggr) += +{\color{darkred} +\frac{d}{dz} +\biggl( +\frac{f}{g} +\biggr)} +\cdot g ++ +\frac{f}{g}\cdot \frac{d}{dz}g. +\] +Jetzt lösen wir nach der {\color{darkred}roten} Ableitung des Quotienten +auf und erhalten +\begin{equation} +\biggl(\frac{f}{g}\biggr)' += +\frac{d}{dz}\biggl(\frac{f}{g}\biggr) += +\frac1g\biggl( +\frac{d}{dz}f - \frac{f}{g}\cdot \frac{d}{dz}g +\biggr) += +\frac{1}{g} +\biggl( +f'-\frac{fg'}{g} +\biggr) += +\frac{f'g-fg'}{g^2}. +\label{buch:integrale:eqn:quotientenregel} +\end{equation} +Dies ist die Quotientenregel. + +Aus der Produktregel folgt natürlich sofort auch die Potenzregel +für die Ableitung der $n$ten Potenz einer Funktion $f\in\mathscr{D}$, +sie lautet: +\begin{equation} +\frac{d}{dz} f^n += +\underbrace{ +f'f^{n-1} + ff'f^{n-2} + f^2f'f^{n-3}+\dots f^{n-1}f' +}_{\displaystyle \text{$n$ Terme}} += +nf^{n-1}f'. +\label{buch:integrale:eqn:potenzregel} +\end{equation} +In dieser Form versteckt sich natürlich auch die Kettenregel, die +Potenzfunktion ist die äussere Funktion, $f$ die innere, $f'$ ist also +die Ableitung er inneren Funktion, wie in der Kettenregel verlangt. +Falls $f$ ein Element von $\mathscr{D}$ ist mit der Eigenschaft +$df/dz=1$, dann entsteht die übliche Produktregel. + +\begin{definition} +Eine Algebra $\mathscr{D}$ von Funktionen mit einem Ableitungsoperator +$d/dz$ heisst eine {\em differentielle Algebra}. +\index{differentielle Algebra}% +\index{Algebra, differentielle}% +In einer differentiellen Algebra gelten die üblichen +Ableitungsregeln. +\end{definition} + +Die Potenzregel war in der Form~\eqref{buch:integrale:eqn:potenzregel} +geschrieben worden, nicht als die Ableitung von $z$. +Der Grund dafür ist, dass wir gar nicht voraussetzen wollen, dass in +unserer differentiellen Algebra eine Funktion existiert, die die +Rolle von $z$ hat. +Dies ist gar nicht nötig, wie das folgende Beispiel zeigt. + +\begin{beispiel} +Als Funktionenmenge $\mathscr{D}$ nehmen wir rationale Funktionen +in zwei Variablen, die wir $\cos x $ und $\sin x$ nennen. +Diese Menge bezeichnen wir mit +$\mathscr{D}=\mathbb{Q}(\cos x,\sin x)$ +Der Ableitungsoperator ist +\begin{align*} +\frac{d}{dx} \cos x &= -\sin x +\\ +\frac{d}{dx} \sin x &= \phantom{-}\cos x. +\end{align*} +Die Funktionen von $\mathbb{Q}(\cos x,\sin x)$ sind also Brüche, +deren Zähler und Nenner Polynome in $\cos x$ und $\sin x$ sind. +Aus den Produkt- und Quotientenregeln und den Ableitungsregeln für +$\cos x$ und $\sin x$ folgt, dass die Ableitung einer Funktion in +$\mathscr{D}$ wieder in $\mathscr{D}$ ist, $\mathscr{D}$ ist eine +differentielle Algebra. +\end{beispiel} + +Die konstanten Funktionen spielen eine besondere Rolle. +Da wir bei der Ableitung nicht von der Vorstellung einer +Funktion mit einem variablen Argument ausgehen wollten und +die Ableitung nicht als Grenzwert definieren wollten, müssen +wir auch bei der Definition der ``Konstanten'' einen neuen +Weg gehen. +In der Analysis sind die Konstanten genau die Funktionen, +deren Ableitung $0$ ist. + +\begin{definition} +\label{buch:integrale:def:konstante} +Ein Element $f\in \mathscr{D}$ mit $df/dz=f'=0$ heissen +{\em Konstante} in $\mathscr{D}$. +\index{Konstante}% +\end{definition} + +Die in der Potenzregel~\eqref{buch:integrale:eqn:potenzregel} +vermisste Funktion $z$ kann man ähnlich zu den Konstanten +zu definieren versuchen. +$z$ müsste ein Element von $\mathscr{D}$ mit $z' = 1$ sein. +Allerdings gibt es viele solche Elemente, ist $c$ eine Konstanten +und $z'=1$, dann ist auch $(z+c)'=1$, $(z+c)$ hat also für +die Zwecke unserer Untersuchung die gleichen Eigenschaften wie +$z$. +Dies deckt sich natürlich auch mit der Erwartung, dass Stammfunktionen +nur bis auf eine Konstante bestimmt sind. +Eine differentielle Algebra muss allerdings kein Element $z$ mit der +Eigenschaft $z'=1$ enthalten. + +\begin{beispiel} +In $\mathscr{D}=\mathbb{Q}(\cos x,\sin x)$ gibt es kein Element $x$. +Ein solches wäre von der Form +\[ +x = \frac{p(\cos x,\sin x)}{q(\cos x,\sin x)}. +\] +Eine solche goniometrische Beziehung würde für $x=\frac{\pi}4$ bedeuten, +dass +\[ +\frac{\pi}4 += +\frac{p(\sqrt{2}/2,\sqrt{2}/2)}{q(\sqrt{2}/2,\sqrt{2}/2)}. +\] +Auf der rechten Seite steht ein Quotient von Polynome, in dessen +Argument nur rationale Zahlen und $\sqrt{2}$ steht. +So ein Ausdruck kann immer in die Form +\[ +\pi += +4\frac{a\sqrt{2}+b}{c\sqrt{2}+d} += +\frac{4(a\sqrt{2}+b)(c\sqrt{2}-d)}{2c^2+d^2} += +r\sqrt{2}+s +\] +gebracht werden. +Die Zahl auf der rechten Seite ist zwar irrational, aber sie ist Nullstelle +des quadratischen Polynoms +\[ +p(x) += +(x-r\sqrt{2}-s)(x+r\sqrt{2}-s) += +x^2 +-2sx +-2r^2+s^2 +\] +mit rationalen Koeffizienten, wie man mit der Lösungsformel für die +quadratische Gleichung nachprüfen kann. +Es ist bekannt, dass $\pi$ als transzendente Zahl nicht Nullstelle +eines Polynoms mit rationalen Koeffizienten ist. +Dieser Widerspruch zeigt, dass $x$ nicht in $\mathbb{Q}(\cos x, \sin x)$ +vorkommen kann. +\end{beispiel} + +In einer differentiellen Algebra kann jetzt die Frage nach der +Existenz einer Stammfunktion gestellt werden. + +\begin{aufgabe} +\label{buch:integrale:aufgabe:existenz-stammfunktion} +Gegeben eine differentielle Algebra $\mathscr{D}$ und ein Element +$f\in\mathscr{D}$, entscheide, ob es ein Element $F\in\mathscr{D}$ +gibt mit der Eigenschaft $F'=f$. +Ein solches $F\in\mathscr{D}$ heisst {\em Stammfunktion} von $f$. +\end{aufgabe} + +\begin{satz} +In einer differentiellen Algebra $\mathscr{D}$ mit $z\in\mathscr{D}$ +hat die Potenzfunktion $f=z^n$ für $n\in\mathbb{N}\setminus\{-1\}$ +ein Stammfunktion, nämlich +\[ +F = \frac{1}{n+1} z^{n+1}. +\] +\label{buch:integrale:satz:potenzstammfunktion} +\end{satz} + +\begin{proof}[Beweis] +Tatsächlich kann man dies sofort nachrechnen, muss allerdings die +Fälle $n+1 >0$ und $n+1<0$ unterscheiden, da die Potenzregel +\eqref{buch:integrale:eqn:potenzregel} nur für natürliche Exponenten +gilt. +Man erhält +\begin{align*} +n+1&>0\colon +& +\frac{d}{dz}\frac{1}{n+1}z^{n+1} +&= +\frac{1}{n+1}(n+1)z^{n+1-1} += +z^n, +\\ +n+1&<0\colon +& +\frac{d}{dz}\frac{1}{n+1}\frac{1}{z^{-(n+1)}} +&= +\frac{1}{n+1}\frac{1'z^{-(n+1)}-1(-(n+1))z^{-n-1-1}}{z^{-2n-2}} +\\ +&& +&= +\frac{1}{n+1} +\frac{(n+1)z^n{-n-2}}{z^{-2n-2}} +\\ +&& +&= +\frac{1}{z^{-n}}=z^n. +\end{align*} +Man beachte, dass in dieser Rechnung nichts anderes als die +algebraischen Eigenschaften der Produkt- und Quotientenregel +verwendet wurden. +\end{proof} + +\subsubsection{Wurzeln} +Die Wurzelfunktionen sollen natürlich als elementare Funktionen +erlaubt sein. +Es ist bekannt, dass $\sqrt{z}\not\in \mathscr{D}=\mathbb{C}(z)$ +ist, ein solches Element müsste also erst noch hinzugefügt werden. +Dabei muss auch seine Ableitung definiert werden. +Auch dabei dürfen wir nicht auf eine Grenzwertüberlegung zurückgreifen, +vielmehr müssen wir die Ableitung auf vollständig algebraische +Weise bestimmen. + +Wir schreiben $f=\sqrt{z}$ und leiten die Gleichung $f^2=z$ nach $z$ ab. +Dabei ergibt sich nach der Potenzregel +\[ +\frac{d}{dz}f^2 = 2f'f = \frac{d}{dz}z=1 +\qquad\Rightarrow\qquad f' = \frac{1}{2f}. +\] +Diese Rechnung lässt sich auch auf $n$-Wurzeln $g=\root{n}\of{z}$ mit +der Gleichung $g^n = z$ verallgemeinern. +Die Ableitung der $n$-ten Wurzel ist +\begin{equation} +\frac{d}{dz}g^n += +ng^{n-1} = \frac{d}{dz}z=1 +\qquad\Rightarrow\qquad +\frac{d}{dz}g = \frac{1}{ng^{n-1}}. +\end{equation} +Es ist also möglich, eine differentielle Algebra $\mathscr{D}$ mit einer +$n$-ten Wurzel $g$ zu einer grösseren differentiellen Algebra $\mathscr{D}(g)$ +zu erweitern, in der wieder alle Regeln für das Rechnen mit Ableitungen +erfüllt sind. + +\subsubsection{Algebraische Elemente} +Die Charakterisierung der Wurzelfunktionen passt zwar zum verlangten +algebraischen Vorgehen, ist aber zu spezielle und nicht gut für die +nachfolgenden Untersuchengen geeignet. +Etwas allgemeiner ist der Begriff der algebraischen Elemente. + +\begin{definition} +\label{buch:integrale:def:algebraisches-element} +Seien $K\subset L$ zwei Körper. +Ein Element $\alpha \in L$ heisst {\em algebraisch} über $K$, +wenn $\alpha$ Nullstelle eines Polynoms $p\in K[X]$ mit Koeffizienten +in $K$ ist. +\index{algebraisch}% +\end{definition} + +Jedes Element $\alpha\in K$ ist algebraisch, da $\alpha$ Nullstelle +von $X-\alpha\in K[X]$ ist. +Die $n$tem Wurzeln eines Elemente $\alpha\in K$ sind ebenfalls algebraisch, +da sie Nullstellen des Polynoms $p(X) = X^n - \alpha$ sind. +Allerdings ist nicht klar, dass diese Wurzeln überhaupt existieren. +Nach dem Satz von Abel~\ref{buch:potenzen:satz:abel} gibt es aber +Nullstellen von Polynomen, die sich nicht als Wurzelausdrücke schreiben +lassen. +Der Begriff der algebraischen Elemente ist also allgemeiner als der +Begriff der Wurzel. + +\begin{definition} +\label{buch:integrale:def:algebraisch-abgeschlossen} +Ein Körper $K$ heisst {\em algebraisch abgeschlossen}, wenn jedes Polynom mit +Koeffizienten in $K$ eine Nullstelle in $K$ hat. +\end{definition} + +Der Körper $\mathbb{C}$ ist nach dem +Fundamentalsatz~\label{buch:potenzen:satz:fundamentalsatz} +der Algebra algebraisch abgeschlossen. +Da wir aber mit Funktionen arbeiten, müssen wir auch Wurzeln +von Funktionen finden können. +Dies ist nicht selbstverständlich, wie das folgende Beispiel zeigt. + +\begin{beispiel} +Es gibt keine stetige Funktion $f\colon \mathbb{C}\to\mathbb{C}$, die +die Gleichung $f(z)^2 = z$ und $f(1)=1$ erfüllt. +Für die Argumente $z(t)= e^{it}$ folgt, dass $f(z(t)) = e^{it/2}$ sein +muss. +Setzt man aber $t=\pm \pi$ ein, ergeben sich die Werte +$f(z(\pm\pi))=e^{\pm i\pi/2}=\pm 1$, die beiden Grenzwerte +für $t\to\pm\pi$ sind also verschieden. +\end{beispiel} + +Die Mathematik hat verschiedene ``Tricks'' entwickelt, wie mit diesem +Problem umgegangen werden kann: Funktionskeime, Garben, Riemannsche +Flächen. +Sie sind alle gleichermassen gut geeignet, das Problem zu lösen. +Für die vorliegende Aufgabe genügt es aber, dass es tatsächlich +immer ein wie auch immer geartetes Element gibt, welches Nullstelle +des Polynoms ist. + +Ist $f$ eine Nullstelle des Polynoms $p(X)$ mit Koeffizienten in +$\mathscr{D}$, dann kann man die Ableitung wie folgt berechnen. +Zunächst leitet man $p(f)$ ab: +\begin{align} +0&= +\frac{d}{dz}(a_nf^n + a_{n-1}f^{n-1}+\ldots+a_1f+a_0) +\notag +\\ +&= +a_n'f^n + a_{n-1}'f^{n-1}+\ldots+a_1'f+a_0' ++ +na_nf^{n-1}f' ++ +(n-1)a_nf^{n-2}f' ++ +\ldots ++ +a_2ff' ++ +a_1f' +\notag +\\ +&= +a_n'f^n + a_{n-1}'f^{n-1}+\ldots+a_1'f+a_0' ++ +( +na_nf^{n-1} ++ +(n-1)a_nf^{n-2} ++ +\ldots ++ +a_2f ++ +a_1 +)f' +\notag +\\ +\Rightarrow +\qquad +f'&=\frac{ +a_n'f^n + a_{n-1}'f^{n-1}+\dots+a_1'f+a_0' +}{ +na_nf^{n-1} ++ +(n-1)a_nf^{n-2} ++ +\dots ++ +a_1 +}. +\label{buch:integrale:eqn:algabl} +\end{align} +Das einzige, was dabei schief gehen könnte ist, dass der Nenner ebenfalls +verschwindet. +Dieses Problem kann man dadurch lösen, dass man als Polynom das +sogenannte Minimalpolynom verwendet. + +\begin{definition} +Das {\em Minimalpolynome} $m(X)$ eines algebraischen Elementes $\alpha$ ist +das Polynom kleinsten Grades, welches $m(\alpha)=0$ erfüllt. +\end{definition} + +Da das Minimalpolynom den kleinstmöglichen Grad hat, kann der Nenner +von~\eqref{buch:integrale:eqn:algabl}, +der noch kleineren Grad hat, unmöglich verschwinden. +Das Minimalpolynom ist auch im wesentlichen eindeutig. +Gäbe es nämlich zwei verschiedene Minimalpolynome $m_1$ und $m_2$, +dann müsste $\alpha$ auch eine Nullstelle des grössten gemeinsamen +Teilers $m_3=\operatorname{ggT}(m_1,m_2)$ sein. +Wären die beiden Polynome wesentlich verschieden, dann hätte $m_3$ +kleineren Grad, im Widerspruch zur Definition des Minimalpolynoms. +Also unterscheiden sich die beiden Polynome $m_1$ und $m_2$ nur um +einen skalaren Faktor. + +\subsubsection{Konjugation, Spur und Norm} +% Konjugation, Spur und Norm +Das Minimalpolynom eines algebraischen Elementes ist nicht +eindeutig bestimmt. +Zum Beispiel ist $\sqrt{2}$ algebraisch über $\mathbb{Q}$, das +Minimalpolynom ist $m(X)=X^2-2\in\mathbb{Q}[X]$. +Es hat aber noch eine zweite Nullstelle $-\sqrt{2}$. +Mit rein algebraischen Mitteln sind die beiden Nullstellen $\pm\sqrt{2}$ +nicht zu unterscheiden, erst die Verwendung der Vergleichsrelation +ermöglicht, sie zu unterscheiden. + +Dasselbe gilt für die imaginäre Einheit $i$, die das Minimalpolynom +$m(X)=X^2+1\in\mathbb{R}[X]$ hat. +Hier gibt es nicht einmal mehr eine Vergleichsrelation, mit der man +die beiden Nullstellen unterscheiden könnte. +In der Tat ändert sich aus algebraischer Sicht nichts, wenn man in +allen Formeln $i$ durch $-i$ ersetzt. + +Etwas komplizierter wird es bei $\root{3}\of{2}$. +Das Polynom $m=x^3-2\in\mathbb{Q}[X]$ hat $\root{3}\of{2}$ als +Nullstelle und dies ist auch tatsächlich das Minimalpolynom. +Das Polynom hat noch zwei weitere Nullstellen +\[ +\alpha_+ = \frac{-1+i\sqrt{3}}{2}\root{3}\of{2} +\qquad\text{und}\qquad +\alpha_- = \frac{-1-i\sqrt{3}}{2}\root{3}\of{2}. +\] +Die beiden Lösungen gehen durch die Vertauschung von $i$ und $-i$ +auseinander hervor. +Betrachtet man dasselbe Polynom aber als Polynom in $\mathbb{R}[X]$, +dann ist es nicht mehr das Minimalpolynom von $\root{3}\of{2}$, da +$X-\root{3}\of{2}\in\mathbb{R}[X]$ kleineren Grad und $\root{3}\of{2}$ +als Nullstelle hat. +Indem man +\[ +m(X)/(X-\root{3}\of{2})=X^2+\root{3}\of{2}X+\root{3}\of{2}^2=m_2(X) +\] +rechnet, bekommt man das Minimalpolynom der beiden Nullstellen $\alpha_+$ +und $\alpha_-$. +Wir lernen aus diesen Beispielen, dass das Minimalpolynom vom Grundkörper +abhängig ist (Die Faktorisierung $(X-\root{3}\of{2})\cdot m_2(X)$ von +$m(X)$ ist in $\mathbb{Q}[X]$ nicht möglich) und dass wir keine +algebraische Möglichkeit haben, die verschiedenen Nullstellen des +Minimalpolynoms zu unterscheiden. + +Die beiden Nullstellen $\alpha_+$ und $\alpha_-$ des Polynoms $m_2(X)$ +erlauben, $m_2(X)=(X-\alpha_+)(X-\alpha_-)$ zu faktorisieren. +Durch Ausmultiplizieren +\[ +(X-\alpha_+)(X-\alpha_-) += +X^2 -(\alpha_++\alpha_-)X+\alpha_+\alpha_- +\] +und Koeffizientenvergleich mit $m_2(X)$ findet man die symmetrischen +Formeln +\[ +\alpha_+ + \alpha_- = \root{3}\of{2} +\qquad\text{und}\qquad +\alpha_+ \alpha_ = \root{3}\of{2}. +\] +Diese Ausdrücke sind nicht mehr abhängig von einer speziellen Wahl +der Nullstellen. + +Das Problem verschärft sich nocheinmal, wenn wir Funktionen betrachten. +Das Polynom $m(X)=X^3-z$ ist das Minimalpolynom der Funktion $\root{3}\of{z}$. +Die komplexe Zahl $z=re^{i\varphi}$ hat aber drei die algebraisch nicht +unterscheidbaren Nullstellen +\[ +\alpha_0(z)=\root{3}\of{r}e^{i\varphi/3}, +\quad +\alpha_1(z)=\root{3}\of{r}e^{i\varphi/3+2\pi/3} +\qquad\text{und}\qquad +\alpha_2(z)=\root{3}\of{r}e^{i\varphi/3+4\pi/3}. +\] +Aus der Faktorisierung $ (X-\alpha_0(z)) (X-\alpha_1(z)) (X-\alpha_2(z))$ +und dem Koeffizientenvergleich mit dem Minimalpolynom kann man wieder +schliessen, dass die Relationen +\[ +\alpha_0(z) + \alpha_1(z) + \alpha_2(z)=0 +\qquad\text{und}\qquad +\alpha_0(z) \alpha_1(z) \alpha_2(z) = z +\] +gelten. + +Wir können also oft keine Aussagen über individuelle Nullstellen +eines Minimalpolynoms machen, sondern nur über deren Summe oder +Produkt. + +\begin{definition} +\index{buch:integrale:def:spur-und-norm} +Sie $m(X)\in K[X]$ das Minimalpolynom eines über $K$ algebraischen +Elements und +\[ +m(X) = a_nX^n + a_{n-1}X^{n-1} + \ldots + a_1X + a_0. +\] +Dann heissen +\[ +\operatorname{Tr}(\alpha) = -a_{n-1} +\qquad\text{und}\qquad +\operatorname{Norm}(\alpha) = (-1)^n a_0 +\] +die {\em Spur} und die {\em Norm} des Elementes $\alpha$. +\index{Spur eines algebraischen Elementes}% +\index{Norm eines algebraischen Elementes}% +\end{definition} + +Die Spur und die Norm können als Spur und Determinante einer Matrix +verstanden werden, diese allgemeineren Definitionen, die man in der +Fachliteratur, z.~B.~in~\cite{buch:lang} nachlesen kann, führen aber +für unsere Zwecke zu weit. + +\begin{hilfssatz} +Die Ableitungen von Spur und Norm sind +\[ +\operatorname{Tr}(\alpha)' += +\operatorname{Tr}(\alpha') +\qquad\text{und}\qquad +\operatorname{Norm}(\alpha)' += +\operatorname{Tr}(\alpha)' +\] +XXX Wirklich? +\end{hilfssatz} + +\subsubsection{Logarithmen und Exponentialfunktionen} +Die Funktion $z^{-1}$ musste im +Satz~\ref{buch:integrale:satz:potenzstammfunktion} +ausgeschlossen werden, sie hat keine Stammfunktion in $\mathbb{C}(z)$. +Aus der Analysis ist bekannt, dass die Logarithmusfunktion $\log z$ +eine Stammfunktion ist. +Der Logarithmus von $z$ aber auch der Logarithmus $\log f(z)$ +einer beliebigen Funktion $f(z)$ oder die Exponentialfunktion $e^{f(z)}$ +sollen ebenfalls elementare Funktionen sein. +Da wir aber auch hier nicht auf die analytischen Eigenschaften zurückgreifen +wollen, brauchen wir ein rein algebraische Definition. + +\begin{definition} +\label{buch:integrale:def:logexp} +Sei $\mathscr{D}$ ein differentielle Algebra und $f\in\mathscr{D}$. +Ein Element $\vartheta\in\mathscr{D}$ heisst ein {\em Logarithmus} +von $f$, geschrieben $\vartheta = \log f$, wenn $f\vartheta' = f'$ gilt. +$\vartheta$ heisst eine Exponentialfunktion von $f$ wenn +$\vartheta'=\vartheta f'$ gilt. +\end{definition} + +Die Formel für die Exponentialfunktion ist etwas vertrauter, sie ist +die bekannte Kettenregel +\begin{equation} +\vartheta' += +\frac{d}{dz} e^f += +e^f \cdot \frac{d}{dz} f += +\vartheta \cdot f'. +\label{buch:integrale:eqn:exponentialableitung} +\end{equation} +Da wir uns vorstellen, dass Logarithmen Umkehrfunktionen von +Exponentialfunktionen sein sollen, +muss die definierende Gleichung genau wie +\eqref{buch:integrale:eqn:exponentialableitung} +aussehen, allerdings mit vertauschten Plätzen von $f$ und $\vartheta$, +also +\begin{equation} +\vartheta' = \vartheta\cdot f' +\qquad +\rightarrow +\qquad +f' = f\cdot \vartheta' +\;\Leftrightarrow\; +\vartheta' = (\log f)' = \frac{f'}{f}. +\label{buch:integrale:eqn:logarithmischeableitung} +\end{equation} +Dies ist die aus der Analysis bekannte Formel für die logarithmische +Ableitung. + +Der Logarithmus von $f$ und die Exponentialfunktion von $f$ sollen +also ebenfalls als elementare Funktionen betrachtet werden. + +\subsubsection{Die trigonometrischen Funktionen} +Die bekannten trigonometrischen Funktionen und ihre Umkehrfunktionen +sollten natürlich auch elementare Funktionen sein. +Dabei kommt uns zur Hilfe, dass sie sich mit Hilfe der Exponentialfunktion +als +\[ +\cos f = \frac{e^{if}+e^{-if}}2 +\qquad\text{und}\qquad +\sin f = \frac{e^{if}-e^{-if}}{2i} +\] +schreiben lassen. +Eine differentielle Algebra, die die Exponentialfunktionen von $if$ und +$-if$ enthält, enthält also automatisch auch die trigonometrischen +Funktionen. +Im Folgenden ist es daher nicht mehr nötig, die trigonometrischen +Funktionen speziell zu untersuchen. + +\subsubsection{Elementare Funktionen} +Damit sind wir nun in der Lage, den Begriff der elementaren Funktion +genau zu fassen. + +\begin{definition} +\label{buch:integrale:def:einfache-elementare-funktion} +Sie $\mathscr{D}$ eine differentielle Algebra über $\mathbb{C}$ und +$\mathscr{D}(\vartheta)$ eine Erweiterung von $\mathscr{D}$ um eine +neue Funktion $\vartheta$, dann heissen $\vartheta$ und die Elemente +von $\mathscr{D}(\vartheta)$ einfach elementar, wenn eine der folgenden +Bedingungen erfüllt ist: +\begin{enumerate} +\item $\vartheta$ ist algebraisch über $\mathscr{D}$, d.~h.~$\vartheta$ +ist eine ``Wurzel''. +\item $\vartheta$ ist ein Logarithmus einer Funktion in $\mathscr{D}$, +d.~h.~es gibt $f\in \mathscr{D}$ mit $f'=f\vartheta'$ +(Definition~\ref{buch:integrale:def:logexp}). +\item $\vartheta$ ist eine Exponentialfunktion einer Funktion in $\mathscr{D}$, +d.~h.~es bit $f\in\mathscr{D}$ mit $\vartheta'=\vartheta f'$ +(Definition~\ref{buch:integrale:def:logexp}). +\end{enumerate} +\end{definition} + +Einfache elementare Funktionen entstehen also ausgehend von einer +differentiellen Algebra, indem man genau einmal eine Wurzel, einen +Logarithmus oder eine Exponentialfunktion hinzufügt. +So etwas wie die zusammengesetzte Funktion $e^{\sqrt{z}}$ ist +damit noch nicht möglich. +Daher erlauben wir, dass man die gesuchten Funktionen in mehreren +Schritten aufbauen kann. + +\begin{definition} +Sei $\mathscr{F}$ eine differentielle Algebra, die die differentielle +Algebra $\mathscr{D}$ enthält, also $\mathscr{D}\subset\mathscr{F}$. +$\mathscr{F}$ und die Elemente von $\mathscr{F}$ heissen einfach, +wenn es endlich viele Elemente $\vartheta_1,\dots,\vartheta_n$ gibt +derart, dass +\[ +\renewcommand{\arraycolsep}{2pt} +\begin{array}{ccccccccccccc} +\mathscr{D} +&\subset& +\mathscr{D}(\vartheta_1) +&\subset& +\mathscr{D}(\vartheta_1,\vartheta_2) +&\subset& +\; +\cdots +\; +&\subset& +\mathscr{D}(\vartheta_1,\vartheta_2,\dots,\vartheta_{n-1}) +&\subset& +\mathscr{D}(\vartheta_1,\vartheta_2,\dots,\vartheta_{n-1},\vartheta_n) +&=& +\mathscr{F} +\\ +\| +&& +\| +&& +\| +&& +&& +\| +&& +\| +&& +\\ +\mathscr{F}_0 +&\subset& +\mathscr{F}_1 +&\subset& +\mathscr{F}_2 +&\subset& +\cdots +&\subset& +\mathscr{F}_{n-1} +&\subset& +\mathscr{F}_{n\mathstrut} +&& +\end{array} +\] +gilt so, dass jedes $\vartheta_{i+1}$ einfach ist über +$\mathscr{F}_i=\mathscr{D}(\vartheta_1,\dots,\vartheta_i)$. +\end{definition} + +In Worten bedeutet dies, dass man den Funktionen von $\mathscr{D}$ +nacheinander Wurzeln, Logarithmen oder Exponentialfunktionen einzelner +Funktionen hinzufügt. +Die Aufgabe~\ref{buch:integrale:aufgabe:existenz-stammfunktion} kann +jetzt so formuliert werden. + +\begin{aufgabe} +\label{buch:integrale:aufgabe:existenz-stammfunktion-dalg} +Gegeben ist eine Differentielle Algebra $\mathscr{D}$ und eine +Funktion $f\in \mathscr{D}$. +Gibt es eine Folge $\vartheta_1,\dots,\vartheta_n$ und eine Funktion +$F\in\mathscr{D}(\vartheta_1,\dots,\vartheta_n)$ derart, dass +$F'=f$. +\end{aufgabe} + +Das folgende Beispiel zeigt, wie man möglicherweise mehrere +Erweiterungsschritte vornehmen muss, um zu einer Stammfunktion +zu kommen. +Es illustriert auch die zentrale Rolle, die der Partialbruchzerlegung +in der weiteren Entwicklung zukommen wird. + +\begin{beispiel} +\label{buch:integrale:beispiel:nichteinfacheelementarefunktion} +Es soll eine Stammfunktion der Funktion +\[ +f(z) += +\frac{z}{(az+b)(cz+d)} +\in +\mathbb{C}(z) +\] +gefunden werden. +In der Analysis lernt man, dass solche Integrale mit der +Partialbruchzerlegung +\[ +\frac{z}{(az+b)(cz+d)} += +\frac{A_1}{az+b}+\frac{A_2}{cz+d} += +\frac{A_1cz+A_1d+A_2az+A_2b}{(az+b)(cz+d)} +\quad\Rightarrow\quad +\left\{ +\renewcommand{\arraycolsep}{2pt} +\begin{array}{rcrcr} +cA_1&+&aA_2&=&1\\ +dA_1&+&bA_2&=&0 +\end{array} +\right. +\] +bestimmt werden. +Die Lösung des Gleichungssystems ergibt +$A_1=b/(bc-ad)$ und $A_2=d/(ad-bc)$. +Die Stammfunktion kann dann aus +\begin{align*} +\int f(z)\,dz +&= +\int\frac{A_1}{az+b}\,dz ++ +\int\frac{A_2}{cz+d}\,dz += +\frac{A_1}{a}\int\frac{a}{az+b}\,dz ++ +\frac{A_2}{c}\int\frac{c}{cz+d}\,dz +\end{align*} +bestimmt werden. +In den Integralen auf der rechten Seite ist der Zähler jeweils die +Ableitung des Nenners, der Integrand hat also die Form $g'/g$. +Genau diese Form tritt in der Definition eines Logarithmus auf. +Die Stammfunktion ist jetzt +\[ +F(z) += +\int f(z)\,dz += +\frac{A_1}{a}\log(az+b) ++ +\frac{A_2}{c}\log(cz+d) += +\frac{b\log(az+b)}{a(bc-ad)} ++ +\frac{d\log(cz+d)}{c(ad-bc)}. +\] +Die beiden Logarithmen kann man nicht durch rein rationale Operationen +ineinander überführen. +Sie müssen daher beide der Algebra $\mathscr{D}$ hinzugefügt werden. +\[ +\left. +\begin{aligned} +\vartheta_1&=\log(az+b)\\ +\vartheta_2&=\log(cz+d) +\end{aligned} +\quad +\right\} +\qquad\Rightarrow\qquad +F(z) \in \mathscr{F}=\mathscr{D}(\vartheta_1,\vartheta_2). +\] +Die Stammfunktion $F(z)$ ist also keine einfache elementare Funktion, +aber $F$ ist immer noch eine elementare Funktion. +\end{beispiel} + +\subsection{Partialbruchzerlegung +\label{buch:integrale:section:partialbruchzerlegung}} +Die Konstruktionen des letzten Abschnitts haben gezeigt, +wie man die Funktionen, die man als Stammfunktionen einer Funktion +zulassen möchte, schrittweise konstruieren kann. +Die Aufgabe~\ref{buch:integrale:aufgabe:existenz-stammfunktion-dalg} +ist eine rein algebraische Formulierung der ursprünglichen +Aufgabe~\ref{buch:integrale:aufgabe:existenz-stammfunktion}. +Schliesslich hat das Beispiel auf +Seite~\pageref{buch:integrale:beispiel:nichteinfacheelementarefunktion} +gezeigt, dass es im allgemeinen mehrere Schritte braucht, um zu einer +elementaren Stammfunktion zu gelangen. +Die Lösung setzt sich aus den Termen der Partialbruchzerlegung. +In diesem Abschnitt soll diese genauer studiert werden. + +In diesem Abschnitt gehen wir immer von einer differentiellen +Algebra über den komplexen Zahlen aus und verlangen, dass die +Konstanten in allen betrachteten differentiellen Algebren +$\mathbb{C}$ sind. + +\subsubsection{Monome} +Die beiden Funktionen $\vartheta-1=\log(az+b)$ und $\vartheta_2=(cz+d)$, +die im Beispiel hinzugefügt werden mussten, verhalten sich ich algebraischer +Hinsicht wie ein Monom: man kann es nicht faktorisieren oder bereits +bekannte Summanden aufspalten. +Solchen Funktionen kommt eine besondere Bedeutung zu. + +\begin{definition} +\label{buch:integrale:def:monom} +Die Funktion $\vartheta$ heisst ein Monom, wenn $\vartheta$ nicht +algebraisch ist über $\mathscr{D}$ und $\mathscr{D}(\vartheta)$ die +gleichen Konstanten enthält wie $\mathscr{D}$. +\end{definition} + +\begin{beispiel} +Als Beispiel beginnen wir mit den komplexen Zahlen $\mathbb{C}$ +und fügen die Funktion $\vartheta_1=z$ hinzu und erhalten +$\mathscr{D}=\mathbb{C}(z)$. +Die Funktionen $z^k$ sind für alle $k$ linear unabhängig, d.~h.~es +gibt keinen Ausdruck +\[ +a_nz^n + a_{n-1}z^{n-1}+\cdots+a_1z+a_0=0. +\] +Dies ist gleichbedeutend damit, dass $z$ nicht algebraisch ist. +Das Monom $z$ ist also auch ein Monom im Sinne der +Definition~\ref{buch:integrale:def:monom}. +\end{beispiel} + +\begin{beispiel} +Wir beginnen wieder mit $\mathbb{C}$ und fügen die Funktion +$e^z$ hinzu. +Gäbe es eine Beziehung +\[ +b_m(e^z)^m + b_{m-1}(e^z)^{m-1}+\dots+b_1e^z + b_0=0 +\] +mit komplexen Koeffizienten $b_i\in\mathbb{C}$, +dann würde daraus durch Einsetzen von $z=1$ die Relation +\[ +b_me^m + b_{m-1}e^{m-1} + \dots + b_1e + b_0=0, +\] +die zeigen würde, dass $e$ eine algebraische Zahl ist. +Es ist aber bekannt, dass $e$ transzendent ist. +Dieser Widersprich zeigt, dass $e^z$ ein Monom ist. +\end{beispiel} + +\begin{beispiel} +Jetzt fügen wir die Exponentialfunktion $\vartheta_2=e^z$ +der differentiellen Algebra $\mathscr{D}=\mathbb{C}(z)$ hinzu +und erhalten $\mathscr{F}_1=\mathscr{D}(e^z) = \mathbb{C}(z,e^z)$. +Gäbe es das Minimalpolynom +\begin{equation} +b_m(z)(e^z)^m + b_{m-1}(z)(e^z)^{m-1}+\dots+b_1(z)e^z + b_0(z)=0 +\label{buch:integrale:beweis:exp-analytisch} +\end{equation} +mit Koeffizienten $b_i\in\mathbb{C}(z)$, dann könnte man mit dem +gemeinsamen Nenner der Koeffizienten durchmultiplizieren und erhielte +eine Relation~\eqref{buch:integrale:beweis:exp-analytisch} mit +Koeffizienten in $\mathbb{C}[z]$. +Dividiert man durch $e^{mz}$ erhält man +\[ +b_m(z) + b_{m-1}(z)\frac{1}{e^z} + \dots + b_1(z)\frac{1}{(e^z)^{m-1}} + b_0(z)\frac{1}{(e^z)^m}=0. +\] +Aus der Analysis weiss man, dass die Exponentialfunktion schneller +anwächst als jedes Polynom, alle Terme auf der rechten Seite +konvergieren daher gegen 0 für $z\to\infty$. +Das bedeutet, dass $b_m(z)\to0$ für $z\to \infty$. +Das Polynom~\eqref{buch:integrale:beweis:exp-analytisch} wäre also gar +nicht das Minimalpolynom. +Dieser Widerspruch zeigt, dass $e^z$ nicht algebraisch ist über +$\mathbb{C}(z)$ und damit ein Monom ist\footnote{Etwas unbefriedigend +an diesem Argument ist, dass man hier wieder rein analytische statt +algebraische Eigenschaften von $e^z$ verwendet. +Gäbe es aber eine minimale Relation wie +\eqref{buch:integrale:beweis:exp-analytisch} +mit Polynomkoeffizienten, dann wäre sie von der Form +\[ +P(z,e^z)=p(z)(e^z)^m + q(z,e^z)=0, +\] +wobei Grad von $e^z$ in $q$ höchstens $m-1$ ist. +Die Ableitung wäre dann +\[ +Q(z,e^z) += +mp(z)(e^z)^m + p'(z)(e^z)^m + r(z,e^z) += +(mp(z) + p'(z))(e^z)^m + r(z,e^z) +=0, +\] +wobei der Grad von $e^z$ in $r$ wieder höchstens $m-1$ ist. +Bildet man $mP(z,e^z) - Q(z,e^z) = 0$ ensteht eine Relation, +in der der Grad des Koeffizienten von $(e^z)^m$ um eins abgenommen hat. +Wiederholt man dies $m$ mal, verschwindet der Term $(e^z)^m$, die +Relation~\eqref{buch:integrale:beweis:exp-analytisch} +war also gar nicht minimal. +Dieser Widerspruch zeigt wieder, dass $e^z$ nicht algebraisch ist, +verwendet aber nur die algebraischen Eigenschaften der differentiellen +Algebra. +}. +\end{beispiel} + +\begin{beispiel} +Wir hätten auch in $\mathbb{Q}$ arbeiten können und $\mathbb{Q}$ +erst die Exponentialfunktion $e^z$ und dann den Logarithmus $z$ von $e^z$ +hinzufügen können. +Es gibt aber noch weitere Logarithmen von $e^z$ zum Beispiel $z+2\pi i$. +Offenbar ist $\psi=z+2\pi i\not\in \mathbb{Q}(z,e^z)$, wir könnten also +auch noch $\psi$ hinzufügen. +Zwar ist $\psi$ auch nicht algebraisch, aber wenn wir $\psi$ hinzufügen, +dann wird aber die Menge der Konstanten grösser, sie umfasst jetzt +$\mathbb{Q}(2\pi i)$. +Die Bedingung in der Definition~\ref{buch:integrale:def:monom}, +dass die Menge der Konstanten nicht grösser werden darf, ist also +verletzt. + +Hätte man mit $\mathbb{Q}(e^z, z+2\pi i)$ begonnen, wäre $z$ aus +dem gleichen Grund kein Monom, aber $z+2\pi i$ wäre eines im Sinne +der Definition~\ref{buch:integrale:def:monom}. +In allen Rechnungen könnte man $\psi=z+2\pi i$ nicht weiter aufteilen, +da $\pi$ oder seine Potenzen keine Elemente von $\mathbb{Q}(e^z)$ sind. +\end{beispiel} + +Da wir im Folgenden davon ausgehen, dass die Konstanten unserer +differentiellen Körper immer $\mathbb{C}$ sind, wird es jeweils +genügen zu untersuchen, ob eine neu hinzuzufügende Funktion algebraisch +ist oder nicht. + +\subsubsection{Ableitungen von Polynomen und rationalen Funktionen von Monomen} +Fügt man einer differentiellen Algebra ein Monom hinzu, dann lässt +sich etwas mehr über Ableitungen von Polynomen oder Brüchen in diesen +Monomen sagen. +Diese Eigenschaften werden später bei der Auflösung der Partialbruchzerlegung +nützlich sein. + +\begin{satz} +\label{buch:integrale:satz:polynom-ableitung-grad} +Sei +\[ +P += +A_nX^n + A_{n-1}X^{n-1} + \dots A_1X+A_0 +\in\mathscr{D}[X] +\] +ein Polynom mit Koeffizienten in einer differentiellen Algebra $\mathscr{D}$ +und $\vartheta$ ein Monom über $\mathscr{D}$. +Dann gilt +\begin{enumerate} +\item +\label{buch:integrale:satz:polynom-ableitung-grad-log} +Falls $\vartheta=\log f$ ist, ist $P(\vartheta)'$ ein +Polynom vom Grad $n$ in $\vartheta$, wenn der Leitkoeffizient $A_n$ +nicht konstant ist, andernfalls ein Polynom vom Grad $n-1$. +\item +\label{buch:integrale:satz:polynom-ableitung-grad-exp} +Falls $\vartheta = \exp f$ ist, dann ist $P(\vartheta)'$ ein Polynom +in $\vartheta$ vom Grad $n$. +\end{enumerate} +\end{satz} + +Der Satz macht also genaue Aussagen darüber, wie sich der Grad eines +Polynoms in $\vartheta$ beim Ableiten ändert. + +\begin{proof}[Beweis] +Für Exponentialfunktion ist $\vartheta'=\vartheta f'$, die Ableitung +fügt also einfach einen Faktor $f'$ hinzu. +Terme der Form $A_k\vartheta^k$ haben die Ableitung +\[ +(A_k\vartheta^k) += +A'_k\vartheta^k + A_kk\vartheta^{k-1}\vartheta' += +A'_k\vartheta^k + A_kk\vartheta^{k-1}\vartheta f' += +(A'_k + kA_k f)\vartheta^k. +\] +Damit wird die Ableitung des Polynoms +\begin{equation} +P(\vartheta)' += +\underbrace{(A'_n+nA_nf')\vartheta^n}_{\displaystyle=(A_n\vartheta^n)'} ++ +(A'_{n-1}+(n-1)A_{n-1}f')\vartheta^{n-1} ++ \dots + +(A'_1+A_1f')\vartheta + A_0'. +\label{buch:integrale:ableitung:polynom} +\end{equation} +Der Grad der Ableitung kann sich also nur ändern, wenn $A_n'+nA_nf'=0$ ist. +Dies bedeutet aber wegen +\( +(A_n\vartheta^n)' += +0 +\), dass $A_n\vartheta^n=c$ eine Konstante ist. +Da alle Konstanten bereits in $\mathscr{D}$ sind, folgt, dass +\[ +\vartheta^n=\frac{c}{A_n} +\qquad\Rightarrow\qquad +\vartheta^n - \frac{c}{A_n}=0, +\] +also wäre $\vartheta$ algebraisch über $\mathscr{D}$, also auch kein Monom. +Dieser Widerspruch zeigt, dass der Leitkoeffizient nicht verschwinden kann. + +Für die erste Aussage ist die Ableitung der einzelnen Terme des Polynoms +\[ +(A_k\vartheta^k)' += +A_k'\vartheta^k + A_kk\vartheta^{k-1}\vartheta' += +A_k'\vartheta^k + A_kk\vartheta^{k-1}\frac{f'}{f} += +\biggl(A_k'\vartheta + kA_k\frac{f'}{f}\biggr)\vartheta^{k-1}. +\] +Die Ableitung des Polynoms ist daher +\[ +P(\vartheta)' += +A_n'\vartheta^n + \biggl(nA_n\frac{f'}{f}+ A'_{n-1}\biggr)\vartheta^{n-1}+\dots +\] +Wenn $A_n$ keine Konstante ist, ist $A_n'\ne 0$ und der Grad von +$P(\vartheta)'$ ist $n$. +Wenn $A_n$ eine Konstante ist, müssen wir noch zeigen, dass der nächste +Koeffizient nicht verschwinden kann. +Wäre der zweite Koeffizient $=0$, dann wäre die Ableitung +\[ +(nA_n\vartheta+A_{n-1})' += +nA_n\vartheta'+A'_{n-1} += +nA_n\frac{f'}{f}+A'_{n-1} += +0, +\] +d.h. $nA_n\vartheta+A_{n-1}=c$ wäre eine Konstante. +Da alle Konstanten schon in $\mathscr{D}$ sind, müsste auch +\[ +\vartheta = \frac{c-A_{n-1}}{nA_n} \in \mathscr{D} +\] +sein, wieder wäre $\vartheta$ kein Monom. +\end{proof} + +Der nächste Satz gibt Auskunft über den führenden Term in +$(\log P(\vartheta))' = P(\vartheta)'/P(\vartheta)$. + +\begin{satz} +\label{buch:integrale:satz:log-polynom-ableitung-grad} +Sei $P$ ein Polynom vom Grad $n$ wie in +\label{buch:integrale:satz:log-polynom-ableitung} +welches zusätzlich normiert ist, also $A_n=1$. +\begin{enumerate} +\item +\label{buch:integrale:satz:log-polynom-ableitung-log} +Ist $\vartheta=\log f$, dann ist +$(\log P(\vartheta))' = P(\vartheta)'/P(\vartheta)$ und $P(\vartheta)'$ +hat Grad $n-1$. +\item +\label{buch:integrale:satz:log-polynom-ableitung-exp} +Ist $\vartheta=\exp f$, dann gibt es ein Polynom $N(\vartheta)$ so, dass +$(\log P(\vartheta))' += +P(\vartheta)'/P(\vartheta) += +N(\vartheta)/P(\vartheta)+nf'$ +ist. +Falls $P(\vartheta)=\vartheta$ ist $N=0$, andernfalls ist $N(\vartheta)$ +ein Polynom vom Grad $0$ das kleinste $k$ so, dass $p<(k+1)q$. +Insbesondere ist dann $kq\le p$. +Nach dem euklidischen Satz für die Division von $P(X)$ durch $Q(X)^k$ +gibt es ein Polynom $P_k(X)$ vom Grad $\le p-qk$ derart, dass +\[ +P(X) = P_k(X)Q(X)^k + R_k(X) +\] +mit einem Rest $R_k(X)$ vom Grad $1$ können mit der Potenzregel +integriert werden, aber für eine Stammfunktion $1/(z-1)$ muss +der Logarithmus $\log(z-1)$ hinzugefügt werden. +Die Stammfunktion +\[ +\int f(z)\,dz += +\int +\frac{1}{z-1} +\,dz ++ +\int +\frac{4}{(z-1)^2} +\,dz ++ +\int +\frac{4}{(z-1)^3} +\,dz += +\log(z-1) +- +\underbrace{\frac{4z-2}{(z-1)^2}}_{\displaystyle\in\mathscr{D}} +\in \mathscr{D}(\log(z-1)) = \mathscr{F} +\] +hat eine sehr spezielle Form. +Sie besteht aus einem Term in $\mathscr{D}$ und einem Logarithmus +einer Funktion von $\mathscr{D}$, also einem Monom über $\mathscr{D}$. + +\subsubsection{Einfach elementare Stammfunktionen} +Der in diesem Abschnitt zu beweisende Satz von Liouville zeigt, +dass die im einführenden Beispiel konstruierte Form der Stammfunktion +eine allgemeine Eigenschaft elementar integrierbarer +Funktionen ist. +Zunächst aber soll dieses Bespiel etwas verallgemeinert werden. + +\begin{satz}[Liouville-Vorstufe für Monome] +\label{buch:integrale:satz:liouville-vorstufe-1} +Sei $\vartheta$ ein Monom über $\mathscr{D}$ und $g\in\mathscr{D}(\vartheta)$ +mit $g'\in\mathscr{D}$. +Dann hat $g$ die Form $v_0 + c_1\vartheta$ mit $v_0\in\mathscr{D}$ und +$c_1\in\mathbb{C}$. +\end{satz} + +\begin{proof}[Beweis] +In Anlehnung an das einführende Beispiel nehmen wir an, dass die +Stammfunktion $g\in\mathscr{D}[\vartheta]$ für ein Monom $\vartheta$ +über $\mathscr{D}$ ist. +Dann hat $g$ die Partialbruchzerlegung +\[ +g += +H(\vartheta) ++ +\sum_{j\le r(i)} \frac{P_{ij}(\vartheta)}{Q_i(\vartheta)^j} +\] +mit irreduziblen normierten Polynomen $Q_i(\vartheta)$ und +Polynomen $P_{ij}(\vartheta)$ vom Grad kleiner als $\deg Q_i(\vartheta)$. +Ausserdem ist $H(\vartheta)$ ein Polynom. +Die Ableitung von $g$ muss jetzt aber wieder in $\mathscr{D}$ sein. +Zu ihrer Berechnung können die Sätze +\ref{buch:integrale:satz:polynom-ableitung-grad}, +\ref{buch:integrale:satz:log-polynom-ableitung-grad} +und +\ref{buch:integrale:satz:partialbruch-monom} +verwendet werden. +Diese besagen, dass in der Partialbruchzerlegung die Exponenten der +Nenner die Quotienten in der Summe nicht kleiner werden. +Die Ableitung $g'\in\mathscr{D}$ darf aber gar keine Nenner mit +$\vartheta$ enthalten, also dürfen die Quotienten gar nicht erst +vorkommen. +$g=H(\vartheta)$ muss also ein Polynom in $\vartheta$ sein. +Die Ableitung des Polynoms darf wegen $g'\in\mathscr{d}$ das Monom +$\vartheta$ ebenfalls nicht mehr enthalten, daher kann es höchstens vom +Grad $1$ sein. +Nach Satz~\ref{buch:integrale:satz:log-polynom-ableitung-grad} +muss ausserdem der Leitkoeffizient von $g$ eine Konstante sein, +das Polynom hat also genau die behauptete Form. +\end{proof} + +\begin{satz}[Liouville-Vorstufe für algebraische Elemente] +\label{buch:integrale:satz:liouville-vorstufe-2} +Sei $\vartheta$ algebraische über $\mathscr{D}$ und +$g\in\mathscr{D}(\vartheta)$ mit $g'\in\mathscr{D}$. +\end{satz} + +\subsubsection{Elementare Stammfunktionen} +Nach den Vorbereitungen über einfach elementare Stammfunktionen +in den Sätzen~\label{buch:integrale:satz:liouville-vorstufe-1} +und +\label{buch:integrale:satz:liouville-vorstufe-2} sind wir jetzt +in der Lage, den allgemeinen Satz von Liouville zu formulieren +und zu beweisen. + +\begin{satz}[Liouville] +Sei $\mathscr{D}$ ein Differentialkörper, $\mathscr{F}$ einfach über +$\mathscr{D}$ mit gleichem Konstantenkörper $\mathbb{C}$. +Wenn $g\in \mathscr{F}$ eine Stammfunktion von $f\in\mathscr{D}$ ist, +also $g'=f$, dann gibt es Zahlen $c_i\in\mathbb{C}$ und +$v_0,v_i\in\mathscr{D}$ derart, dass +\begin{equation} +g = v_0 + \sum_{i=1}^k c_i \log v_i +\qquad\Rightarrow\qquad +g' = v_0' + \sum_{i=1}^k c_i \frac{v_i'}{v_i} = f +\label{buch:integrale:satz:liouville-fform} +\end{equation} +gilt. +\end{satz} + +Der Satz hat zur Folge, dass eine elementare Stammfunktion für $f$ +nur dann existieren kann, wenn sich $f$ in der speziellen Form +\eqref{buch:integrale:satz:liouville-fform} +schreiben lässt. +Die Aufgabe~\ref{buch:integrale:aufgabe:existenz-stammfunktion-dalg} +lässt sich damit jetzt lösen. + + +\begin{proof}[Beweis] +Wenn die Stammfunktion $g\in\mathscr{D}$ ist, dann hat $g$ die Form +\eqref{buch:integrale:satz:liouville-fform} mit $v_0=g$, die Summe +wird nicht benötigt. + +Wir verwenden Induktion nach der Anzahl der Elemente, die zu $\mathscr{D}$ +hinzugefügt werden müssen, um einen Differentialkörper +$\mathscr{F}=\mathscr{D}(\vartheta_1,\dots,\vartheta_n)$ zu konstruieren, +der $g$ enthält. +Da $f\in\mathscr{D}\subset\mathscr{D}(\vartheta_1)$ ist, können wir die +Induktionsannahme auf die Erweiterung +\[ +\mathscr{D}(\vartheta_1)\subset\mathscr{D}(\vartheta_1,\vartheta_2) +\subset\cdots\subset \mathscr{D}(\vartheta_1,\cdots,\vartheta_n)=\mathscr{F} +\] +anwenden, die durch Hinzufügen von nur $n-1$ Elemente +$\vartheta_2,\dots,\vartheta_n$ aus $\mathscr{D}(\vartheta_1)$ den +Differentialkörper $\mathscr{F}$ erreicht, der $g$ enthält. +Sie besagt, dass sich $g$ schreiben lässt als +\[ +g = w_0 + \sum_{i=1}^{k_1} c_i\log w_i +\qquad\text{mit $c_i\in\mathbb{C}$ und $w_0,w_i\in\mathscr{D}(\vartheta_1)$.} +\] +Wir müssen jetzt zeigen, dass sich dieser Ausdruck umformen lässt +in den Ausdruck der Form~\eqref{buch:integrale:satz:liouville-fform}. + +Der Term $w_0\in\mathscr{D}(\vartheta_1)$ hat eine Partialbruchzerlegung +\[ +H(\vartheta_1) ++ +\sum_{j\le r(l)} \frac{P_{lj}(\vartheta_1)}{Q_l(\vartheta_1)^j} +\] +in der Variablen $\vartheta_1$. + +Da $w_i\in\mathscr{D}(\vartheta_1)$ ist, kann man Zähler und Nenner +von $w_i$ als Produkt irreduzibler normierter Polynome schreiben: +\[ +w_i += +\frac{h_i Z_{i1}(\vartheta_1)^{s_{i1}}\cdots Z_{im(i)}^{s_{im(i)}} +}{ +N_{i1}(\vartheta_1)^{t_{i1}}\cdots N_{in(i)}(\vartheta_1)^{t_{in(i)}} +} +\] +Der Logarithmus hat die Form +\begin{align*} +\log w_i +&= \log h_i + +s_{i1} +\log Z_{i1}(\vartheta_1) ++ +\cdots ++ +s_{im(i)} +\log Z_{im(i)} +- +t_{i1} +\log +N_{i1}(\vartheta_1) +- +\cdots +- +t_{in(i)} +\log +N_{in(i)}(\vartheta_1). +\end{align*} +$g$ kann also geschrieben werden als eine Summe von Polynomen, Brüchen, +wie sie in der Partialbruchzerlegung vorkommen, Logarithmen von irreduziblen +normierten Polynomen und Logarithmen von Elementen von $\mathscr{D}$. + +Die Ableitung $g'$ muss jetzt aber wieder in $\mathscr{D}$ sein, beim +Ableiten müssen also alle Terme verschwinden, die $\vartheta_1$ enthalten. +Dabei spielt es eine Rolle, ob $\vartheta_1$ ein Monom oder algebraisch ist. +\begin{enumerate} +\item +Wenn $\vartheta_1$ ein Monom ist, dann kann man wie im Beweis des +Satzes~\ref{buch:integrale:satz:liouville-vorstufe-1} argumentieren, +dass die Brüchterme gar nicht vorkommen und +$H(\vartheta_1)=v_0+c_1\vartheta_1$ sein muss. +Die Ableitung Termen der Form $\log Z(\vartheta_1)$ ist ein Bruchterm +mit dem irreduziblen Nenner $Z(\vartheta_1)$, die ebenfalls verschwinden +müssen. +Ist $\vartheta_1$ eine Exponentialfunktion, dann ist +$\vartheta_1' \in \mathscr{D}(\vartheta_1)\setminus\mathscr{D}$, also muss +$c_1=0$ sein. +Ist $\vartheta_1$ ein Logarithmus, also $\vartheta_1=\log v_1$, dann +kommen nur noch Terme der in +\eqref{buch:integrale:satz:liouville-fform} +erlaubten Form vor. + +\item +Wenn $\vartheta_1$ algebraisch vom Grad $m$ ist, dann ist +\[ +g' = w_0' + \sum_{i=1}^{k_1} d_i\frac{w_i'}{w_i} = f. +\] +Weder $w_0$ noch $\log w_i$ sind in $\mathscr{D}(\vartheta_1)$. +Aber wenn man $\vartheta_1$ durch die $m$ konjugierten Elemente +ersetzt und alle summiert, dann ist +\[ +mf += +\operatorname{Tr}(w_0) + \sum_{i=1}^{k_1} d_i \log\operatorname{Norm}(w_i). +\] +Da die Spur und die Norm in $\mathscr{D}$ sind, folgt, dass +\[ +f += +\underbrace{\frac{1}{m} +\operatorname{Tr}(w_0)}_{\displaystyle= v_0} ++ +\sum_{i=1}^{k_1} \underbrace{\frac{d_i}{m}}_{\displaystyle=c_i} +\log +\underbrace{ \operatorname{Norm}(w_i)}_{\displaystyle=v_i} += +v_0 + \sum_{i=1}^{k_1} c_i\log v_i +\] +die verlangte Form hat. +\qedhere +\end{enumerate} +\end{proof} + +\subsection{Die Fehlerfunktion ist keine elementare Funktion +\label{buch:integrale:section:fehlernichtelementar}} +% \url{https://youtu.be/bIdPQTVF5n4} +Mit Hilfe des Satzes von Liouville kann man jetzt beweisen, dass +die Fehlerfunktion keine elementare Funktion ist. +Dazu braucht man die folgende spezielle Form des Satzes. + +\begin{satz} +\label{buch:integrale:satz:elementarestammfunktion} +Wenn $f(x)$ und $g(x)$ rationale Funktionen von $x$ sind, dann +ist die Stammfunktion von $f(x)e^{g(x)}$ genau dann eine +elementare Funktion, wenn es eine rationale Funktion gibt, die +Lösung der Differentialgleichung +\[ +r'(x) + g'(x)r(x)=f(x) +\] +ist. +\end{satz} + +\begin{satz} +Die Funktion $x\mapsto e^{-x^2}$ hat keine elementare Stammfunktion. +\label{buch:iintegrale:satz:expx2} +\end{satz} + +\begin{proof}[Beweis] +Unter Anwendung des Satzes~\ref{buch:integrale:satz:elementarestammfunktion} +auf $f(x)=1$ und $g(x)=-x^2$ folgt, $e^{-x^2}$ genau dann eine rationale +Stammfunktion hat, wenn es eine rationale Funktion $r(x)$ gibt, die +Lösung der Differentialgleichung +\begin{equation} +r'(x) -2xr(x)=1 +\label{buch:integrale:expx2dgl} +\end{equation} +ist. + +Zunächst halten wir fest, dass $r(x)$ kein Polynom sein kann. +Wäre nämlich +\[ +r(x) += +a_0 + a_1x + \dots + a_nx^n += +\sum_{k=0}^n a_kx^k +\quad\Rightarrow\quad +r'(x) += +a_1 + 2a_2x + \dots + na_nx^{n-1} += +\sum_{k=1}^n +ka_kx^{k-1} +\] +ein Polynom, dann ergäbe sich beim Einsetzen in die Differentialgleichung +\begin{align*} +1 +&= +r'(x)-2xr(x) +\\ +&= +a_1 + 2a_2x + 3a_3x^2 + \dots + (n-1)a_{n-1}x^{n-2} + na_nx^{n-1} +\\ +&\qquad +- +2a_0x -2a_1x^2 -2a_2x^3 - \dots - 2a_{n-1}x^n - 2a_nx^{n+1} +\\ +& +\hspace{0.7pt} +\renewcommand{\arraycolsep}{1.8pt} +\begin{array}{crcrcrcrcrcrcrcr} +=&a_1&+&2a_2x&+&3a_3x^2&+&\dots&+&(n-1)a_{n-1}x^{n-2}&+&na_{n }x^{n-1}& & & & \\ + & &-&2a_0x&-&2a_1x^2&-&\dots&-& 2a_{n-3}x^{n-2}&-&2a_{n-2}x^{n-1}&-&2a_{n-1}x^n&-&2a_nx^{n+1} +\end{array} +\\ +&= +a_1 ++ +(2a_2-2a_0)x ++ +(3a_3-2a_1)x^2 +%+ +%(4a_4-2a_2)x^3 ++ +\dots ++ +(na_n-2a_{n-2})x^{n-1} +- +2a_{n-1}x^n +- +2a_nx^{n+1}. +\end{align*} +Koeffizientenvergleich zeigt, dass $a_1=1$ sein muss. +Aus den letzten zwei Termen liest man ebenfalls mittels Koeffizientenvergleich +ab, dass $a_n=0$ und $a_{n-1}=0$ sein müssen. +Aus den Koeffizienten $(ka_k-2a_{k-2})=0$ folgt, dass +$a_{k-2}=\frac{k}{2}a_k$ für alle $k>1$ sein muss, diese Koeffizienten +verschwinden also auch, inklusive $a_1=0$. +Dies ist allerdings im Widerspruch zu $a_1=1$. +Es folgt, dass $r(x)$ kein Polynom sein kann. + +Der Nenner der rationalen Funktion $r(x)$ hat also mindestens eine Nullstelle +$\alpha$, man kann daher $r(x)$ auch schreiben als +\[ +r(x) = \frac{s(x)}{(x-\alpha)^n}, +\] +wobei die rationale Funktion $s(x)$ keine Nullstellen und keine Pole hat. +Einsetzen in die Differentialgleichung ergibt: +\[ +1 += +r'(x) -2xr(x) += +\frac{s'(x)}{(x-\alpha)^n} +-n +\frac{s(x)}{(x-\alpha)^{n+1}} +- +\frac{2xs(x)}{(x-\alpha)^n}. +\] +Multiplizieren mit $(x-\alpha)^{n+1}$ gibt +\[ +(x-\alpha)^{n+1} += +s'(x)(x-\alpha) +- +ns(x) +- +2xs(x)(x-\alpha) +\] +Setzt man $x=\alpha$ ein, verschwinden alle Terme ausser dem mittleren +auf der rechten Seite, es bleibt +\[ +ns(\alpha) = 0. +\] +Dies widerspricht aber der Wahl der rationalen Funktion $s(x)$, für die +$\alpha$ keine Nullstelle ist. + +Somit kann es keine rationale Funktion $r(x)$ geben, die eine Lösung der +Differentialgleichung~\eqref{buch:integrale:expx2dgl} ist und +die Funktion $e^{-x^2}$ hat keine elementare Stammfunktion. +\end{proof} + +Der Satz~\ref{buch:iintegrale:satz:expx2} rechtfertigt die Einführung +der Fehlerfunktion $\operatorname{erf}(x)$ als neue spezielle Funktion, +mit deren Hilfe die Funktion $e^{-x^2}$ integriert werden kann. + + + diff --git a/buch/chapters/060-integral/diffke.tex b/buch/chapters/060-integral/diffke.tex new file mode 100644 index 0000000..53b46ad --- /dev/null +++ b/buch/chapters/060-integral/diffke.tex @@ -0,0 +1,20 @@ +% +% diffke.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue +% +\subsection{Differentialkörper und ihre Erweiterungen +\label{buch:integral:subsection:diffke}} +% +\subsubsection{Derivation} +% Ableitungsaxiome + +\subsubsection{Ableitungsregeln} +% Ableitungsregeln + +\subsubsection{Konstantenkörper} +% Konstantenkörper + +\subsubsection{Logarithmus und Exponentialfunktion} +% Logarithmus und Exponentialfunktion + diff --git a/buch/chapters/060-integral/elementar.tex b/buch/chapters/060-integral/elementar.tex new file mode 100644 index 0000000..2962178 --- /dev/null +++ b/buch/chapters/060-integral/elementar.tex @@ -0,0 +1,7 @@ +% +% elementar.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue +% +\subsection{Elementare Funktionen +\label{buch:integral:subsection:elementar}} diff --git a/buch/chapters/060-integral/erweiterungen.tex b/buch/chapters/060-integral/erweiterungen.tex new file mode 100644 index 0000000..f88f6e3 --- /dev/null +++ b/buch/chapters/060-integral/erweiterungen.tex @@ -0,0 +1,12 @@ +% +% erweiterungen.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue +% +\subsection{Körpererweiterungen +\label{buch:integral:subsection:koerpererweiterungen}} +% +% algebraische Zahl-Erweiterungen +% rationale Funktionen als Körpererweiterungen +% Erweiterungen mit algebraischen Funktionen +% diff --git a/buch/chapters/060-integral/iproblem.tex b/buch/chapters/060-integral/iproblem.tex new file mode 100644 index 0000000..85db464 --- /dev/null +++ b/buch/chapters/060-integral/iproblem.tex @@ -0,0 +1,93 @@ +% +% iproblem.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue +% +\subsection{Das Integrationsproblem +\label{buch:integral:subsection:integrationsproblem}} +\index{Integrationsproblem}% +Die Ableitung ist ein einem Differentialkörper mit Hilfe der Ableitungsregeln +immer ausführbar, ganz ähnlich wie die Berechnung von Potenzen in einem Körper +immer ausführbar ist. +Die Umkehrung, also eine sogenannte Stammfunktion zu finden, ist dagegen +deutlich schwieriger. + +\begin{definition} +\index{Stammfunktion} +Eine {\em Stammfunktion} einer Funktion $f\in\mathscr{K}$ im Funktionenkörper +$\mathscr{K}$ ist eine Funktion $F\in\mathscr{K}$ derart, dass $F'=f$. +Wir schreiben auch $F=\int f$. +\end{definition} + +Zwei Stammfunktionen $F_1$ und $F_2$ einer Funktion $f\in\mathscr{K}$ +haben die Eigenschaft +\[ +\left.\begin{aligned} +F_1' &= f \\ +F_2' &= f +\end{aligned}\quad\right\} +\qquad +\Rightarrow +\qquad +(F_1-F_2)' = 0 +\qquad\Rightarrow\qquad +F_1-F_2\in\mathscr{C}, +\] +die beiden Stammfunktionen unterscheiden sich daher nur durch eine +Konstante. + +\subsubsection{Stammfunktion von Polynomen} +Für Polynome ist das Problem leicht lösbar. +Aus der Ableitungsregel +\[ +\frac{d}{dx} x^n = nx^{n-1} +\] +folgt, dass +\[ +\int x^n = \frac{1}{n+1} x^{n+1} +\] +eine Stammfunktion von $x^n$ ist. +Da $\int$ linear ist, ergibt sich damit auch eine Stammfunktion für +ein beliebiges Polynom +\[ +g(x) += +g_0 + g_1x + g_2x^2 + \dots g_nx^n += +\sum_{k=0}^n g_kx^k +\in\mathbb{Q}(x) +\] +angeben: +\begin{equation} +\int g(x) += +g_0x + \frac12g_1x^2 + \frac13g_2x^3 + \dots \frac{1}{n+1}g_nx^{n+1} += +\sum_{k=0}^n +\frac{g_k}{k+1}x^{k+1}. +\label{buch:integral:iproblem:eqn:polyintegral} +\end{equation} + +\subsubsection{Körpererweiterungen} +Obwohl die Ableitung in einem Differentialkörper immer ausgeführt werden +kann, gibt es keine Garantie, dass es eine Stammfunktion im gleichen +Körper gibt. +Im kleinsten denkbaren Funktionenkörper $\mathbb{Q}(x)$ +haben die negativen Potenzen linearer Funktionen die Stammfunktionen +\[ +\int +\frac{1}{(x-\alpha)^k} += +\frac{1}{(-k+1)(x-\alpha)^{k-1}} +\] +für $k\ne 1$, sind also wieder in $\mathbb{Q}(x)$. +Für $k=1$ ist aber +\[ +\int \frac{1}{x-\alpha} += +\log(x-\alpha), +\] +es braucht also eine Körpererweiterung um $\log(x-\alpha)$, damit +$(x-\alpha)^{-1}$ eine Stammfunktion in $\mathbb{Q}(x,\log(x-\alpha))$ +hat. + diff --git a/buch/chapters/060-integral/irat.tex b/buch/chapters/060-integral/irat.tex new file mode 100644 index 0000000..2d03b7b --- /dev/null +++ b/buch/chapters/060-integral/irat.tex @@ -0,0 +1,140 @@ +% +% irat.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue +% +\subsection{Integration rationaler Funktionen +\label{buch:integral:subsection:rationalefunktionen}} +Für die Integration der rationalen Funktionen lernt man in einem +Analysis-Kurs üblicherweise ein Lösungsverfahren. +Dies zeigt zunächst, dass rationale Funktionen immer eine Stammfunktion +in einem geeigneten Erweiterungskörper haben. +Es deutet aber auch an, dass Stammfunktionen eine ziemlich spezielle +Form haben, die später als +Satz von Liouville~\ref{buch:integral:satz:liouville} +ein besondere Rolle spielen wird. + +% +% Aufgabenstellung +% +\subsubsection{Aufgabenstellung} +In diesem Abschnitt ist eine rationale Funktion $f(x)\in\mathbb{Q}(x)$ +gegeben, deren Stammfunktion bestimmt werden soll. +Als rationale Funktion kann sie als Bruch +\begin{equation} +f(x) = \frac{p(x)}{q(x)} +\label{buch:integral:irat:eqn:quotient} +\end{equation} +mit Polynomen $p(x),q(x)\in\mathbb{Q}[x]$ geschrieben werden. +Gesucht ist ein Erweiterungskörper $\mathscr{K}\supset \mathbb{Q}(x)$ +derart und eine Stammfunktion $F\in\mathscr{K}$ von $f$, also $F'=f$. + +% +% Polynomdivision +% +\subsubsection{Polynomdivision} +Der Quotient~\eqref{buch:integral:irat:eqn:quotient} kann durch Polynomdivision +mit Rest vereinfacht werden in einen polynomialen Teil und einen echten +Bruch: +\begin{equation} +f(x) += +g(x) ++ +\frac{a(x)}{b(x)} +\label{buch:integral:irat:eqn:polydiv} +\end{equation} +mit Polynomen $g(x),a(x),b(x)\in\mathbb[Q](x)$ und $\deg a(x) < \deg b(x)$. +Für den ersten Summanden liefert +\eqref{buch:integral:iproblem:eqn:polyintegral} eine Stammfunktion. +Im Folgenden bleibt also nur noch der zweite Term zu behandeln. + +% +% Partialbruchzerlegung +% +\subsubsection{Partialbruchzerlegung} +Zur Berechnung des Integral des Bruchs +in~\eqref{buch:integral:irat:eqn:polydiv} wird die Partialbruchzerlegung +benötigt. +Der Einfachheit halber nehmen wir an, dass wir den Körper $\mathbb{Q}(x)$ +mit alle Nullstellen $\beta_i$ des Nenner-Polynoms $b(x)$ zu einem Körper +$\mathscr{K}$ erweitert haben, in dem Nenner in Linearfaktoren zerfällt. +Unter diesen Voraussetzungen hat die Partialbruchzerlegung die Form +\begin{equation} +\frac{a(x)}{b(x)} += +\sum_{i=1}^m +\sum_{k=1}^{k_i} +\frac{A_{ik}}{(x-\beta_i)^k}, +\label{buch:integral:irat:eqn:partialbruch} +\end{equation} +wobei $k_i$ die Vielfachheit der Nullstelle $\beta_i$ ist. +Die Koeffizienten $A_{ik}$ können zum Beispiel mit Hilfe eines linearen +Gleichungssystems bestimmt werden. + +Um eine Stammfunktion zu finden, muss man also Stammfunktionen für +jeden einzelnen Summanden bestimmen. +Für Exponenten $k>1$ im Nenner eines Terms der +Partialbruchzerlegung~\eqref{buch:integral:irat:eqn:partialbruch} +kann dazu die Regel +\[ +\int \frac{A_{ik}}{(x-\beta_i)^k} += +\frac{A_{ik}}{(-k+1)(x-\beta_i)^{k-1}} +\] +verwendet werden. +Diese Stammfunktion liegt wieder in $\mathbb{Q}(x)$ liegt. + +% +% Körpererweiterungen +% +\subsubsection{Körpererweiterung} +Für $k=1$ ist eine logarithmische Erweiterung um die Funktion +\begin{equation} +\int \frac{A_{i1}}{x-\alpha_i} += +A_{i1} +\log(x-\alpha_i) +\label{buch:integral:irat:eqn:logs} +\end{equation} +nötig. +Es gibt also eine Stammfunktion in einem Erweiterungskörper, sofern +er zusätzlich alle logarithmischen Funktionen +in~\ref{buch:integral:irat:eqn:logs} enthält. +Sie hat die Form +\[ +\sum_{i=1}^m A_{i1} \log(x-\beta_i), +\] +wobei $A_{i1}\in\mathbb{Q}$ ist. + +Setzt man alle vorher schon gefundenen Teile der Stammfunktion zusammen, +kann man sehen, dass die Stammfunktion die Form +\begin{equation} +F(x) = v_0(x) + \sum_{i=1}^m c_i \log v_i(x) +\label{buch:integral:irat:eqn:liouvillstammfunktion} +\end{equation} +haben muss. +Dabei ist $v_0(x)\in\mathbb{Q}(x)$ und besteht aus der Stammfunktion +des polynomiellen Teils und den Stammfunktionen der Terme der Partialbruchzerlegung mit Exponenten $k>1$. +Die logarithmischen Terme bestehen aus den Konstanten $c_i=A_{i1}$ +und den Logarithmusfunktionen $v_i(x)=x-\beta_i\in\mathbb{Q}(x)$. +Die Funktion $f(x)$ muss daher die Form +\[ +f(x) += +v_0'(x) ++ +\sum_{i=1}^m c_i\frac{v'_i(x)}{v_i(x)} +\] +gehabt haben. +Die Form~\eqref{buch:integral:irat:eqn:liouvillstammfunktion} +der Stammfunktion ist nicht eine Spezialität der rationalen Funktionen. +Sie wird auch bei grösseren Funktionenkörpern immer wieder auftreten +und ist als Satz von Liouville bekannt. + +% +% Minimale algebraische Erweiterung +% +\subsubsection{Minimale algebraische Erweiterung} +XXX Rothstein-Trager + diff --git a/buch/chapters/060-integral/logexp.tex b/buch/chapters/060-integral/logexp.tex new file mode 100644 index 0000000..7cbb906 --- /dev/null +++ b/buch/chapters/060-integral/logexp.tex @@ -0,0 +1,27 @@ +% +% logexp.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue +% +\subsection{Log-Exp-Notation für elementare Funktionen +\label{buch:integral:subsection:logexp}} +Die Integration rationaler Funktionen hat bereits gezeigt, dass +eine Stammfunktion nicht immer im Körper der rationalen Funktionen +existiert. +Es kann notwendig sein, dem Körper logarithmische Erweiterungen der Form +$\log(x-\alpha)$ hinzuzufügen. + +Es können jedoch noch ganz andere neue Funktionen auftreten, wie die +folgende Zusammenstellung einiger Stammfunktionen zeigt: +\begin{align*} +\int\frac{dx}{1+x^2} +&= +\arctan x +\\ +\end{align*} + + + + + + diff --git a/buch/chapters/060-integral/rational.tex b/buch/chapters/060-integral/rational.tex new file mode 100644 index 0000000..19f2ad9 --- /dev/null +++ b/buch/chapters/060-integral/rational.tex @@ -0,0 +1,8 @@ +% +% rational.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue +% +\subsection{Rationale Funktionen und Funktionenkörper +\label{buch:integral:subsection:rational}} + diff --git a/buch/chapters/060-integral/risch.tex b/buch/chapters/060-integral/risch.tex index 6c8ff96..1ba746a 100644 --- a/buch/chapters/060-integral/risch.tex +++ b/buch/chapters/060-integral/risch.tex @@ -6,7 +6,8 @@ \section{Der Risch-Algorithmus \label{buch:integral:section:risch}} \rhead{Risch-Algorithmus} - +\input{chapters/060-integral/logexp.tex} +\input{chapters/060-integral/elementar.tex} diff --git a/buch/chapters/060-integral/sqrat.tex b/buch/chapters/060-integral/sqrat.tex new file mode 100644 index 0000000..71eb39b --- /dev/null +++ b/buch/chapters/060-integral/sqrat.tex @@ -0,0 +1,8 @@ +% +% sqrat.tex +% +% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue +% +\subsection{Integranden der Form $R(x,\sqrt{ax^2+bx+c})$ +\label{buch:integral:subsection:rxy}} + -- cgit v1.2.1 From a5b447ef1ab21d9dcb88d696862c75b81e994a32 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 23 May 2022 12:36:40 +0200 Subject: more rational integration stuff --- buch/chapters/060-integral/irat.tex | 8 +-- buch/chapters/060-integral/sqrat.tex | 110 +++++++++++++++++++++++++++++++++++ 2 files changed, 114 insertions(+), 4 deletions(-) (limited to 'buch') diff --git a/buch/chapters/060-integral/irat.tex b/buch/chapters/060-integral/irat.tex index 2d03b7b..4c472ea 100644 --- a/buch/chapters/060-integral/irat.tex +++ b/buch/chapters/060-integral/irat.tex @@ -83,7 +83,7 @@ kann dazu die Regel \frac{A_{ik}}{(-k+1)(x-\beta_i)^{k-1}} \] verwendet werden. -Diese Stammfunktion liegt wieder in $\mathbb{Q}(x)$ liegt. +Diese Stammfunktion liegt wieder in $\mathscr{K}(x)$ liegt. % % Körpererweiterungen @@ -105,7 +105,7 @@ Sie hat die Form \[ \sum_{i=1}^m A_{i1} \log(x-\beta_i), \] -wobei $A_{i1}\in\mathbb{Q}$ ist. +wobei $A_{i1}\in\mathscr{K}$ ist. Setzt man alle vorher schon gefundenen Teile der Stammfunktion zusammen, kann man sehen, dass die Stammfunktion die Form @@ -114,10 +114,10 @@ F(x) = v_0(x) + \sum_{i=1}^m c_i \log v_i(x) \label{buch:integral:irat:eqn:liouvillstammfunktion} \end{equation} haben muss. -Dabei ist $v_0(x)\in\mathbb{Q}(x)$ und besteht aus der Stammfunktion +Dabei ist $v_0(x)\in\mathscr{K}(x)$ und besteht aus der Stammfunktion des polynomiellen Teils und den Stammfunktionen der Terme der Partialbruchzerlegung mit Exponenten $k>1$. Die logarithmischen Terme bestehen aus den Konstanten $c_i=A_{i1}$ -und den Logarithmusfunktionen $v_i(x)=x-\beta_i\in\mathbb{Q}(x)$. +und den Logarithmusfunktionen $v_i(x)=x-\beta_i\in\mathscr{K}(x)$. Die Funktion $f(x)$ muss daher die Form \[ f(x) diff --git a/buch/chapters/060-integral/sqrat.tex b/buch/chapters/060-integral/sqrat.tex index 71eb39b..38b1504 100644 --- a/buch/chapters/060-integral/sqrat.tex +++ b/buch/chapters/060-integral/sqrat.tex @@ -5,4 +5,114 @@ % \subsection{Integranden der Form $R(x,\sqrt{ax^2+bx+c})$ \label{buch:integral:subsection:rxy}} +Für rationale Funktionen lässt sich immer eine Stammfunktion in einem +Erweiterungskörper angeben, der durch hinzufügen einzelner logarithmischer +Funktionen entsteht. +Die dabei verwendeten Techniken lassen sich verallgemeinern. +Zur Illustration und Motivation des später beschriebenen Risch-Algorithmus +stellen wir uns in diesem Abschnitt der Aufgabe, Integrale +mit einem Integranden zu berechnen, der eine rationale Funktion von $x$ +und $\sqrt{ax^2+bx+c}$ ist. + +% +% Aufgabenstellung +% +\subsubsection{Aufgabenstellung} +Eine rationale Funktion von $x$ und $\sqrt{ax^2+bx+c}$ ist ein +Element des Differentialkörpers, den man aus $\mathbb{Q}(x)$ durch +hinzufügen des Elementes +\[ +y=\sqrt{ax^2+bx+c} +\] +erhält. +Eine Funktion $f\in\mathbb{Q}(x,y)$ kann geschrieben werden als Bruch +\begin{equation} +f += +\frac{ +\tilde{p}_0 + \tilde{p}_1y + \dots + \tilde{p}_n y^n +}{ +\tilde{q}_0 + \tilde{q}_1y + \dots + \tilde{q}_m y^m +} +\label{buch:integral:sqrat:eqn:ftilde} +\end{equation} +mit rationalen Koeffizienten $\tilde{p}_i,\tilde{q}_i\in\mathbb{Q}(x)$. +Gesucht ist eine Stammfunktion von $f$. + +% +% Algebraische Vereinfachungen +% +\subsubsection{Algebraische Vereinfachungen} +Da $x^2=ax^2+bx+c$ ein Polynom ist, sind auch alle geraden Potenzen +von $y$ Polynome in $\mathbb{Q}(x)$, +und die ungeraden Potenzen von $y$ lassen sich als Produkt aus einem +Polynom und dem Faktor $y$ schreiben. +Der Integrand~\eqref{buch:integral:sqrat:eqn:ftilde} +lässt sich daher vereinfachen zu einem Bruch der Form +\begin{equation} +f(x) += +\frac{p_0+p_1y}{q_0+q_1y}, +\label{buch:integral:sqrat:eqn:moebius} +\end{equation} +wobei $p_i$ und $q_i$ rationale Funktionen in $\mathbb{Q}(x)$ sind. + +% +% Rationalisieren +% +\subsubsection{Rationalisieren} +Unschön an der Form~\eqref{buch:integral:sqrat:eqn:moebius} ist die +Tatsache, dass $y$ sowohl im Nenner wie auch im Zähler auftreten kann. +Da aber $y$ die quadratische Identität $y^2=ax^2+bx+c$ erfüllt, +kann das $y$ im Nenner durch Erweitern mit $q_0-q_1y$ zum verschwinden +gebracht werden. +Die Rechnung ergibt +\begin{align*} +\frac{p_0+p_1y}{q_0+q_1y} +&= +\frac{p_0+p_1y}{q_0+q_1y} +\cdot +\frac{q_0-q_1y}{q_0-q_1y} += +\frac{(p_0+p_1y)(q_0-q_1y)}{q_0^2-q_1^2y^2} +\\ +&= +\frac{p_0q_0-p_1q_1(ax^2+bx+c)}{q_0^2-q_1^2(ax^2+bx+c)} ++ +\frac{q_0p_1-q_1p_0}{q_0^2-q_1^2(ax^2+bx+c)} y. +\end{align*} +Die Quotienten enthalten $y$ nicht mehr, sind also in $\mathbb{Q}(x)$. +In der späteren Rechnung stellt sich heraus, dass es praktischer ist, +das $y$ im Nenner zu haben, was man durch erweitern mit $y$ wieder +unter Ausnützung von $y^2=ax^2+bx+c$ erreichen kann. +Die zu integrierende Funktion kann also in der Form +\begin{equation} +f(x) += +W_1 + W_2\frac{1}{y} +\end{equation} +geschrieben werden mit rationalen Funktionen +$W_1,W_2\in\mathbb{Q}(x)$. +Eine Stammfunktion von $W_1$ kann mit der Methode von +Abschnitt~\ref{buch:integral:subsection:rationalefunktionen} +gefunden werden. +Im Folgenden kümmern wir uns daher nur noch um $W_1$. + +\subsubsection{Polynomdivision} + +\subsubsection{Integranden der Form $p(x)/y$} + +\subsubsection{Partialbruchzerlegung} + +\begin{equation} +\int +\frac{1}{(x-\alpha)^k \sqrt{ax^2+bx+c}} +\label{buch:integral:sqrat:eqn:2teart} +\end{equation} + +\subsubsection{Integrale der Form \eqref{buch:integral:sqrat:eqn:2teart}} + + + + -- cgit v1.2.1 From e8bb3fd399f2261c9b430ffa319626950499d4c1 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 23 May 2022 23:06:11 +0200 Subject: new spherical graph --- buch/papers/kugel/images/Makefile | 12 ++- buch/papers/kugel/images/spherecurve.m | 160 ++++++++++++++++++++++++++++ buch/papers/kugel/images/spherecurve.maxima | 13 +++ buch/papers/kugel/images/spherecurve.pov | 73 +++++++++++++ 4 files changed, 257 insertions(+), 1 deletion(-) create mode 100644 buch/papers/kugel/images/spherecurve.m create mode 100644 buch/papers/kugel/images/spherecurve.maxima create mode 100644 buch/papers/kugel/images/spherecurve.pov (limited to 'buch') diff --git a/buch/papers/kugel/images/Makefile b/buch/papers/kugel/images/Makefile index e8bf919..6187fed 100644 --- a/buch/papers/kugel/images/Makefile +++ b/buch/papers/kugel/images/Makefile @@ -3,7 +3,7 @@ # # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -all: curvature.jpg +all: curvature.jpg spherecurve.jpg curvature.inc: curvgraph.m octave curvgraph.m @@ -13,3 +13,13 @@ curvature.png: curvature.pov curvature.inc curvature.jpg: curvature.png convert curvature.png -density 300 -units PixelsPerInch curvature.jpg + +spherecurve.inc: spherecurve.m + octave spherecurve.m + +spherecurve.png: spherecurve.pov spherecurve.inc + povray +A0.1 +W1920 +H1080 +Ospherecurve.png spherecurve.pov + +spherecurve.jpg: spherecurve.png + convert spherecurve.png -density 300 -units PixelsPerInch spherecurve.jpg + diff --git a/buch/papers/kugel/images/spherecurve.m b/buch/papers/kugel/images/spherecurve.m new file mode 100644 index 0000000..ea9c901 --- /dev/null +++ b/buch/papers/kugel/images/spherecurve.m @@ -0,0 +1,160 @@ +# +# spherecurv.m +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global a; +a = 5; +global A; +A = 10; + +phisteps = 400; +hphi = 2 * pi / phisteps; +thetasteps = 200; +htheta = pi / thetasteps; + +function retval = f(z) + global a; + global A; + retval = A * exp(a * (z^2 - 1)); +endfunction + +function retval = g(z) + global a; + retval = -f(z) * 2 * a * (2 * a * z^4 + (3 - 2*a) * z^2 - 1); + # 2 + # - a 2 4 2 2 a z + #(%o6) - %e (4 a z + (6 a - 4 a ) z - 2 a) %e +endfunction + +phi = (1 + sqrt(5)) / 2; + +global axes; +axes = [ + 0, 0, 1, -1, phi, -phi; + 1, -1, phi, phi, 0, 0; + phi, phi, 0, 0, 1, 1; +]; +axes = axes / (sqrt(phi^2+1)); + +function retval = kugel(theta, phi) + retval = [ + cos(phi) * sin(theta); + sin(phi) * sin(theta); + cos(theta) + ]; +endfunction + +function retval = F(v) + global axes; + s = 0; + for i = (1:6) + z = axes(:,i)' * v; + s = s + f(z); + endfor + retval = s / 6; +endfunction + +function retval = F2(theta, phi) + v = kugel(theta, phi); + retval = F(v); +endfunction + +function retval = G(v) + global axes; + s = 0; + for i = (1:6) + s = s + g(axes(:,i)' * v); + endfor + retval = s / 6; +endfunction + +function retval = G2(theta, phi) + v = kugel(theta, phi); + retval = G(v); +endfunction + +function retval = cnormalize(u) + utop = 11; + ubottom = -30; + retval = (u - ubottom) / (utop - ubottom); + if (retval > 1) + retval = 1; + endif + if (retval < 0) + retval = 0; + endif +endfunction + +global umin; +umin = 0; +global umax; +umax = 0; + +function color = farbe(v) + global umin; + global umax; + u = G(v); + if (u < umin) + umin = u; + endif + if (u > umax) + umax = u; + endif + u = cnormalize(u); + color = [ u, 0.5, 1-u ]; + color = color/max(color); +endfunction + +function dreieck(fn, v0, v1, v2) + fprintf(fn, " mesh {\n"); + c = (v0 + v1 + v2) / 3; + c = c / norm(c); + color = farbe(c); + v0 = v0 * (1 + F(v0)); + v1 = v1 * (1 + F(v1)); + v2 = v2 * (1 + F(v2)); + fprintf(fn, "\ttriangle {\n"); + fprintf(fn, "\t <%.6f,%.6f,%.6f>,\n", v0(1,1), v0(3,1), v0(2,1)); + fprintf(fn, "\t <%.6f,%.6f,%.6f>,\n", v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "\t <%.6f,%.6f,%.6f>\n", v2(1,1), v2(3,1), v2(2,1)); + fprintf(fn, "\t}\n"); + fprintf(fn, "\tpigment { color rgb<%.4f,%.4f,%.4f> }\n", + color(1,1), color(1,2), color(1,3)); + fprintf(fn, "\tfinish { metallic specular 0.5 }\n"); + fprintf(fn, " }\n"); +endfunction + +fn = fopen("spherecurve.inc", "w"); + + for i = (1:phisteps) + # Polkappe nord + v0 = [ 0; 0; 1 ]; + v1 = kugel(htheta, (i-1) * hphi); + v2 = kugel(htheta, i * hphi); + fprintf(fn, " // i = %d\n", i); + dreieck(fn, v0, v1, v2); + + # Polkappe sued + v0 = [ 0; 0; -1 ]; + v1 = kugel(pi-htheta, (i-1) * hphi); + v2 = kugel(pi-htheta, i * hphi); + dreieck(fn, v0, v1, v2); + endfor + + for j = (1:thetasteps-2) + for i = (1:phisteps) + v0 = kugel( j * htheta, (i-1) * hphi); + v1 = kugel((j+1) * htheta, (i-1) * hphi); + v2 = kugel( j * htheta, i * hphi); + v3 = kugel((j+1) * htheta, i * hphi); + fprintf(fn, " // i = %d, j = %d\n", i, j); + dreieck(fn, v0, v1, v2); + dreieck(fn, v1, v2, v3); + endfor + endfor + +fclose(fn); + +umin +umax diff --git a/buch/papers/kugel/images/spherecurve.maxima b/buch/papers/kugel/images/spherecurve.maxima new file mode 100644 index 0000000..1e9077c --- /dev/null +++ b/buch/papers/kugel/images/spherecurve.maxima @@ -0,0 +1,13 @@ +/* + * spherecurv.maxima + * + * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule + */ +f: exp(-a * sin(theta)^2); + +g: ratsimp(diff(sin(theta) * diff(f, theta), theta)/sin(theta)); +g: subst(z, cos(theta), g); +g: subst(sqrt(1-z^2), sin(theta), g); +ratsimp(g); + +f: ratsimp(subst(sqrt(1-z^2), sin(theta), f)); diff --git a/buch/papers/kugel/images/spherecurve.pov b/buch/papers/kugel/images/spherecurve.pov new file mode 100644 index 0000000..86c3745 --- /dev/null +++ b/buch/papers/kugel/images/spherecurve.pov @@ -0,0 +1,73 @@ +// +// curvature.pov +// +// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.14; + +camera { + location <10, 10, -40> + look_at <0, 0, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-10, 10, -40> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +// +// draw an arrow from to with thickness with +// color +// +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +arrow(<-2.7,0,0>, <2.7,0,0>, 0.03, White) +arrow(<0,-2.7,0>, <0,2.7,0>, 0.03, White) +arrow(<0,0,-2.7>, <0,0,2.7>, 0.03, White) + +#include "spherecurve.inc" + -- cgit v1.2.1 From 6ee6a7b0cf91469c7a79827293b8e3b880a6a0aa Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 24 May 2022 11:45:50 +0200 Subject: add C++ program to compute the surface --- buch/papers/kugel/images/Makefile | 9 +- buch/papers/kugel/images/spherecurve.cpp | 292 +++++++++++++++++++++++++++++++ buch/papers/kugel/images/spherecurve.m | 4 +- buch/papers/kugel/images/spherecurve.pov | 4 +- 4 files changed, 303 insertions(+), 6 deletions(-) create mode 100644 buch/papers/kugel/images/spherecurve.cpp (limited to 'buch') diff --git a/buch/papers/kugel/images/Makefile b/buch/papers/kugel/images/Makefile index 6187fed..4226dab 100644 --- a/buch/papers/kugel/images/Makefile +++ b/buch/papers/kugel/images/Makefile @@ -14,12 +14,17 @@ curvature.png: curvature.pov curvature.inc curvature.jpg: curvature.png convert curvature.png -density 300 -units PixelsPerInch curvature.jpg -spherecurve.inc: spherecurve.m +spherecurve2.inc: spherecurve.m octave spherecurve.m spherecurve.png: spherecurve.pov spherecurve.inc - povray +A0.1 +W1920 +H1080 +Ospherecurve.png spherecurve.pov + povray +A0.1 +W1080 +H1080 +Ospherecurve.png spherecurve.pov spherecurve.jpg: spherecurve.png convert spherecurve.png -density 300 -units PixelsPerInch spherecurve.jpg +spherecurve: spherecurve.cpp + g++ -o spherecurve -g -Wall -O spherecurve.cpp + +spherecurve.inc: spherecurve + ./spherecurve diff --git a/buch/papers/kugel/images/spherecurve.cpp b/buch/papers/kugel/images/spherecurve.cpp new file mode 100644 index 0000000..eff8c33 --- /dev/null +++ b/buch/papers/kugel/images/spherecurve.cpp @@ -0,0 +1,292 @@ +/* + * spherecurve.cpp + * + * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule + */ +#include +#include +#include +#include +#include + +inline double sqr(double x) { return x * x; } + +/** + * \brief Class for 3d vectors (also used as colors) + */ +class vector { + double X[3]; +public: + vector() { X[0] = X[1] = X[2] = 0; } + vector(double a) { X[0] = X[1] = X[2] = a; } + vector(double x, double y, double z) { + X[0] = x; X[1] = y; X[2] = z; + } + vector(double theta, double phi) { + double s = sin(theta); + X[0] = cos(phi) * s; + X[1] = sin(phi) * s; + X[2] = cos(theta); + } + vector(const vector& other) { + for (int i = 0; i < 3; i++) { + X[i] = other.X[i]; + } + } + vector operator+(const vector& other) const { + return vector(X[0] + other.X[0], + X[1] + other.X[1], + X[2] + other.X[2]); + } + vector operator*(double l) const { + return vector(X[0] * l, X[1] * l, X[2] * l); + } + double operator*(const vector& other) const { + double s = 0; + for (int i = 0; i < 3; i++) { + s += X[i] * other.X[i]; + } + return s; + } + double norm() const { + double s = 0; + for (int i = 0; i < 3; i++) { + s += sqr(X[i]); + } + return sqrt(s); + } + vector normalize() const { + double l = norm(); + return vector(X[0]/l, X[1]/l, X[2]/l); + } + double max() const { + return std::max(X[0], std::max(X[1], X[2])); + } + double l0norm() const { + double l = 0; + for (int i = 0; i < 3; i++) { + if (fabs(X[i]) > l) { + l = fabs(X[i]); + } + } + return l; + } + vector l0normalize() const { + double l = l0norm(); + vector result(X[0]/l, X[1]/l, X[2]/l); + return result; + } + const double& operator[](int i) const { return X[i]; } + double& operator[](int i) { return X[i]; } +}; + +/** + * \brief Derived 3d vector class implementing color + * + * The constructor in this class converts a single value into a + * color on a suitable gradient. + */ +class color : public vector { +public: + static double utop; + static double ubottom; + static double green; +public: + color(double u) { + u = (u - ubottom) / (utop - ubottom); + if (u > 1) { + u = 1; + } + if (u < 0) { + u = 0; + } + u = pow(u,2); + (*this)[0] = u; + (*this)[1] = green; + (*this)[2] = 1-u; + double l = l0norm(); + for (int i = 0; i < 3; i++) { + (*this)[i] /= l; + } + } +}; + +double color::utop = 12; +double color::ubottom = -31; +double color::green = 0.5; + +/** + * \brief Surface model + * + * This class contains the definitions of the functions to plot + * and the parameters to + */ +class surfacefunction { + static vector axes[6]; + + double _a; + double _A; + + double _umin; + double _umax; +public: + double a() const { return _a; } + double A() const { return _A; } + + double umin() const { return _umin; } + double umax() const { return _umax; } + + surfacefunction(double a, double A) : _a(a), _A(A), _umin(0), _umax(0) { + } + + double f(double z) { + return A() * exp(a() * (sqr(z) - 1)); + } + + double g(double z) { + return -f(z) * 2*a() * ((2*a()*sqr(z) + (3-2*a()))*sqr(z) - 1); + } + + double F(const vector& v) { + double s = 0; + for (int i = 0; i < 6; i++) { + s += f(axes[i] * v); + } + return s / 6; + } + + double G(const vector& v) { + double s = 0; + for (int i = 0; i < 6; i++) { + s += g(axes[i] * v); + } + return s / 6; + } +protected: + color farbe(const vector& v) { + double u = G(v); + if (u < _umin) { + _umin = u; + } + if (u > _umax) { + _umax = u; + } + return color(u); + } +}; + +static double phi = (1 + sqrt(5)) / 2; +static double sl = sqrt(sqr(phi) + 1); +vector surfacefunction::axes[6] = { + vector( 0. , -1./sl, phi/sl ), + vector( 0. , 1./sl, phi/sl ), + vector( 1./sl, phi/sl, 0. ), + vector( -1./sl, phi/sl, 0. ), + vector( phi/sl, 0. , 1./sl ), + vector( -phi/sl, 0. , 1./sl ) +}; + +/** + * \brief Class to construct the plot + */ +class surface : public surfacefunction { + FILE *outfile; + + int _phisteps; + int _thetasteps; + double _hphi; + double _htheta; +public: + int phisteps() const { return _phisteps; } + int thetasteps() const { return _thetasteps; } + double hphi() const { return _hphi; } + double htheta() const { return _htheta; } + void phisteps(int s) { _phisteps = s; _hphi = 2 * M_PI / s; } + void thetasteps(int s) { _thetasteps = s; _htheta = M_PI / s; } + + surface(const std::string& filename, double a, double A) + : surfacefunction(a, A) { + outfile = fopen(filename.c_str(), "w"); + phisteps(400); + thetasteps(200); + } + + ~surface() { + fclose(outfile); + } + +private: + void triangle(const vector& v0, const vector& v1, const vector& v2) { + fprintf(outfile, " mesh {\n"); + vector c = (v0 + v1 + v2) * (1./3.); + vector color = farbe(c.normalize()); + vector V0 = v0 * (1 + F(v0)); + vector V1 = v1 * (1 + F(v1)); + vector V2 = v2 * (1 + F(v2)); + fprintf(outfile, "\ttriangle {\n"); + fprintf(outfile, "\t <%.6f,%.6f,%.6f>,\n", + V0[0], V0[2], V0[1]); + fprintf(outfile, "\t <%.6f,%.6f,%.6f>,\n", + V1[0], V1[2], V1[1]); + fprintf(outfile, "\t <%.6f,%.6f,%.6f>\n", + V2[0], V2[2], V2[1]); + fprintf(outfile, "\t}\n"); + fprintf(outfile, "\tpigment { color rgb<%.4f,%.4f,%.4f> }\n", + color[0], color[1], color[2]); + fprintf(outfile, "\tfinish { metallic specular 0.5 }\n"); + fprintf(outfile, " }\n"); + } + + void northcap() { + vector v0(0, 0, 1); + for (int i = 1; i <= phisteps(); i++) { + fprintf(outfile, " // northcap i = %d\n", i); + vector v1(htheta(), (i - 1) * hphi()); + vector v2(htheta(), i * hphi()); + triangle(v0, v1, v2); + } + } + + void southcap() { + vector v0(0, 0, -1); + for (int i = 1; i <= phisteps(); i++) { + fprintf(outfile, " // southcap i = %d\n", i); + vector v1(M_PI - htheta(), (i - 1) * hphi()); + vector v2(M_PI - htheta(), i * hphi()); + triangle(v0, v1, v2); + } + } + + void zone() { + for (int j = 1; j < thetasteps() - 1; j++) { + for (int i = 1; i <= phisteps(); i++) { + fprintf(outfile, " // zone j = %d, i = %d\n", + j, i); + vector v0( j * htheta(), (i-1) * hphi()); + vector v1((j+1) * htheta(), (i-1) * hphi()); + vector v2( j * htheta(), i * hphi()); + vector v3((j+1) * htheta(), i * hphi()); + triangle(v0, v1, v2); + triangle(v1, v2, v3); + } + } + } +public: + void draw() { + northcap(); + southcap(); + zone(); + } +}; + +/** + * \brief main function + */ +int main(int argc, char *argv[]) { + surface S("spherecurve.inc", 5, 10); + color::green = 0.3; + S.draw(); + std::cout << "umin: " << S.umin() << std::endl; + std::cout << "umax: " << S.umax() << std::endl; + return EXIT_SUCCESS; +} diff --git a/buch/papers/kugel/images/spherecurve.m b/buch/papers/kugel/images/spherecurve.m index ea9c901..99d5c9a 100644 --- a/buch/papers/kugel/images/spherecurve.m +++ b/buch/papers/kugel/images/spherecurve.m @@ -1,5 +1,5 @@ # -# spherecurv.m +# spherecurve.m # # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # @@ -125,7 +125,7 @@ function dreieck(fn, v0, v1, v2) fprintf(fn, " }\n"); endfunction -fn = fopen("spherecurve.inc", "w"); +fn = fopen("spherecurve2.inc", "w"); for i = (1:phisteps) # Polkappe nord diff --git a/buch/papers/kugel/images/spherecurve.pov b/buch/papers/kugel/images/spherecurve.pov index 86c3745..b1bf4b8 100644 --- a/buch/papers/kugel/images/spherecurve.pov +++ b/buch/papers/kugel/images/spherecurve.pov @@ -11,12 +11,12 @@ global_settings { assumed_gamma 1 } -#declare imagescale = 0.14; +#declare imagescale = 0.13; camera { location <10, 10, -40> look_at <0, 0, 0> - right 16/9 * x * imagescale + right x * imagescale up y * imagescale } -- cgit v1.2.1 From d08813723e1cab4bca4a527218610023775a4634 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 24 May 2022 11:53:56 +0200 Subject: better color coding --- buch/papers/kugel/images/spherecurve.cpp | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'buch') diff --git a/buch/papers/kugel/images/spherecurve.cpp b/buch/papers/kugel/images/spherecurve.cpp index eff8c33..8ddf5e5 100644 --- a/buch/papers/kugel/images/spherecurve.cpp +++ b/buch/papers/kugel/images/spherecurve.cpp @@ -102,7 +102,7 @@ public: } u = pow(u,2); (*this)[0] = u; - (*this)[1] = green; + (*this)[1] = green * u * (1 - u); (*this)[2] = 1-u; double l = l0norm(); for (int i = 0; i < 3; i++) { @@ -284,7 +284,7 @@ public: */ int main(int argc, char *argv[]) { surface S("spherecurve.inc", 5, 10); - color::green = 0.3; + color::green = 1.0; S.draw(); std::cout << "umin: " << S.umin() << std::endl; std::cout << "umax: " << S.umax() << std::endl; -- cgit v1.2.1 From c9c9f97f5cf1bbe669acfdb8aae1e6c81f8faed9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 24 May 2022 16:23:27 +0200 Subject: Integrale von R(x,y) --- buch/chapters/060-integral/differentialkoerper.tex | 12 +- buch/chapters/060-integral/sqrat.tex | 365 ++++++++++++++++++++- 2 files changed, 374 insertions(+), 3 deletions(-) (limited to 'buch') diff --git a/buch/chapters/060-integral/differentialkoerper.tex b/buch/chapters/060-integral/differentialkoerper.tex index 66bb0c1..a071ae2 100644 --- a/buch/chapters/060-integral/differentialkoerper.tex +++ b/buch/chapters/060-integral/differentialkoerper.tex @@ -3,9 +3,19 @@ % % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % -\section{Differentialkörper +\section{Differentialkörper und das Integrationsproblem \label{buch:integrale:section:dkoerper}} \rhead{Differentialkörper} +Die Einführung einer neuen Funktion $\operatorname{erf}(x)$ wurde +durch die Behauptung gerechtfertigt, dass es für den Integranden +$e^{-x^2}$ keine Stammfunktion in geschlossener Form gäbe. +Die Fehlerfunktion ist bei weitem nicht die einzige mit dieser +Eigenschaft. +Doch woher weiss man, dass es keine solche Funktion gibt, und +was heisst überhaupt ``Stammfunktion in geschlossener Form''? +In diesem Abschnitt wird daher ein algebraischer Rahmen entwickelt, +in dem diese Frage sinnvoll gestellt werden kann. + \input{chapters/060-integral/rational.tex} \input{chapters/060-integral/erweiterungen.tex} \input{chapters/060-integral/diffke.tex} diff --git a/buch/chapters/060-integral/sqrat.tex b/buch/chapters/060-integral/sqrat.tex index 38b1504..20f1ef7 100644 --- a/buch/chapters/060-integral/sqrat.tex +++ b/buch/chapters/060-integral/sqrat.tex @@ -90,6 +90,7 @@ Die zu integrierende Funktion kann also in der Form f(x) = W_1 + W_2\frac{1}{y} +\label{buch:integral:sqint:eqn:w1w2y} \end{equation} geschrieben werden mit rationalen Funktionen $W_1,W_2\in\mathbb{Q}(x)$. @@ -98,20 +99,380 @@ Abschnitt~\ref{buch:integral:subsection:rationalefunktionen} gefunden werden. Im Folgenden kümmern wir uns daher nur noch um $W_1$. +% +% Polynomdivision +% \subsubsection{Polynomdivision} +Die Funktion $W_2$ in \eqref{buch:integral:sqint:eqn:w1w2y} ist eine +rationale Funktion $W_2\in \mathbb{K}(x)$, also ein Bruch mit Polynomen +in $x$ als Zähler und Nenner. +Durch Polynomdivision mit Rest können wir $W_2$ schreiben als +\[ +W_1 = \varphi + W_3, +\] +wobei $\varphi$ ein Polynom in $x$ ist und $W_3$ eine rationale +Funktion, deren Zählergrad kleiner ist als der Nennergrad. +Zur Bestimmung der Stammfunktion bleibt jetzt nur noch +\begin{equation} +\int W_2\frac{1}{y} += +\int \frac{\varphi}{y} ++ +\int W_3\frac1{y} +\label{buch:integral:sqint:eqn:Wy} +\end{equation} +zu berechnen. -\subsubsection{Integranden der Form $p(x)/y$} +% +% Integranden der Form $\varphi(x)/y$ +% +\subsubsection{Integranden der Form $\varphi(x)/y$} +Der erste Term in~\eqref{buch:integral:sqint:eqn:Wy} ist ein Integral eines +Quotienten eines Polynoms geteilt durch $y$. +Solche Integrale können, wie im Folgenden gezeigt werden soll, reduziert +werden auf das Integral von $1/y$. +Genauer gilt der folgende Satz. -\subsubsection{Partialbruchzerlegung} +\begin{satz} +\label{buch:integral:sqint:satz:polyy} +Sei $\varphi\in\mathcal{K}(x)$ ein Polynom in $x$, dann gibt +es ein Polynom $\psi\in\mathcal{K}(x)$ vom Grad $\deg\psi < \deg\varphi$, +und $A\in\mathcal{K}$ derart, dass +\begin{equation} +\int \frac{\varphi}{y} += +\psi y + A\int\frac{1}{y}. +\label{buch:integral:sqint:eqn:phipsi} +\end{equation} +\end{satz} + +\begin{proof}[Beweis] +Wir schreiben die Polynome in der Form +\begin{align*} +\varphi +&= +\varphi_mx^m + \varphi_{m-1}x^{m-1} + \dots + \varphi_2x^2 + \varphi_1x + \varphi_0 +\\ +\psi +&= +\phantom{\varphi_mx^m+\mathstrut} +\psi_{m-1}x^{m-1} + \dots + \psi_2x^2 + \psi_1x + \psi_0 +\intertext{mit der Ableitung} +\psi' +&= +\phantom{\varphi_mx^m+\mathstrut} +\psi_{m-1}(m-1)x^{m-2} + \dots + 2\psi_2x + \psi_1. +\end{align*} +Wir leiten die Gleichung~\eqref{buch:integral:sqint:eqn:phipsi} +nach $x$ ab und erhalten +\begin{align*} +\frac{\varphi}{y} +&= +\psi'y + \psi y' + \frac{A}{y} += +\psi'y + \psi \frac{ax+b/2}{y} + \frac{A}{y}. +\intertext{Durch Multiplikation mit $y$ wird die Gleichung wesentlich +vereinfacht zu} +\varphi +&= +\psi' y^2 + \psi y' y + A += +\psi' \cdot(ax^2+bx+c) + \psi\cdot (ax+b/2) + A. +\end{align*} +Auf beiden Seiten stehen Polynome, man kann daher versuchen, die +Koeffizienten von $\psi$ mit Hilfe eines Koeffizientenvergleichs zu +bestimmen. +Dazu müssen die Produkte auf der rechten Seite ausmultipliziert werden. +So ergeben sich die Gleichungen +\begin{equation} +\renewcommand{\arraycolsep}{2pt} +\begin{array}{lcrcrcrcrcrcrcr} +\varphi_m +&=& +(m-1)\psi_{m-1} a &+& & & +&+& +\psi_{m-1} a & & & & +\\ +\varphi_{m-1} +&=& +(m-2)\psi_{m-2}a +&+& +(m-1)\psi_{m-1}b +& & +&+& +\psi_{m-2}a +&+& +\psi_{m-1}\frac{b}2 +& & +\\ +\varphi_{m-2} +&=& +(m-3)\psi_{m-3}a +&+& +(m-2)\psi_{m-2}b +&+& +(m-1)\psi_{m-1}c +&+& +\psi_{m-3}a +&+& +\psi_{m-2}\frac{b}2 +& & +\\ +&\vdots&&&&&&&&&&& +\\ +\varphi_2 +&=& +\psi_{1\phantom{-m}}a +&+& +2\psi_{2\phantom{-m}}b +&+& +3\psi_{3\phantom{-m}}c +&+& +\psi_{1\phantom{-m}}a +&+& +\psi_{2\phantom{-m}}\frac{b}2 +& & +\\ +\varphi_1 +&=& +& & +\psi_{1\phantom{-m}}b +& & +2\psi_{2\phantom{-m}}c +&+& +\psi_{0\phantom{-m}}a +&+& +\psi_{1\phantom{-m}}\frac{b}2 +\\ +\varphi_0 +&=& +& & +& & +\psi_{1\phantom{-m}}c +& & +&+& +\psi_{0\phantom{-m}}\frac{b}2 +&+&A +\end{array} +\end{equation} +In jeder Gleichung kommen hächstens drei der Koeffizienten von $\psi$ vor. +Fasst man sie zusammen und stellt die Terme etwas um, +erhält man die einfacheren Gleichungen +\begin{equation} +\renewcommand{\arraycolsep}{2pt} +\renewcommand{\arraystretch}{1.3} +\begin{array}{lcrcrcrcrcrcrcr} +\varphi_m +&=& +(m-0){\color{red}\psi_{m-1}}a & & & & +& & +\\ +\varphi_{m-1} +&=& +(m-1+\frac12)\psi_{m-1}b +&+& +(m-1){\color{red}\psi_{m-2}}a +& & +& & +\\ +\varphi_{m-2} +&=& +(m-1)\psi_{m-1}c +&+& +(m-2+\frac12)\psi_{m-2}b +&+& +(m-2){\color{red}\psi_{m-3}}a +& & +\\ +&\vdots&&&&&&&&&&& +\\ +\varphi_2 +&=& +3\psi_{3\phantom{-m}}c +&+& +(2+\frac12)\psi_{2\phantom{-m}}b +&+& +2{\color{red}\psi_{1\phantom{-m}}}a +& & +\\ +\varphi_1 +&=& +2\psi_{2\phantom{-m}}c +&+& +(1+\frac12)\psi_{1\phantom{-m}}b +&+& +{\color{red}\psi_{0\phantom{-m}}}a +& & +\\ +\varphi_0 +&=& +\psi_{1\phantom{-m}}c +& & +&+& +(0+\frac12) \psi_{0\phantom{-m}}b +&+&{\color{red}A} +\end{array} +\end{equation} +Die erste Gleichung kann wegen $a\ne 0$ nach $\psi_{m-1}$ aufgelöst werden, +dadurch ist $\psi_{m-1}$ bestimmt. +In allen folgenden Gleichungen taucht jeweils ein neuer Koeffizient +von $\psi$ auf, der rot hervorgehoben ist. +Wieder wegen $a\ne 0$ kann die Gleichung immer nach dieser Variablen +aufgelöst werden. +Die Gleichungen zeigen daher, dass die Koeffizienten des Polynoms $\psi$ +in absteigender Folge und die Konstanten $A$ eindeutig bestimmt werden. +\end{proof} + +Mit diesem Satz ist das Integral über den Teil $\varphi/y$ auf den +Fall des Integrals von $1/y$ reduziert. +Letzteres wird im nächsten Abschnitt berechnet. +% +% Das Integral von $1/y$ +% +\subsubsection{Das Integral von $1/y$} +Eine Stammfunktion von $1/y$ kann mit etwas Geschick bekannten +Interationstechnikgen gefunden werden. +Durch Ableitung der Funktion +\[ +F += +\frac{1}{\sqrt{a}}\log\biggl(x+\frac{b}{2a}+\frac{y}{\sqrt{y}}\biggr) +\] +kann man nachprüfen, dass $F$ eine Stammfunktion von $1/y$ ist, +also +\begin{equation} +\int +\frac{1}{y} += +\frac{1}{\sqrt{a}}\log\biggl(x+\frac{b}{2a}+\frac{y}{\sqrt{y}}\biggr). +\end{equation} + +% +% Partialbruchzerlegung +% +\subsubsection{Partialbruchzerlegung} +In der rationalen Funktion $W_3$ in \eqref{buch:integral:sqint:eqn:Wy} +hat der Zähler kleineren Grad als der Nenner, sie kann daher wieder +in Partialbrüche +\[ +W_3 += +\sum_{i=1}^n +\sum_{k=1}^{k_i} +\frac{A_{ik}}{(x-\alpha_i)^k} +\] +mit den Nullstellen $\alpha_i$ des Nenners von $W_3$ mit Vielfachheiten +$k_i$ zerlegt werden. +Die Stammfunktion von $W_3/y$ wird damit zu +\begin{equation} +\int W_3\frac{1}{y} += +\sum_{i=1}^n +\sum_{k=1}^{k_i} +A_{ik} +\int +\frac{1}{(x-\alpha_i)^ky} += +\sum_{i=1}^n +\sum_{k=1}^{k_i} +A_{ik} +\int +\frac{1}{(x-\alpha_i)^k \sqrt{ax^2+bx+c}}. +\end{equation} +Die Stammfunktion ist damit reduziert auf Integrale der Form \begin{equation} \int \frac{1}{(x-\alpha)^k \sqrt{ax^2+bx+c}} \label{buch:integral:sqrat:eqn:2teart} \end{equation} +mit $k>0$. +% +% Integrale der Form \eqref{buch:integral:sqrat:eqn:2teart} +% \subsubsection{Integrale der Form \eqref{buch:integral:sqrat:eqn:2teart}} +Die Integrale~\eqref{buch:integral:sqrat:eqn:2teart} +können mit Hilfe der Substution +\[ +t=\frac{1}{x-\alpha} +\qquad\text{oder}\qquad +x=\frac1t+\alpha +\] +In ein Integral verwandelt werden, für welches bereits eine +Berechnungsmethode entwickelt wurde. +Dazu berechnet man +\begin{align*} +y^2 +&= a\biggl(\frac1t+\alpha\biggr)^2 + b\biggl(\frac1t+\alpha\biggr) + c +\\ +&= +a\biggl(\frac{1}{t^2}+2\frac{\alpha}{t}+\alpha^2\biggr) ++\frac{b}{t}+b\alpha+c += +\frac{1}{t^2}\bigl( +\underbrace{a+(2a\alpha+b)t+(a\alpha^2+c)t^2}_{\displaystyle=Y^2} +\bigr) +\intertext{und damit} +y&=\frac{Y}{t}. +\end{align*} +Führt man die Substition +$dx = -dt/t^2$ im Integral aus, erhält man +\begin{align*} +\int\frac{dx}{(x-\alpha)^ky} +&= +- +\int +t^k\cdot\frac{t}{Y}\frac{dt}{t^2} += +-\int\frac{t^{k-1}}{Y}\,dt. +\end{align*} +Das letzte Integral ist wieder von der Form, die in +Satz~\ref{buch:integral:sqint:satz:polyy} behandelt wurde. +Insbesondere gibt es ein Polynom $\psi$ vom Grad $k-2$ und +eine Konstante $A$ derart, dass +\[ +\int\frac{1}{(x-\alpha)^ky} += +\psi Y + A\int\frac{1}{Y} +\] +ist. +Damit ist das Integral von $R(x,y)$ vollständig bestimmt. +\subsubsection{Beobachtungen} +Die eben dargestellte Berechnung des Integrals von $R(x,y)$ zeigt einige +Gemeinsamkeiten mit der entsprechenden Rechnung für rationale +Integranden, aber auch einige wesentliche Unterschiede. +Wieder zeigt sich, dass Polynomdivision und Partialbruchzerlegung +die zentralen Werkzeuge sind, mit denen der Integrand zerlegt und +leichter integrierbare Funktionen umgeformt werden kann. +Andererseits ist der in +Satz~\ref{buch:integral:sqint:satz:polyy} +zusammengefasste Schritt eine wesentliche zusätzliche Vereinfachung, +die keine Entsprechung bei rationalen Integranden hat. + +Die gefunden Form der Stammfunktion hat jedoch die allgemeine +Form +\[ +\int R(x,y) += +v_0 + +C +\log\biggl(x+\frac{b}{2a}+\frac{y}{\sqrt{y}}\biggr) ++ +\sum_{i=1}^n c_i +\log v_i, +\] +die ganz der bei rationalen Integranden gefunden Form entspricht. +Darin ist $v_0$ die Summe der angefallenen rationalen Teilintegrale, +also $v_0\in\mathcal{K}(x,y)$. +Die $v_i\in\mathcal{K}(x,y)$ sind die entsprechenden Logarithmusfunktionen, +die bei der Berechnung der Integrale \eqref{buch:integral:sqrat:eqn:2teart} +auftreten. +Insbesondere liefert die Rechnung eine Körpererweiterung von +$\mathcal{K}(x,y)$ um die logarithmische Funktionen +$\log(x+b/2a+y/\sqrt{y})$ und $\log v_i$, in der $R(x,y)$ eine +Stammfunktion hat. -- cgit v1.2.1 From 2dd23cdeef2889a5b3210e324c159ab462bb267c Mon Sep 17 00:00:00 2001 From: "ENEZ-PC\\erdem" Date: Tue, 24 May 2022 16:20:10 +0200 Subject: Korrektur (noch nicht fertig) --- buch/papers/nav/einleitung.tex | 2 +- buch/papers/nav/flatearth.tex | 3 +- buch/papers/nav/main.tex | 1 + buch/papers/nav/nautischesdreieck.tex | 89 +++++++++++++++---------------- buch/papers/nav/packages.tex | 3 +- buch/papers/nav/sincos.tex | 6 ++- buch/papers/nav/trigo.tex | 99 ++++++++++++++++++++++------------- 7 files changed, 116 insertions(+), 87 deletions(-) (limited to 'buch') diff --git a/buch/papers/nav/einleitung.tex b/buch/papers/nav/einleitung.tex index aafa107..8eb4481 100644 --- a/buch/papers/nav/einleitung.tex +++ b/buch/papers/nav/einleitung.tex @@ -3,7 +3,7 @@ \section{Einleitung} Heutzutage ist die Navigation ein Teil des Lebens. Man sendet dem Kollegen seinen eigenen Standort, um sich das ewige Erklären zu sparen oder gibt die Adresse des Ziels ein, damit man seinen Aufenthaltsort zum Beispiel auf einer riesigen Wiese am See findet. -Dies wird durch Technologien wie Funknavigation, welches ein auf Langzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt. +Dies wird durch Technologien wie Funknavigation, welches ein auf Laufzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist, oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt. Vor all diesen technologischen Fortschritten gab es lediglich die Astronavigation, welche heute noch auf Schiffen verwendet wird im Falle eines Stromausfalls. Aber wie funktioniert die Navigation mit den Sternen? Welche Hilfsmittel benötigt man, welche Rolle spielt die Mathematik und weshalb kann die Erde nicht flach sein? In diesem Kapitel werden genau diese Fragen mithilfe des nautischen Dreiecks, der sphärischen Trigonometrie und einigen Hilfsmitteln und Messgeräten beantwortet. \ No newline at end of file diff --git a/buch/papers/nav/flatearth.tex b/buch/papers/nav/flatearth.tex index 5bfc1b7..3b08e8d 100644 --- a/buch/papers/nav/flatearth.tex +++ b/buch/papers/nav/flatearth.tex @@ -17,10 +17,9 @@ Eratosthenes konnte etwa 100 Jahre später den Erdumfang berechnen. Er beobachtete, dass die Sonne in Syene mittags im Zenit steht und gleichzeitig in Alexandria unter einem Winkel einfällt. Mithilfe der Trigonometrie konnte er mit dem Abstand der Städte und dem Einfallswinkel den Umfang berechnen. -Der Kartograph Gerhard Mercator projizierte die Erdkugel auf ein Papier und erstellte so eine winkeltreue Karte. +Der Kartograph Gerhard Mercator projizierte die Erdkugel wie in Abbildung 21.1 dargestellt auf ein Papier und erstellte so eine winkeltreue Karte. Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen. - Dies sieht man zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. Grönland ist auf der Weltkarte so gross wie Afrika. In der Anwendung Google Earth jedoch ist Grönland etwa so gross wie Algerien. diff --git a/buch/papers/nav/main.tex b/buch/papers/nav/main.tex index e16dc2a..4c52547 100644 --- a/buch/papers/nav/main.tex +++ b/buch/papers/nav/main.tex @@ -19,3 +19,4 @@ \printbibliography[heading=subbibliography] \end{refsection} + diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex index c239d64..36e9c99 100644 --- a/buch/papers/nav/nautischesdreieck.tex +++ b/buch/papers/nav/nautischesdreieck.tex @@ -4,49 +4,26 @@ Die Himmelskugel ist eine gedachte Kugel, welche die Erde und dessen Beobachter Der Zenit ist jener Punkt, der vom Erdmittelpunkt durch denn eigenen Standort an die Himmelskugel verlängert wird. Ein Gestirn ist ein Planet oder ein Fixstern, zu welchen es diverse Jahrbücher mit allen astronomischen Eigenschaften gibt. Der Himmelspol ist der Nordpol an die Himmelskugel projiziert. -Das nautische Dreieck definiert sich durch folgende Ecken: Zenit, Gestirn und Himmelspol. +Das nautische Dreieck hat die Ecken Zenit, Gestirn und Himmelspol, wie man in der Abbildung 21.5 sehen kann. Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der Himmelskugel zu bestimmen. -Für die Definition gilt: -\begin{center} - \begin{tabular}{ c c c } - Winkel && Name / Beziehung \\ - \hline - $\alpha$ && Rektaszension \\ - $\delta$ && Deklination \\ - $\theta$ && Sternzeit von Greenwich\\ - $\phi$ && Geographische Breite\\ - $\tau=\theta-\alpha$ && Stundenwinkel und Längengrad des Gestirns. \\ - $a$ && Azimut\\ - $h$ && Höhe - \end{tabular} -\end{center} - -\begin{itemize} - \item Seitenlänge Zenit zu Himmelspol $= \frac{\pi}{2} - \phi $ - \item Seitenlänge Himmelspol zu Gestirn $= \frac{\pi}{2} - \delta$ - \item Seitenlänge Zenit zu Gestirn $= \frac{\pi}{2} - h$ - \item Winkel von Zenit zu Himmelsnordpol zu Gestirn$=\pi - \alpha$ - \item Winkel von Himmelsnordpol zu Zenit und Gestirn$= \tau$ -\end{itemize} - - -\subsection{Zusammenhang des nautischen Dreiecks und des Kugeldreiecks auf der Erdkugel} +\subsection{Das Bilddreieck} \begin{figure} \begin{center} - \includegraphics[height=5cm,width=8cm]{papers/nav/bilder/kugel3.png} + \includegraphics[width=8cm]{papers/nav/bilder/kugel3.png} \caption[Nautisches Dreieck]{Nautisches Dreieck} \end{center} \end{figure} - -Wie man in der Abbildung 21.4 sieht, liegt das nautische Dreieck auf der Himmelskugel mit den Ecken Zenit, Gestirn und Himmelsnordpol. -Das selbe Dreieck kann man aber auch auf die Erdkugel projizieren. Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. Die Projektion auf der Erdkugel hat die Ecken Nordpol, Standort und Bildpunkt. + Man kann das nautische Dreieck auf die Erdkugel projizieren. +Dieses Dreieck nennt man dann Bilddreieck. +Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. +Die Projektion auf der Erdkugel hat die Ecken Nordpol, Standort und Bildpunkt. \section{Standortbestimmung ohne elektronische Hilfsmittel} Um den eigenen Standort herauszufinden, wird in diesem Kapitel die Projektion des nautische Dreiecks auf die Erdkugel zur Hilfe genommen. Mithilfe eines Sextanten, einem Jahrbuch und der sphärischen Trigonometrie kann man dann die Längen- und Breitengrade des eigenen Standortes bestimmen. -Was ein Sextant und ein Jahrbuch ist, wird im Kapitel 21.6 erklärt. +Was ein Sextant und ein Jahrbuch ist, wird im Abschnitt 21.6.3 erklärt. \begin{figure} \begin{center} \includegraphics[width=10cm]{papers/nav/bilder/dreieck.pdf} @@ -59,8 +36,8 @@ Was ein Sextant und ein Jahrbuch ist, wird im Kapitel 21.6 erklärt. \subsection{Ecke $P$ und $A$} Unser eigener Standort ist der gesuchte Ecke $P$ und die Ecke $A$ ist in unserem Fall der Nordpol. -Der Vorteil ander Idee des nautischen Dreiecks ist, dass eine Ecke immer der Nordpol ist. -Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so simpel. +Der Vorteil an der Idee des nautischen Dreiecks ist, dass eine Ecke immer der Nordpol ist. +Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so einfach. \subsection{Ecke $B$ und $C$ - Bildpunkt $X$ und $Y$} Für die Standortermittlung benötigt man als weiteren Punkt ein Gestirn bzw. seinen Bildpunkt auf der Erdkugel. @@ -69,8 +46,8 @@ Es gibt diverse Gestirne, die man nutzen kann wie zum Beispiel die Sonne, der Mo Die Bildpunkte von den beiden Gestirnen $X$ und $Y$ bilden die beiden Ecken $B$ und $C$ im Dreieck der Abbildung 21.5. \subsection{Ephemeriden} -Zu all diesen Gestirnen gibt es Ephemeriden, die man auch Jahrbücher nennt. -In diesen findet man Begriffe wie Rektaszension, Deklination und Sternzeit. +Zu all diesen Gestirnen gibt es Ephemeriden. +Diese enthalten die Rektaszensionen und Deklinationen in Abhängigkeit von der Zeit. \begin{figure} \begin{center} @@ -83,25 +60,24 @@ In diesen findet man Begriffe wie Rektaszension, Deklination und Sternzeit. Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und entspricht dem Breitengrad des Gestirns. \subsubsection{Rektaszension und Sternzeit} -Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt steht und geht vom Koordinatensystem der Himmelskugel aus. -Der Frühlungspunkt ist der Nullpunkt auf dem Himmelsäquator. +Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt, welcher der Nullpunkt auf dem Himmelsäquator ist, steht und geht vom Koordinatensystem der Himmelskugel aus. + Die Tatsache, dass sich die Himmelskugel ca. vier Minuten schneller um die eigene Achse dreht als die Erdkugel, stellt hier ein kleines Problem dar. Die Lösung ist die Sternzeit. -Mit dieser können wir die schnellere Drehung der Himmelskugel ausgleichen und können die -Am Frühlingspunkt (21. März) 12:00 Uhr ist die Sternzeit -$\theta = 0$. +Mit dieser können wir die schnellere Drehung der Himmelskugel ausgleichen und können die am Frühlingspunkt (21. März) 12:00 Uhr ist die Sternzeit $\theta = 0$. Die Sternzeit geht vom Frühlungspunkt aus, an welchem die Sonne den Himmelsäquator schneidet. Für die Standortermittlung auf der Erdkugel ist es am einfachsten, wenn man die Sternzeit von Greenwich berechnet. -Für die Sternzeit von Greenwich $\theta $braucht man als erstes das Julianische Datum $T$ vom aktuellen Tag, welches sich leicht recherchieren lässt. +Für die Sternzeit von Greenwich $\theta$ braucht man als erstes das Julianische Datum $T$ vom aktuellen Tag, welches sich leicht nachschlagen lässt. Im Anschluss berechnet man die Sternzeit von Greenwich \[\theta = 6^h 41^m 50^s,54841 + 8640184^s,812866 \cdot T + 0^s,093104 \cdot T^2 - 0^s,0000062 \cdot T^3.\] -Wenn man die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ mithilfe der Rektaszension und Sternzeit von Greenwich bestimmen. +Wenn man die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ bestimmen, wobei $\alpha$ die Rektaszension und $\theta$ die Sternzeit von Greenwich ist. Dies gilt analog auch für das zweite Gestirn. \subsubsection{Sextant} -Ein Sextant ist ein nautisches Messinstrument, mit dem man den Winkel zwischen der Blickrichtung zu weit entfernten Objekten bestimmen kann, insbesondere den Winkelabstand zu einem Gestirn vom Horizont. Man nutze ihn vor allem für die astronomische Navigation auf See. +Ein Sextant ist ein nautisches Messinstrument, mit dem man den Winkel zwischen der Blickrichtung zu weit entfernten Objekten bestimmen kann. Es wird vor allem der Winkelabstand zu Gestirnen gemessen. +Man benutzt ihn vor allem für die astronomische Navigation auf See. \begin{figure} \begin{center} @@ -109,7 +85,32 @@ Ein Sextant ist ein nautisches Messinstrument, mit dem man den Winkel zwischen d \caption[Sextant]{Sextant} \end{center} \end{figure} - +\subsubsection{Eingeschaften} +Für das nautische Dreieck gibt es folgende Eigenschaften: +\begin{center} + \begin{tabular}{ l c l } + Legende && Name / Beziehung \\ + \hline + $\alpha$ && Rektaszension \\ + $\delta$ && Deklination \\ + $\theta$ && Sternzeit von Greenwich\\ + $\phi$ && Geographische Breite\\ + $\tau=\theta-\alpha$ && Stundenwinkel und Längengrad des Gestirns. \\ + $a$ && Azimut\\ + $h$ && Höhe + \end{tabular} +\end{center} +\begin{center} + \begin{tabular}{ l c l } + Eigenschaften \\ + \hline + Seitenlänge Zenit zu Himmelspol= && $\frac{\pi}{2} - \phi$ \\ + Seitenlänge Himmelspol zu Gestirn= && $\frac{\pi}{2} - \delta$ \\ + Seitenlänge Himmelspol zu Gestirn= && $\frac{\pi}{2} - h$ \\ + Winkel von Zenit zu Himmelsnordpol zu Gestirn= && $\pi-\alpha$\\ + Winkel von Himmelsnordpol zu Zenit zu Gestirn= && $\tau$\\ + \end{tabular} +\end{center} \subsection{Bestimmung des eigenen Standortes $P$} Nun hat man die Koordinaten der beiden Gestirne und man weiss die Koordinaten des Nordpols. Damit wir unseren Standort bestimmen können, bilden wir zuerst das Dreieck $ABC$, dann das Dreieck $BPC$ und zum Schluss noch das Dreieck $ABP$. diff --git a/buch/papers/nav/packages.tex b/buch/papers/nav/packages.tex index 5b87303..f2e6132 100644 --- a/buch/papers/nav/packages.tex +++ b/buch/papers/nav/packages.tex @@ -8,4 +8,5 @@ % following example %\usepackage{packagename} -\usepackage{amsmath} \ No newline at end of file +\usepackage{amsmath} +\usepackage{cancel} \ No newline at end of file diff --git a/buch/papers/nav/sincos.tex b/buch/papers/nav/sincos.tex index d56d482..a1653e8 100644 --- a/buch/papers/nav/sincos.tex +++ b/buch/papers/nav/sincos.tex @@ -7,12 +7,14 @@ Jedoch konnten sie dieses Problem nicht lösen. Die Geschichte der sphärischen Trigonometrie ist daher eng mit der Astronomie verknüpft. Ca. 350 vor Christus dachten die Griechen über Kugelgeometrie nach und sie wurde zu einer Hilfswissenschaft der Astronomen. Zwischen 190 v. Chr. und 120 v. Chr. lebte ein griechischer Astronom names Hipparchos. -Dieser entwickelte unter anderem die Chordentafeln, welche die Chord - Funktionen, auch Chord genannt, beinhalten. +Dieser entwickelte unter anderem die Chordentafeln, welche die Chord - Funktionen, auch Chord genannt, beinhalten und im Abschnitt 3.1.1 beschrieben sind. Chord ist der Vorgänger der Sinusfunktion und galt damals als wichtigste Grundlage der Trigonometrie. -In dieser Zeit wurden auch die ersten Sternenkarten angefertigt, jedoch kannte man damals die Sinusfunktion noch nicht. +In dieser Zeit wurden auch die ersten Sternenkarten angefertigt. Damals kannte man die Sinusfunktionen noch nicht. Aus Indien stammten die ersten Ansätze zu den Kosinussätzen. Aufbauend auf den indischen und griechischen Forschungen entwickeln die Araber um das 9. Jahrhundert den Sinussatz. +Die Definition der trigonometrischen Funktionen ermöglicht nur, rechtwinklige Dreiecke zu berechnen. +Die Beziehung zwischen Seiten und Winkeln sind komplizierter und als Sinus- und Kosinussätze bekannt. Doch ein paar weitere Jahrhunderte vergingen bis zu diesem Thema wieder verstärkt Forschung betrieben wurde, da im 15. Jahrhundert grosse Entdeckungsreisen, hauptsächlich per Schiff, erfolgten und die Orientierung mit Sternen vermehrt an Wichtigkeit gewann. Man nutzte für die Kartographie nun die Kugelgeometrie, um die Genauigkeit zu erhöhen. Der Sinussatz, die Tangensfunktion und der neu entwickelte Seitenkosinussatz wurden in dieser Zeit bereits verwendet und im darauffolgenden Jahrhundert folgte der Winkelkosinussatz. diff --git a/buch/papers/nav/trigo.tex b/buch/papers/nav/trigo.tex index ce367f6..aca8bd2 100644 --- a/buch/papers/nav/trigo.tex +++ b/buch/papers/nav/trigo.tex @@ -1,16 +1,13 @@ \section{Sphärische Trigonometrie} -In der sphärischen Trigonometrie gibt es eine Symetrie zwischen Seiten und Winkel, also zu jedem Satz über Seiten und Winkel gibt es einen entsprechenden Satz, mit dem man Winkel durch Seiten und Seiten durch Winkel ersetzt hat. -Dabei gibt es folgenden Zusammenhang zwischen der ebenen- und sphärischen Trigonometrie: - \subsection{Das Kugeldreieck} -Damit man die Definition des Kugeldreiecks versteht, müssen wir zuerst Begriffe wie "Grosskreisebene" und "Grosskreisbögen" verstehen. -Ein Grosskreis ist ein größtmöglicher Kreis auf einer Kugeloberfläche. +Damit man die Definition des Kugeldreiecks versteht, müssen wir zuerst Begriffe wie Grosskreisebene und Grosskreisbögen verstehen. +Ein Grosskreis ist ein grösstmöglicher Kreis auf einer Kugeloberfläche. Sein Mittelpunkt fällt immer mit dem Mittelpunkt der Kugel zusammen und ein Schnitt auf dem Großkreis teilt die Kugel in jedem Fall in zwei gleich grosse Hälften. Da es unendlich viele Möglichkeiten gibt, eine Kugel so zu zerschneiden, dass die Schnittebene den Kugelmittelpunkt trifft, gibt es auch unendlich viele Grosskreise. -Grosskreisbögen sind die Verbindungslinien zwischen zwei Punkten auf der Kugel, welche auch "Seiten" eines Kugeldreiecks gennant werden. +Grosskreisbögen sind die kürzesten Verbindungslinien zwischen zwei Punkten auf der Kugel. -Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden, so entsteht ein Kugeldreieck $ABC$. +Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden werden, so entsteht ein Kugeldreieck $ABC$. Für ein Kugeldreieck gilt, dass die Summe der drei Seiten kleiner als $2\pi$ aber grösser als 0 ist. $A$, $B$ und $C$ sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten (siehe Abbildung 21.2). @@ -19,18 +16,6 @@ Laut dieser Definition ist die Seite $c$ der Winkel $AMB$, wobei der Punkt $M$ d Man kann bei Kugeldreiecken nicht so einfach unterscheiden, was Innen oder Aussen ist. Wenn man drei Eckpunkte miteinander verbindet, ergeben sich immer 16 Kugeldreiecke. -Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha, \beta, \gamma < \pi$ nennt man Eulersches Dreieck. - -Es gibt einen Zusammenhang zwischen der ebenen- und sphärischen Trigonometrie, wobei folgend $a$ eine Seite beschreibt: -\begin{center} - \begin{tabular}{ccc} - Eben & $\leftrightarrow$ & sphärisch \\ - \hline - $a$ & $\leftrightarrow$ & $\sin \ a$ \\ - - $a^2$ & $\leftrightarrow$ & $-\cos \ a$ \\ - \end{tabular} -\end{center} \begin{figure} \begin{center} @@ -41,8 +26,11 @@ Es gibt einen Zusammenhang zwischen der ebenen- und sphärischen Trigonometrie, \end{figure} \subsection{Rechtwinkliges Dreieck und rechtseitiges Dreieck} +In der sphärischen Trigonometrie gibt es eine Symetrie zwischen Seiten und Winkel, also zu jedem Satz über Seiten und Winkel gibt es einen entsprechenden Satz, mit dem man Winkel durch Seiten und Seiten durch Winkel ersetzt hat. + Wie auch im ebenen Dreieck gibt es beim Kugeldreieck auch ein rechtwinkliges Kugeldreieck, bei dem ein Winkel $\frac{\pi}{2}$ ist. -Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss. +Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss, wie man in der Abbildung 21.3 sehen kann. + \begin{figure} \begin{center} @@ -51,7 +39,7 @@ Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine S \end{center} \end{figure} -\subsection{Winkelsumme} +\subsection{Winkelsumme und Flächeninhalt} \begin{figure} \begin{center} @@ -64,9 +52,9 @@ Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine S Die Winkel eines Kugeldreiecks sind die, welche die Halbtangenten in den Eckpunkten einschliessen. Für die Summe der Innenwinkel gilt \begin{align} - \alpha+\beta+\gamma &= \frac{F}{r^2} + \pi \ \text{und} \ \alpha+\beta+\gamma > \pi, \nonumber + \alpha+\beta+\gamma &= \frac{F}{r^2} + \pi \quad \text{und} \quad \alpha+\beta+\gamma > \pi, \nonumber \end{align} -wobei F der Flächeninhalt des Kugeldreiecks ist. +wobei $F$ der Flächeninhalt des Kugeldreiecks ist. \subsubsection{Sphärischer Exzess} Der sphärische Exzess \begin{align} @@ -77,32 +65,69 @@ beschreibt die Abweichung der Innenwinkelsumme von $\pi$ und ist proportional zu \subsubsection{Flächeninnhalt} Mithilfe des Radius $r$ und dem sphärischen Exzess $\epsilon$ gilt für den Flächeninhalt \[ F=\frac{\pi \cdot r^2}{\frac{\pi}{2}} \cdot \epsilon\]. -\subsection{Sphärischer Sinussatz} -In jedem Dreieck ist das Verhältnis des Sinus einer Seite zum Sinus des Gegenwinkels konstant. -Das bedeutet, dass +\subsection{Seiten und Winkelberechnung} +Es gibt in der sphärischen Trigonometrie eigentlich gar keinen Satz des Pythagoras, wie man ihn aus der zweidimensionalen Geometrie kennt. +Es gibt aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks, nicht aber für das rechtseitige Kugeldreieck, in eine Beziehung bringt und zum jetzigen Punkt noch unklar ist, weshalb dieser Satz so aussieht. +Die Approximation folgt noch. +Es gilt nämlich: +\begin{align} + \cos c = \cos a \cdot \cos b \quad \text{wenn} \nonumber & + \quad \alpha = \frac{\pi}{2} \nonumber +\end{align} + +\subsubsection{Approximation von kleinen Dreiecken} +Die Sätze in der ebenen Trigonometrie sind eigentlich Approximationen der sphärischen Trigonometrie. +So ist der Sinussatz in der Ebene nur eine Annäherung des sphärischen Sinussatzes. Das Gleiche gilt für den Kosinussatz und dem Satz des Pythagoras. +So kann mit dem Taylorpolynom 2. Grades den Sinus und den Kosinus vom Sphärischen in die Ebene approximieren: +\begin{align} + \sin(a) &\approx a \nonumber \intertext{und} + \cos(a)&\approx 1-\frac{a^2}{2}.\nonumber +\end{align} +Es gibt ebenfalls folgende Approximierung der Seiten von der Sphäre in die Ebene: +\begin{align} + a &\approx \sin(a) \nonumber \intertext{und} + a^2 &\approx 1-\cos(a). \nonumber +\end{align} +Die Korrespondenzen zwischen der ebenen- und sphärischen Trigonometrie werden in den kommenden Abschnitten erläutert. + +\subsubsection{Sphärischer Satz des Pythagoras} +Die Korrespondenz \[ a^2 \approx 1-cos(a)\] liefert unter Anderem einen entsprechenden Satz des Pythagoras, nämlich + +\begin{align} + \cos(a)\cdot \cos(b) &= \cos(c) \\ + \bigg[1-\frac{a^2}{2}\bigg] \cdot \bigg[1-\frac{b^2}{2}\bigg] &= 1-\frac{c^2}{2} \\ + \xcancel{1}- \frac{a^2}{2} - \frac{b^2}{2} + \xcancel{\frac{a^2b^2}{4}}&= \xcancel{1}- \frac{c^2}{2} \intertext{Höhere Potenzen vernachlässigen} + -a^2-b^2 &=-c^2\\ + a^2+b^2&=c^2 +\end{align} + +\subsubsection{Sphärischer Sinussatz} +Den sphärischen Sinussatz \begin{align} \frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} \nonumber \end{align} -auch beim Kugeldreieck gilt. +kann man ebenfalls mit der Korrespondenz \[a \approx \sin(a) \] zum entsprechenden ebenen Sinussatz \[\frac{a}{\sin (\alpha)} =\frac{b}{\sin (\beta)} = \frac{c}{\sin (\gamma)}\] approximieren. -\subsection{Sphärische Kosinussätze} -Auch in der sphärischen Trigonometrie gibt es den Seitenkosinussatz + +\subsubsection{Sphärische Kosinussätze} +In der sphärischen Trigonometrie gibt es den Seitenkosinussatz \begin{align} \cos \ a = \cos b \cdot \cos c + \sin b \cdot \sin c \cdot \cos \alpha \nonumber \end{align} %Seitenkosinussatz und den Winkelkosinussatz \begin{align} - \cos \gamma = -\cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \cdot \cos c. \nonumber -\end{align} + \cos \gamma = -\cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \cdot \cos c, \nonumber +\end{align} der nur in der sphärischen Trigonometrie vorhanden ist. -\subsection{Sphärischer Satz des Pythagoras für das rechtwinklige Kugeldreieck} -Es gibt in der sphärischen Trigonometrie eigentlich garkeinen "Satz des Pythagoras", wie man ihn aus der zweidimensionalen Geometrie kennt. -In der sphärischen Trigonometrie gibt es aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks, nicht aber für das rechtseitige Kugeldreieck, in eine Beziehung bringt. -Es gilt nämlich: +Analog gibt es auch beim Seitenkosinussatz eine Korrespondenz zu \[ a^2 \leftrightarrow 1-\cos(a),\] die den ebenen Kosinussatz herleiten lässt, nämlich \begin{align} - \cos c = \cos a \cdot \cos b \ \text{wenn} \nonumber & - \alpha = \frac{\pi}{2} \nonumber + \cos(a)&= \cos(b)\cdot \cos(c) + \sin(b) \cdot \sin(c)\cdot \cos(\alpha) \\ + 1-\frac{a^2}{2} &= \bigg[1-\frac{b^2}{2}\bigg]\bigg[1-\frac{c^2}{2}\bigg]+bc\cdot\cos(\alpha) \\ + \xcancel{1}-\frac{a^2}{2} &= \xcancel{1}-\frac{b^2}{2}-\frac{c^2}{2} \xcancel{+\frac{b^2c^2}{4}}+bc \cdot \cos(\alpha)\intertext{Höhere Potenzen vernachlässigen} + a^2&=b^2+c^2-2bc \cdot \cos(\alpha) \end{align} + + \ No newline at end of file -- cgit v1.2.1 From 537a80724031881b7ca7e84873d8f189fe70db45 Mon Sep 17 00:00:00 2001 From: "ENEZ-PC\\erdem" Date: Tue, 24 May 2022 16:23:00 +0200 Subject: Revert "Korrektur (noch nicht fertig)" This reverts commit ebe0085df81f3190423e14e6a48fc9d17550e417. --- buch/papers/nav/einleitung.tex | 2 +- buch/papers/nav/flatearth.tex | 3 +- buch/papers/nav/main.tex | 1 - buch/papers/nav/nautischesdreieck.tex | 89 ++++++++++++++++--------------- buch/papers/nav/packages.tex | 3 +- buch/papers/nav/sincos.tex | 6 +-- buch/papers/nav/trigo.tex | 99 +++++++++++++---------------------- 7 files changed, 87 insertions(+), 116 deletions(-) (limited to 'buch') diff --git a/buch/papers/nav/einleitung.tex b/buch/papers/nav/einleitung.tex index 8eb4481..aafa107 100644 --- a/buch/papers/nav/einleitung.tex +++ b/buch/papers/nav/einleitung.tex @@ -3,7 +3,7 @@ \section{Einleitung} Heutzutage ist die Navigation ein Teil des Lebens. Man sendet dem Kollegen seinen eigenen Standort, um sich das ewige Erklären zu sparen oder gibt die Adresse des Ziels ein, damit man seinen Aufenthaltsort zum Beispiel auf einer riesigen Wiese am See findet. -Dies wird durch Technologien wie Funknavigation, welches ein auf Laufzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist, oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt. +Dies wird durch Technologien wie Funknavigation, welches ein auf Langzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt. Vor all diesen technologischen Fortschritten gab es lediglich die Astronavigation, welche heute noch auf Schiffen verwendet wird im Falle eines Stromausfalls. Aber wie funktioniert die Navigation mit den Sternen? Welche Hilfsmittel benötigt man, welche Rolle spielt die Mathematik und weshalb kann die Erde nicht flach sein? In diesem Kapitel werden genau diese Fragen mithilfe des nautischen Dreiecks, der sphärischen Trigonometrie und einigen Hilfsmitteln und Messgeräten beantwortet. \ No newline at end of file diff --git a/buch/papers/nav/flatearth.tex b/buch/papers/nav/flatearth.tex index 3b08e8d..5bfc1b7 100644 --- a/buch/papers/nav/flatearth.tex +++ b/buch/papers/nav/flatearth.tex @@ -17,9 +17,10 @@ Eratosthenes konnte etwa 100 Jahre später den Erdumfang berechnen. Er beobachtete, dass die Sonne in Syene mittags im Zenit steht und gleichzeitig in Alexandria unter einem Winkel einfällt. Mithilfe der Trigonometrie konnte er mit dem Abstand der Städte und dem Einfallswinkel den Umfang berechnen. -Der Kartograph Gerhard Mercator projizierte die Erdkugel wie in Abbildung 21.1 dargestellt auf ein Papier und erstellte so eine winkeltreue Karte. +Der Kartograph Gerhard Mercator projizierte die Erdkugel auf ein Papier und erstellte so eine winkeltreue Karte. Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen. + Dies sieht man zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. Grönland ist auf der Weltkarte so gross wie Afrika. In der Anwendung Google Earth jedoch ist Grönland etwa so gross wie Algerien. diff --git a/buch/papers/nav/main.tex b/buch/papers/nav/main.tex index 4c52547..e16dc2a 100644 --- a/buch/papers/nav/main.tex +++ b/buch/papers/nav/main.tex @@ -19,4 +19,3 @@ \printbibliography[heading=subbibliography] \end{refsection} - diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex index 36e9c99..c239d64 100644 --- a/buch/papers/nav/nautischesdreieck.tex +++ b/buch/papers/nav/nautischesdreieck.tex @@ -4,26 +4,49 @@ Die Himmelskugel ist eine gedachte Kugel, welche die Erde und dessen Beobachter Der Zenit ist jener Punkt, der vom Erdmittelpunkt durch denn eigenen Standort an die Himmelskugel verlängert wird. Ein Gestirn ist ein Planet oder ein Fixstern, zu welchen es diverse Jahrbücher mit allen astronomischen Eigenschaften gibt. Der Himmelspol ist der Nordpol an die Himmelskugel projiziert. -Das nautische Dreieck hat die Ecken Zenit, Gestirn und Himmelspol, wie man in der Abbildung 21.5 sehen kann. +Das nautische Dreieck definiert sich durch folgende Ecken: Zenit, Gestirn und Himmelspol. Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der Himmelskugel zu bestimmen. -\subsection{Das Bilddreieck} +Für die Definition gilt: +\begin{center} + \begin{tabular}{ c c c } + Winkel && Name / Beziehung \\ + \hline + $\alpha$ && Rektaszension \\ + $\delta$ && Deklination \\ + $\theta$ && Sternzeit von Greenwich\\ + $\phi$ && Geographische Breite\\ + $\tau=\theta-\alpha$ && Stundenwinkel und Längengrad des Gestirns. \\ + $a$ && Azimut\\ + $h$ && Höhe + \end{tabular} +\end{center} + +\begin{itemize} + \item Seitenlänge Zenit zu Himmelspol $= \frac{\pi}{2} - \phi $ + \item Seitenlänge Himmelspol zu Gestirn $= \frac{\pi}{2} - \delta$ + \item Seitenlänge Zenit zu Gestirn $= \frac{\pi}{2} - h$ + \item Winkel von Zenit zu Himmelsnordpol zu Gestirn$=\pi - \alpha$ + \item Winkel von Himmelsnordpol zu Zenit und Gestirn$= \tau$ +\end{itemize} + + +\subsection{Zusammenhang des nautischen Dreiecks und des Kugeldreiecks auf der Erdkugel} \begin{figure} \begin{center} - \includegraphics[width=8cm]{papers/nav/bilder/kugel3.png} + \includegraphics[height=5cm,width=8cm]{papers/nav/bilder/kugel3.png} \caption[Nautisches Dreieck]{Nautisches Dreieck} \end{center} \end{figure} - Man kann das nautische Dreieck auf die Erdkugel projizieren. -Dieses Dreieck nennt man dann Bilddreieck. -Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. -Die Projektion auf der Erdkugel hat die Ecken Nordpol, Standort und Bildpunkt. + +Wie man in der Abbildung 21.4 sieht, liegt das nautische Dreieck auf der Himmelskugel mit den Ecken Zenit, Gestirn und Himmelsnordpol. +Das selbe Dreieck kann man aber auch auf die Erdkugel projizieren. Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. Die Projektion auf der Erdkugel hat die Ecken Nordpol, Standort und Bildpunkt. \section{Standortbestimmung ohne elektronische Hilfsmittel} Um den eigenen Standort herauszufinden, wird in diesem Kapitel die Projektion des nautische Dreiecks auf die Erdkugel zur Hilfe genommen. Mithilfe eines Sextanten, einem Jahrbuch und der sphärischen Trigonometrie kann man dann die Längen- und Breitengrade des eigenen Standortes bestimmen. -Was ein Sextant und ein Jahrbuch ist, wird im Abschnitt 21.6.3 erklärt. +Was ein Sextant und ein Jahrbuch ist, wird im Kapitel 21.6 erklärt. \begin{figure} \begin{center} \includegraphics[width=10cm]{papers/nav/bilder/dreieck.pdf} @@ -36,8 +59,8 @@ Was ein Sextant und ein Jahrbuch ist, wird im Abschnitt 21.6.3 erklärt. \subsection{Ecke $P$ und $A$} Unser eigener Standort ist der gesuchte Ecke $P$ und die Ecke $A$ ist in unserem Fall der Nordpol. -Der Vorteil an der Idee des nautischen Dreiecks ist, dass eine Ecke immer der Nordpol ist. -Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so einfach. +Der Vorteil ander Idee des nautischen Dreiecks ist, dass eine Ecke immer der Nordpol ist. +Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so simpel. \subsection{Ecke $B$ und $C$ - Bildpunkt $X$ und $Y$} Für die Standortermittlung benötigt man als weiteren Punkt ein Gestirn bzw. seinen Bildpunkt auf der Erdkugel. @@ -46,8 +69,8 @@ Es gibt diverse Gestirne, die man nutzen kann wie zum Beispiel die Sonne, der Mo Die Bildpunkte von den beiden Gestirnen $X$ und $Y$ bilden die beiden Ecken $B$ und $C$ im Dreieck der Abbildung 21.5. \subsection{Ephemeriden} -Zu all diesen Gestirnen gibt es Ephemeriden. -Diese enthalten die Rektaszensionen und Deklinationen in Abhängigkeit von der Zeit. +Zu all diesen Gestirnen gibt es Ephemeriden, die man auch Jahrbücher nennt. +In diesen findet man Begriffe wie Rektaszension, Deklination und Sternzeit. \begin{figure} \begin{center} @@ -60,24 +83,25 @@ Diese enthalten die Rektaszensionen und Deklinationen in Abhängigkeit von der Z Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und entspricht dem Breitengrad des Gestirns. \subsubsection{Rektaszension und Sternzeit} -Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt, welcher der Nullpunkt auf dem Himmelsäquator ist, steht und geht vom Koordinatensystem der Himmelskugel aus. - +Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt steht und geht vom Koordinatensystem der Himmelskugel aus. +Der Frühlungspunkt ist der Nullpunkt auf dem Himmelsäquator. Die Tatsache, dass sich die Himmelskugel ca. vier Minuten schneller um die eigene Achse dreht als die Erdkugel, stellt hier ein kleines Problem dar. Die Lösung ist die Sternzeit. -Mit dieser können wir die schnellere Drehung der Himmelskugel ausgleichen und können die am Frühlingspunkt (21. März) 12:00 Uhr ist die Sternzeit $\theta = 0$. +Mit dieser können wir die schnellere Drehung der Himmelskugel ausgleichen und können die +Am Frühlingspunkt (21. März) 12:00 Uhr ist die Sternzeit +$\theta = 0$. Die Sternzeit geht vom Frühlungspunkt aus, an welchem die Sonne den Himmelsäquator schneidet. Für die Standortermittlung auf der Erdkugel ist es am einfachsten, wenn man die Sternzeit von Greenwich berechnet. -Für die Sternzeit von Greenwich $\theta$ braucht man als erstes das Julianische Datum $T$ vom aktuellen Tag, welches sich leicht nachschlagen lässt. +Für die Sternzeit von Greenwich $\theta $braucht man als erstes das Julianische Datum $T$ vom aktuellen Tag, welches sich leicht recherchieren lässt. Im Anschluss berechnet man die Sternzeit von Greenwich \[\theta = 6^h 41^m 50^s,54841 + 8640184^s,812866 \cdot T + 0^s,093104 \cdot T^2 - 0^s,0000062 \cdot T^3.\] -Wenn man die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ bestimmen, wobei $\alpha$ die Rektaszension und $\theta$ die Sternzeit von Greenwich ist. +Wenn man die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ mithilfe der Rektaszension und Sternzeit von Greenwich bestimmen. Dies gilt analog auch für das zweite Gestirn. \subsubsection{Sextant} -Ein Sextant ist ein nautisches Messinstrument, mit dem man den Winkel zwischen der Blickrichtung zu weit entfernten Objekten bestimmen kann. Es wird vor allem der Winkelabstand zu Gestirnen gemessen. -Man benutzt ihn vor allem für die astronomische Navigation auf See. +Ein Sextant ist ein nautisches Messinstrument, mit dem man den Winkel zwischen der Blickrichtung zu weit entfernten Objekten bestimmen kann, insbesondere den Winkelabstand zu einem Gestirn vom Horizont. Man nutze ihn vor allem für die astronomische Navigation auf See. \begin{figure} \begin{center} @@ -85,32 +109,7 @@ Man benutzt ihn vor allem für die astronomische Navigation auf See. \caption[Sextant]{Sextant} \end{center} \end{figure} -\subsubsection{Eingeschaften} -Für das nautische Dreieck gibt es folgende Eigenschaften: -\begin{center} - \begin{tabular}{ l c l } - Legende && Name / Beziehung \\ - \hline - $\alpha$ && Rektaszension \\ - $\delta$ && Deklination \\ - $\theta$ && Sternzeit von Greenwich\\ - $\phi$ && Geographische Breite\\ - $\tau=\theta-\alpha$ && Stundenwinkel und Längengrad des Gestirns. \\ - $a$ && Azimut\\ - $h$ && Höhe - \end{tabular} -\end{center} -\begin{center} - \begin{tabular}{ l c l } - Eigenschaften \\ - \hline - Seitenlänge Zenit zu Himmelspol= && $\frac{\pi}{2} - \phi$ \\ - Seitenlänge Himmelspol zu Gestirn= && $\frac{\pi}{2} - \delta$ \\ - Seitenlänge Himmelspol zu Gestirn= && $\frac{\pi}{2} - h$ \\ - Winkel von Zenit zu Himmelsnordpol zu Gestirn= && $\pi-\alpha$\\ - Winkel von Himmelsnordpol zu Zenit zu Gestirn= && $\tau$\\ - \end{tabular} -\end{center} + \subsection{Bestimmung des eigenen Standortes $P$} Nun hat man die Koordinaten der beiden Gestirne und man weiss die Koordinaten des Nordpols. Damit wir unseren Standort bestimmen können, bilden wir zuerst das Dreieck $ABC$, dann das Dreieck $BPC$ und zum Schluss noch das Dreieck $ABP$. diff --git a/buch/papers/nav/packages.tex b/buch/papers/nav/packages.tex index f2e6132..5b87303 100644 --- a/buch/papers/nav/packages.tex +++ b/buch/papers/nav/packages.tex @@ -8,5 +8,4 @@ % following example %\usepackage{packagename} -\usepackage{amsmath} -\usepackage{cancel} \ No newline at end of file +\usepackage{amsmath} \ No newline at end of file diff --git a/buch/papers/nav/sincos.tex b/buch/papers/nav/sincos.tex index a1653e8..d56d482 100644 --- a/buch/papers/nav/sincos.tex +++ b/buch/papers/nav/sincos.tex @@ -7,14 +7,12 @@ Jedoch konnten sie dieses Problem nicht lösen. Die Geschichte der sphärischen Trigonometrie ist daher eng mit der Astronomie verknüpft. Ca. 350 vor Christus dachten die Griechen über Kugelgeometrie nach und sie wurde zu einer Hilfswissenschaft der Astronomen. Zwischen 190 v. Chr. und 120 v. Chr. lebte ein griechischer Astronom names Hipparchos. -Dieser entwickelte unter anderem die Chordentafeln, welche die Chord - Funktionen, auch Chord genannt, beinhalten und im Abschnitt 3.1.1 beschrieben sind. +Dieser entwickelte unter anderem die Chordentafeln, welche die Chord - Funktionen, auch Chord genannt, beinhalten. Chord ist der Vorgänger der Sinusfunktion und galt damals als wichtigste Grundlage der Trigonometrie. -In dieser Zeit wurden auch die ersten Sternenkarten angefertigt. Damals kannte man die Sinusfunktionen noch nicht. +In dieser Zeit wurden auch die ersten Sternenkarten angefertigt, jedoch kannte man damals die Sinusfunktion noch nicht. Aus Indien stammten die ersten Ansätze zu den Kosinussätzen. Aufbauend auf den indischen und griechischen Forschungen entwickeln die Araber um das 9. Jahrhundert den Sinussatz. -Die Definition der trigonometrischen Funktionen ermöglicht nur, rechtwinklige Dreiecke zu berechnen. -Die Beziehung zwischen Seiten und Winkeln sind komplizierter und als Sinus- und Kosinussätze bekannt. Doch ein paar weitere Jahrhunderte vergingen bis zu diesem Thema wieder verstärkt Forschung betrieben wurde, da im 15. Jahrhundert grosse Entdeckungsreisen, hauptsächlich per Schiff, erfolgten und die Orientierung mit Sternen vermehrt an Wichtigkeit gewann. Man nutzte für die Kartographie nun die Kugelgeometrie, um die Genauigkeit zu erhöhen. Der Sinussatz, die Tangensfunktion und der neu entwickelte Seitenkosinussatz wurden in dieser Zeit bereits verwendet und im darauffolgenden Jahrhundert folgte der Winkelkosinussatz. diff --git a/buch/papers/nav/trigo.tex b/buch/papers/nav/trigo.tex index aca8bd2..ce367f6 100644 --- a/buch/papers/nav/trigo.tex +++ b/buch/papers/nav/trigo.tex @@ -1,13 +1,16 @@ \section{Sphärische Trigonometrie} +In der sphärischen Trigonometrie gibt es eine Symetrie zwischen Seiten und Winkel, also zu jedem Satz über Seiten und Winkel gibt es einen entsprechenden Satz, mit dem man Winkel durch Seiten und Seiten durch Winkel ersetzt hat. +Dabei gibt es folgenden Zusammenhang zwischen der ebenen- und sphärischen Trigonometrie: + \subsection{Das Kugeldreieck} -Damit man die Definition des Kugeldreiecks versteht, müssen wir zuerst Begriffe wie Grosskreisebene und Grosskreisbögen verstehen. -Ein Grosskreis ist ein grösstmöglicher Kreis auf einer Kugeloberfläche. +Damit man die Definition des Kugeldreiecks versteht, müssen wir zuerst Begriffe wie "Grosskreisebene" und "Grosskreisbögen" verstehen. +Ein Grosskreis ist ein größtmöglicher Kreis auf einer Kugeloberfläche. Sein Mittelpunkt fällt immer mit dem Mittelpunkt der Kugel zusammen und ein Schnitt auf dem Großkreis teilt die Kugel in jedem Fall in zwei gleich grosse Hälften. Da es unendlich viele Möglichkeiten gibt, eine Kugel so zu zerschneiden, dass die Schnittebene den Kugelmittelpunkt trifft, gibt es auch unendlich viele Grosskreise. -Grosskreisbögen sind die kürzesten Verbindungslinien zwischen zwei Punkten auf der Kugel. +Grosskreisbögen sind die Verbindungslinien zwischen zwei Punkten auf der Kugel, welche auch "Seiten" eines Kugeldreiecks gennant werden. -Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden werden, so entsteht ein Kugeldreieck $ABC$. +Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden, so entsteht ein Kugeldreieck $ABC$. Für ein Kugeldreieck gilt, dass die Summe der drei Seiten kleiner als $2\pi$ aber grösser als 0 ist. $A$, $B$ und $C$ sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten (siehe Abbildung 21.2). @@ -16,6 +19,18 @@ Laut dieser Definition ist die Seite $c$ der Winkel $AMB$, wobei der Punkt $M$ d Man kann bei Kugeldreiecken nicht so einfach unterscheiden, was Innen oder Aussen ist. Wenn man drei Eckpunkte miteinander verbindet, ergeben sich immer 16 Kugeldreiecke. +Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha, \beta, \gamma < \pi$ nennt man Eulersches Dreieck. + +Es gibt einen Zusammenhang zwischen der ebenen- und sphärischen Trigonometrie, wobei folgend $a$ eine Seite beschreibt: +\begin{center} + \begin{tabular}{ccc} + Eben & $\leftrightarrow$ & sphärisch \\ + \hline + $a$ & $\leftrightarrow$ & $\sin \ a$ \\ + + $a^2$ & $\leftrightarrow$ & $-\cos \ a$ \\ + \end{tabular} +\end{center} \begin{figure} \begin{center} @@ -26,11 +41,8 @@ Wenn man drei Eckpunkte miteinander verbindet, ergeben sich immer 16 Kugeldreiec \end{figure} \subsection{Rechtwinkliges Dreieck und rechtseitiges Dreieck} -In der sphärischen Trigonometrie gibt es eine Symetrie zwischen Seiten und Winkel, also zu jedem Satz über Seiten und Winkel gibt es einen entsprechenden Satz, mit dem man Winkel durch Seiten und Seiten durch Winkel ersetzt hat. - Wie auch im ebenen Dreieck gibt es beim Kugeldreieck auch ein rechtwinkliges Kugeldreieck, bei dem ein Winkel $\frac{\pi}{2}$ ist. -Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss, wie man in der Abbildung 21.3 sehen kann. - +Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss. \begin{figure} \begin{center} @@ -39,7 +51,7 @@ Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine S \end{center} \end{figure} -\subsection{Winkelsumme und Flächeninhalt} +\subsection{Winkelsumme} \begin{figure} \begin{center} @@ -52,9 +64,9 @@ Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine S Die Winkel eines Kugeldreiecks sind die, welche die Halbtangenten in den Eckpunkten einschliessen. Für die Summe der Innenwinkel gilt \begin{align} - \alpha+\beta+\gamma &= \frac{F}{r^2} + \pi \quad \text{und} \quad \alpha+\beta+\gamma > \pi, \nonumber + \alpha+\beta+\gamma &= \frac{F}{r^2} + \pi \ \text{und} \ \alpha+\beta+\gamma > \pi, \nonumber \end{align} -wobei $F$ der Flächeninhalt des Kugeldreiecks ist. +wobei F der Flächeninhalt des Kugeldreiecks ist. \subsubsection{Sphärischer Exzess} Der sphärische Exzess \begin{align} @@ -65,69 +77,32 @@ beschreibt die Abweichung der Innenwinkelsumme von $\pi$ und ist proportional zu \subsubsection{Flächeninnhalt} Mithilfe des Radius $r$ und dem sphärischen Exzess $\epsilon$ gilt für den Flächeninhalt \[ F=\frac{\pi \cdot r^2}{\frac{\pi}{2}} \cdot \epsilon\]. +\subsection{Sphärischer Sinussatz} +In jedem Dreieck ist das Verhältnis des Sinus einer Seite zum Sinus des Gegenwinkels konstant. +Das bedeutet, dass -\subsection{Seiten und Winkelberechnung} -Es gibt in der sphärischen Trigonometrie eigentlich gar keinen Satz des Pythagoras, wie man ihn aus der zweidimensionalen Geometrie kennt. -Es gibt aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks, nicht aber für das rechtseitige Kugeldreieck, in eine Beziehung bringt und zum jetzigen Punkt noch unklar ist, weshalb dieser Satz so aussieht. -Die Approximation folgt noch. -Es gilt nämlich: -\begin{align} - \cos c = \cos a \cdot \cos b \quad \text{wenn} \nonumber & - \quad \alpha = \frac{\pi}{2} \nonumber -\end{align} - -\subsubsection{Approximation von kleinen Dreiecken} -Die Sätze in der ebenen Trigonometrie sind eigentlich Approximationen der sphärischen Trigonometrie. -So ist der Sinussatz in der Ebene nur eine Annäherung des sphärischen Sinussatzes. Das Gleiche gilt für den Kosinussatz und dem Satz des Pythagoras. -So kann mit dem Taylorpolynom 2. Grades den Sinus und den Kosinus vom Sphärischen in die Ebene approximieren: -\begin{align} - \sin(a) &\approx a \nonumber \intertext{und} - \cos(a)&\approx 1-\frac{a^2}{2}.\nonumber -\end{align} -Es gibt ebenfalls folgende Approximierung der Seiten von der Sphäre in die Ebene: -\begin{align} - a &\approx \sin(a) \nonumber \intertext{und} - a^2 &\approx 1-\cos(a). \nonumber -\end{align} -Die Korrespondenzen zwischen der ebenen- und sphärischen Trigonometrie werden in den kommenden Abschnitten erläutert. - -\subsubsection{Sphärischer Satz des Pythagoras} -Die Korrespondenz \[ a^2 \approx 1-cos(a)\] liefert unter Anderem einen entsprechenden Satz des Pythagoras, nämlich - -\begin{align} - \cos(a)\cdot \cos(b) &= \cos(c) \\ - \bigg[1-\frac{a^2}{2}\bigg] \cdot \bigg[1-\frac{b^2}{2}\bigg] &= 1-\frac{c^2}{2} \\ - \xcancel{1}- \frac{a^2}{2} - \frac{b^2}{2} + \xcancel{\frac{a^2b^2}{4}}&= \xcancel{1}- \frac{c^2}{2} \intertext{Höhere Potenzen vernachlässigen} - -a^2-b^2 &=-c^2\\ - a^2+b^2&=c^2 -\end{align} - -\subsubsection{Sphärischer Sinussatz} -Den sphärischen Sinussatz \begin{align} \frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} \nonumber \end{align} -kann man ebenfalls mit der Korrespondenz \[a \approx \sin(a) \] zum entsprechenden ebenen Sinussatz \[\frac{a}{\sin (\alpha)} =\frac{b}{\sin (\beta)} = \frac{c}{\sin (\gamma)}\] approximieren. +auch beim Kugeldreieck gilt. - -\subsubsection{Sphärische Kosinussätze} -In der sphärischen Trigonometrie gibt es den Seitenkosinussatz +\subsection{Sphärische Kosinussätze} +Auch in der sphärischen Trigonometrie gibt es den Seitenkosinussatz \begin{align} \cos \ a = \cos b \cdot \cos c + \sin b \cdot \sin c \cdot \cos \alpha \nonumber \end{align} %Seitenkosinussatz und den Winkelkosinussatz \begin{align} - \cos \gamma = -\cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \cdot \cos c, \nonumber -\end{align} der nur in der sphärischen Trigonometrie vorhanden ist. + \cos \gamma = -\cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \cdot \cos c. \nonumber +\end{align} -Analog gibt es auch beim Seitenkosinussatz eine Korrespondenz zu \[ a^2 \leftrightarrow 1-\cos(a),\] die den ebenen Kosinussatz herleiten lässt, nämlich +\subsection{Sphärischer Satz des Pythagoras für das rechtwinklige Kugeldreieck} +Es gibt in der sphärischen Trigonometrie eigentlich garkeinen "Satz des Pythagoras", wie man ihn aus der zweidimensionalen Geometrie kennt. +In der sphärischen Trigonometrie gibt es aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks, nicht aber für das rechtseitige Kugeldreieck, in eine Beziehung bringt. +Es gilt nämlich: \begin{align} - \cos(a)&= \cos(b)\cdot \cos(c) + \sin(b) \cdot \sin(c)\cdot \cos(\alpha) \\ - 1-\frac{a^2}{2} &= \bigg[1-\frac{b^2}{2}\bigg]\bigg[1-\frac{c^2}{2}\bigg]+bc\cdot\cos(\alpha) \\ - \xcancel{1}-\frac{a^2}{2} &= \xcancel{1}-\frac{b^2}{2}-\frac{c^2}{2} \xcancel{+\frac{b^2c^2}{4}}+bc \cdot \cos(\alpha)\intertext{Höhere Potenzen vernachlässigen} - a^2&=b^2+c^2-2bc \cdot \cos(\alpha) + \cos c = \cos a \cdot \cos b \ \text{wenn} \nonumber & + \alpha = \frac{\pi}{2} \nonumber \end{align} - - \ No newline at end of file -- cgit v1.2.1 From 7776e5829bf5da82b6b3fc5478ed05c6c9a66d29 Mon Sep 17 00:00:00 2001 From: "ENEZ-PC\\erdem" Date: Tue, 24 May 2022 16:23:21 +0200 Subject: Revert "Revert "Korrektur (noch nicht fertig)"" This reverts commit 2fd00f1b2f0d123fdb1fb1a93b5e4d361587329c. --- buch/papers/nav/einleitung.tex | 2 +- buch/papers/nav/flatearth.tex | 3 +- buch/papers/nav/main.tex | 1 + buch/papers/nav/nautischesdreieck.tex | 89 +++++++++++++++---------------- buch/papers/nav/packages.tex | 3 +- buch/papers/nav/sincos.tex | 6 ++- buch/papers/nav/trigo.tex | 99 ++++++++++++++++++++++------------- 7 files changed, 116 insertions(+), 87 deletions(-) (limited to 'buch') diff --git a/buch/papers/nav/einleitung.tex b/buch/papers/nav/einleitung.tex index aafa107..8eb4481 100644 --- a/buch/papers/nav/einleitung.tex +++ b/buch/papers/nav/einleitung.tex @@ -3,7 +3,7 @@ \section{Einleitung} Heutzutage ist die Navigation ein Teil des Lebens. Man sendet dem Kollegen seinen eigenen Standort, um sich das ewige Erklären zu sparen oder gibt die Adresse des Ziels ein, damit man seinen Aufenthaltsort zum Beispiel auf einer riesigen Wiese am See findet. -Dies wird durch Technologien wie Funknavigation, welches ein auf Langzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt. +Dies wird durch Technologien wie Funknavigation, welches ein auf Laufzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist, oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt. Vor all diesen technologischen Fortschritten gab es lediglich die Astronavigation, welche heute noch auf Schiffen verwendet wird im Falle eines Stromausfalls. Aber wie funktioniert die Navigation mit den Sternen? Welche Hilfsmittel benötigt man, welche Rolle spielt die Mathematik und weshalb kann die Erde nicht flach sein? In diesem Kapitel werden genau diese Fragen mithilfe des nautischen Dreiecks, der sphärischen Trigonometrie und einigen Hilfsmitteln und Messgeräten beantwortet. \ No newline at end of file diff --git a/buch/papers/nav/flatearth.tex b/buch/papers/nav/flatearth.tex index 5bfc1b7..3b08e8d 100644 --- a/buch/papers/nav/flatearth.tex +++ b/buch/papers/nav/flatearth.tex @@ -17,10 +17,9 @@ Eratosthenes konnte etwa 100 Jahre später den Erdumfang berechnen. Er beobachtete, dass die Sonne in Syene mittags im Zenit steht und gleichzeitig in Alexandria unter einem Winkel einfällt. Mithilfe der Trigonometrie konnte er mit dem Abstand der Städte und dem Einfallswinkel den Umfang berechnen. -Der Kartograph Gerhard Mercator projizierte die Erdkugel auf ein Papier und erstellte so eine winkeltreue Karte. +Der Kartograph Gerhard Mercator projizierte die Erdkugel wie in Abbildung 21.1 dargestellt auf ein Papier und erstellte so eine winkeltreue Karte. Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen. - Dies sieht man zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. Grönland ist auf der Weltkarte so gross wie Afrika. In der Anwendung Google Earth jedoch ist Grönland etwa so gross wie Algerien. diff --git a/buch/papers/nav/main.tex b/buch/papers/nav/main.tex index e16dc2a..4c52547 100644 --- a/buch/papers/nav/main.tex +++ b/buch/papers/nav/main.tex @@ -19,3 +19,4 @@ \printbibliography[heading=subbibliography] \end{refsection} + diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex index c239d64..36e9c99 100644 --- a/buch/papers/nav/nautischesdreieck.tex +++ b/buch/papers/nav/nautischesdreieck.tex @@ -4,49 +4,26 @@ Die Himmelskugel ist eine gedachte Kugel, welche die Erde und dessen Beobachter Der Zenit ist jener Punkt, der vom Erdmittelpunkt durch denn eigenen Standort an die Himmelskugel verlängert wird. Ein Gestirn ist ein Planet oder ein Fixstern, zu welchen es diverse Jahrbücher mit allen astronomischen Eigenschaften gibt. Der Himmelspol ist der Nordpol an die Himmelskugel projiziert. -Das nautische Dreieck definiert sich durch folgende Ecken: Zenit, Gestirn und Himmelspol. +Das nautische Dreieck hat die Ecken Zenit, Gestirn und Himmelspol, wie man in der Abbildung 21.5 sehen kann. Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der Himmelskugel zu bestimmen. -Für die Definition gilt: -\begin{center} - \begin{tabular}{ c c c } - Winkel && Name / Beziehung \\ - \hline - $\alpha$ && Rektaszension \\ - $\delta$ && Deklination \\ - $\theta$ && Sternzeit von Greenwich\\ - $\phi$ && Geographische Breite\\ - $\tau=\theta-\alpha$ && Stundenwinkel und Längengrad des Gestirns. \\ - $a$ && Azimut\\ - $h$ && Höhe - \end{tabular} -\end{center} - -\begin{itemize} - \item Seitenlänge Zenit zu Himmelspol $= \frac{\pi}{2} - \phi $ - \item Seitenlänge Himmelspol zu Gestirn $= \frac{\pi}{2} - \delta$ - \item Seitenlänge Zenit zu Gestirn $= \frac{\pi}{2} - h$ - \item Winkel von Zenit zu Himmelsnordpol zu Gestirn$=\pi - \alpha$ - \item Winkel von Himmelsnordpol zu Zenit und Gestirn$= \tau$ -\end{itemize} - - -\subsection{Zusammenhang des nautischen Dreiecks und des Kugeldreiecks auf der Erdkugel} +\subsection{Das Bilddreieck} \begin{figure} \begin{center} - \includegraphics[height=5cm,width=8cm]{papers/nav/bilder/kugel3.png} + \includegraphics[width=8cm]{papers/nav/bilder/kugel3.png} \caption[Nautisches Dreieck]{Nautisches Dreieck} \end{center} \end{figure} - -Wie man in der Abbildung 21.4 sieht, liegt das nautische Dreieck auf der Himmelskugel mit den Ecken Zenit, Gestirn und Himmelsnordpol. -Das selbe Dreieck kann man aber auch auf die Erdkugel projizieren. Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. Die Projektion auf der Erdkugel hat die Ecken Nordpol, Standort und Bildpunkt. + Man kann das nautische Dreieck auf die Erdkugel projizieren. +Dieses Dreieck nennt man dann Bilddreieck. +Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. +Die Projektion auf der Erdkugel hat die Ecken Nordpol, Standort und Bildpunkt. \section{Standortbestimmung ohne elektronische Hilfsmittel} Um den eigenen Standort herauszufinden, wird in diesem Kapitel die Projektion des nautische Dreiecks auf die Erdkugel zur Hilfe genommen. Mithilfe eines Sextanten, einem Jahrbuch und der sphärischen Trigonometrie kann man dann die Längen- und Breitengrade des eigenen Standortes bestimmen. -Was ein Sextant und ein Jahrbuch ist, wird im Kapitel 21.6 erklärt. +Was ein Sextant und ein Jahrbuch ist, wird im Abschnitt 21.6.3 erklärt. \begin{figure} \begin{center} \includegraphics[width=10cm]{papers/nav/bilder/dreieck.pdf} @@ -59,8 +36,8 @@ Was ein Sextant und ein Jahrbuch ist, wird im Kapitel 21.6 erklärt. \subsection{Ecke $P$ und $A$} Unser eigener Standort ist der gesuchte Ecke $P$ und die Ecke $A$ ist in unserem Fall der Nordpol. -Der Vorteil ander Idee des nautischen Dreiecks ist, dass eine Ecke immer der Nordpol ist. -Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so simpel. +Der Vorteil an der Idee des nautischen Dreiecks ist, dass eine Ecke immer der Nordpol ist. +Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so einfach. \subsection{Ecke $B$ und $C$ - Bildpunkt $X$ und $Y$} Für die Standortermittlung benötigt man als weiteren Punkt ein Gestirn bzw. seinen Bildpunkt auf der Erdkugel. @@ -69,8 +46,8 @@ Es gibt diverse Gestirne, die man nutzen kann wie zum Beispiel die Sonne, der Mo Die Bildpunkte von den beiden Gestirnen $X$ und $Y$ bilden die beiden Ecken $B$ und $C$ im Dreieck der Abbildung 21.5. \subsection{Ephemeriden} -Zu all diesen Gestirnen gibt es Ephemeriden, die man auch Jahrbücher nennt. -In diesen findet man Begriffe wie Rektaszension, Deklination und Sternzeit. +Zu all diesen Gestirnen gibt es Ephemeriden. +Diese enthalten die Rektaszensionen und Deklinationen in Abhängigkeit von der Zeit. \begin{figure} \begin{center} @@ -83,25 +60,24 @@ In diesen findet man Begriffe wie Rektaszension, Deklination und Sternzeit. Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und entspricht dem Breitengrad des Gestirns. \subsubsection{Rektaszension und Sternzeit} -Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt steht und geht vom Koordinatensystem der Himmelskugel aus. -Der Frühlungspunkt ist der Nullpunkt auf dem Himmelsäquator. +Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt, welcher der Nullpunkt auf dem Himmelsäquator ist, steht und geht vom Koordinatensystem der Himmelskugel aus. + Die Tatsache, dass sich die Himmelskugel ca. vier Minuten schneller um die eigene Achse dreht als die Erdkugel, stellt hier ein kleines Problem dar. Die Lösung ist die Sternzeit. -Mit dieser können wir die schnellere Drehung der Himmelskugel ausgleichen und können die -Am Frühlingspunkt (21. März) 12:00 Uhr ist die Sternzeit -$\theta = 0$. +Mit dieser können wir die schnellere Drehung der Himmelskugel ausgleichen und können die am Frühlingspunkt (21. März) 12:00 Uhr ist die Sternzeit $\theta = 0$. Die Sternzeit geht vom Frühlungspunkt aus, an welchem die Sonne den Himmelsäquator schneidet. Für die Standortermittlung auf der Erdkugel ist es am einfachsten, wenn man die Sternzeit von Greenwich berechnet. -Für die Sternzeit von Greenwich $\theta $braucht man als erstes das Julianische Datum $T$ vom aktuellen Tag, welches sich leicht recherchieren lässt. +Für die Sternzeit von Greenwich $\theta$ braucht man als erstes das Julianische Datum $T$ vom aktuellen Tag, welches sich leicht nachschlagen lässt. Im Anschluss berechnet man die Sternzeit von Greenwich \[\theta = 6^h 41^m 50^s,54841 + 8640184^s,812866 \cdot T + 0^s,093104 \cdot T^2 - 0^s,0000062 \cdot T^3.\] -Wenn man die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ mithilfe der Rektaszension und Sternzeit von Greenwich bestimmen. +Wenn man die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ bestimmen, wobei $\alpha$ die Rektaszension und $\theta$ die Sternzeit von Greenwich ist. Dies gilt analog auch für das zweite Gestirn. \subsubsection{Sextant} -Ein Sextant ist ein nautisches Messinstrument, mit dem man den Winkel zwischen der Blickrichtung zu weit entfernten Objekten bestimmen kann, insbesondere den Winkelabstand zu einem Gestirn vom Horizont. Man nutze ihn vor allem für die astronomische Navigation auf See. +Ein Sextant ist ein nautisches Messinstrument, mit dem man den Winkel zwischen der Blickrichtung zu weit entfernten Objekten bestimmen kann. Es wird vor allem der Winkelabstand zu Gestirnen gemessen. +Man benutzt ihn vor allem für die astronomische Navigation auf See. \begin{figure} \begin{center} @@ -109,7 +85,32 @@ Ein Sextant ist ein nautisches Messinstrument, mit dem man den Winkel zwischen d \caption[Sextant]{Sextant} \end{center} \end{figure} - +\subsubsection{Eingeschaften} +Für das nautische Dreieck gibt es folgende Eigenschaften: +\begin{center} + \begin{tabular}{ l c l } + Legende && Name / Beziehung \\ + \hline + $\alpha$ && Rektaszension \\ + $\delta$ && Deklination \\ + $\theta$ && Sternzeit von Greenwich\\ + $\phi$ && Geographische Breite\\ + $\tau=\theta-\alpha$ && Stundenwinkel und Längengrad des Gestirns. \\ + $a$ && Azimut\\ + $h$ && Höhe + \end{tabular} +\end{center} +\begin{center} + \begin{tabular}{ l c l } + Eigenschaften \\ + \hline + Seitenlänge Zenit zu Himmelspol= && $\frac{\pi}{2} - \phi$ \\ + Seitenlänge Himmelspol zu Gestirn= && $\frac{\pi}{2} - \delta$ \\ + Seitenlänge Himmelspol zu Gestirn= && $\frac{\pi}{2} - h$ \\ + Winkel von Zenit zu Himmelsnordpol zu Gestirn= && $\pi-\alpha$\\ + Winkel von Himmelsnordpol zu Zenit zu Gestirn= && $\tau$\\ + \end{tabular} +\end{center} \subsection{Bestimmung des eigenen Standortes $P$} Nun hat man die Koordinaten der beiden Gestirne und man weiss die Koordinaten des Nordpols. Damit wir unseren Standort bestimmen können, bilden wir zuerst das Dreieck $ABC$, dann das Dreieck $BPC$ und zum Schluss noch das Dreieck $ABP$. diff --git a/buch/papers/nav/packages.tex b/buch/papers/nav/packages.tex index 5b87303..f2e6132 100644 --- a/buch/papers/nav/packages.tex +++ b/buch/papers/nav/packages.tex @@ -8,4 +8,5 @@ % following example %\usepackage{packagename} -\usepackage{amsmath} \ No newline at end of file +\usepackage{amsmath} +\usepackage{cancel} \ No newline at end of file diff --git a/buch/papers/nav/sincos.tex b/buch/papers/nav/sincos.tex index d56d482..a1653e8 100644 --- a/buch/papers/nav/sincos.tex +++ b/buch/papers/nav/sincos.tex @@ -7,12 +7,14 @@ Jedoch konnten sie dieses Problem nicht lösen. Die Geschichte der sphärischen Trigonometrie ist daher eng mit der Astronomie verknüpft. Ca. 350 vor Christus dachten die Griechen über Kugelgeometrie nach und sie wurde zu einer Hilfswissenschaft der Astronomen. Zwischen 190 v. Chr. und 120 v. Chr. lebte ein griechischer Astronom names Hipparchos. -Dieser entwickelte unter anderem die Chordentafeln, welche die Chord - Funktionen, auch Chord genannt, beinhalten. +Dieser entwickelte unter anderem die Chordentafeln, welche die Chord - Funktionen, auch Chord genannt, beinhalten und im Abschnitt 3.1.1 beschrieben sind. Chord ist der Vorgänger der Sinusfunktion und galt damals als wichtigste Grundlage der Trigonometrie. -In dieser Zeit wurden auch die ersten Sternenkarten angefertigt, jedoch kannte man damals die Sinusfunktion noch nicht. +In dieser Zeit wurden auch die ersten Sternenkarten angefertigt. Damals kannte man die Sinusfunktionen noch nicht. Aus Indien stammten die ersten Ansätze zu den Kosinussätzen. Aufbauend auf den indischen und griechischen Forschungen entwickeln die Araber um das 9. Jahrhundert den Sinussatz. +Die Definition der trigonometrischen Funktionen ermöglicht nur, rechtwinklige Dreiecke zu berechnen. +Die Beziehung zwischen Seiten und Winkeln sind komplizierter und als Sinus- und Kosinussätze bekannt. Doch ein paar weitere Jahrhunderte vergingen bis zu diesem Thema wieder verstärkt Forschung betrieben wurde, da im 15. Jahrhundert grosse Entdeckungsreisen, hauptsächlich per Schiff, erfolgten und die Orientierung mit Sternen vermehrt an Wichtigkeit gewann. Man nutzte für die Kartographie nun die Kugelgeometrie, um die Genauigkeit zu erhöhen. Der Sinussatz, die Tangensfunktion und der neu entwickelte Seitenkosinussatz wurden in dieser Zeit bereits verwendet und im darauffolgenden Jahrhundert folgte der Winkelkosinussatz. diff --git a/buch/papers/nav/trigo.tex b/buch/papers/nav/trigo.tex index ce367f6..aca8bd2 100644 --- a/buch/papers/nav/trigo.tex +++ b/buch/papers/nav/trigo.tex @@ -1,16 +1,13 @@ \section{Sphärische Trigonometrie} -In der sphärischen Trigonometrie gibt es eine Symetrie zwischen Seiten und Winkel, also zu jedem Satz über Seiten und Winkel gibt es einen entsprechenden Satz, mit dem man Winkel durch Seiten und Seiten durch Winkel ersetzt hat. -Dabei gibt es folgenden Zusammenhang zwischen der ebenen- und sphärischen Trigonometrie: - \subsection{Das Kugeldreieck} -Damit man die Definition des Kugeldreiecks versteht, müssen wir zuerst Begriffe wie "Grosskreisebene" und "Grosskreisbögen" verstehen. -Ein Grosskreis ist ein größtmöglicher Kreis auf einer Kugeloberfläche. +Damit man die Definition des Kugeldreiecks versteht, müssen wir zuerst Begriffe wie Grosskreisebene und Grosskreisbögen verstehen. +Ein Grosskreis ist ein grösstmöglicher Kreis auf einer Kugeloberfläche. Sein Mittelpunkt fällt immer mit dem Mittelpunkt der Kugel zusammen und ein Schnitt auf dem Großkreis teilt die Kugel in jedem Fall in zwei gleich grosse Hälften. Da es unendlich viele Möglichkeiten gibt, eine Kugel so zu zerschneiden, dass die Schnittebene den Kugelmittelpunkt trifft, gibt es auch unendlich viele Grosskreise. -Grosskreisbögen sind die Verbindungslinien zwischen zwei Punkten auf der Kugel, welche auch "Seiten" eines Kugeldreiecks gennant werden. +Grosskreisbögen sind die kürzesten Verbindungslinien zwischen zwei Punkten auf der Kugel. -Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden, so entsteht ein Kugeldreieck $ABC$. +Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden werden, so entsteht ein Kugeldreieck $ABC$. Für ein Kugeldreieck gilt, dass die Summe der drei Seiten kleiner als $2\pi$ aber grösser als 0 ist. $A$, $B$ und $C$ sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten (siehe Abbildung 21.2). @@ -19,18 +16,6 @@ Laut dieser Definition ist die Seite $c$ der Winkel $AMB$, wobei der Punkt $M$ d Man kann bei Kugeldreiecken nicht so einfach unterscheiden, was Innen oder Aussen ist. Wenn man drei Eckpunkte miteinander verbindet, ergeben sich immer 16 Kugeldreiecke. -Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha, \beta, \gamma < \pi$ nennt man Eulersches Dreieck. - -Es gibt einen Zusammenhang zwischen der ebenen- und sphärischen Trigonometrie, wobei folgend $a$ eine Seite beschreibt: -\begin{center} - \begin{tabular}{ccc} - Eben & $\leftrightarrow$ & sphärisch \\ - \hline - $a$ & $\leftrightarrow$ & $\sin \ a$ \\ - - $a^2$ & $\leftrightarrow$ & $-\cos \ a$ \\ - \end{tabular} -\end{center} \begin{figure} \begin{center} @@ -41,8 +26,11 @@ Es gibt einen Zusammenhang zwischen der ebenen- und sphärischen Trigonometrie, \end{figure} \subsection{Rechtwinkliges Dreieck und rechtseitiges Dreieck} +In der sphärischen Trigonometrie gibt es eine Symetrie zwischen Seiten und Winkel, also zu jedem Satz über Seiten und Winkel gibt es einen entsprechenden Satz, mit dem man Winkel durch Seiten und Seiten durch Winkel ersetzt hat. + Wie auch im ebenen Dreieck gibt es beim Kugeldreieck auch ein rechtwinkliges Kugeldreieck, bei dem ein Winkel $\frac{\pi}{2}$ ist. -Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss. +Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss, wie man in der Abbildung 21.3 sehen kann. + \begin{figure} \begin{center} @@ -51,7 +39,7 @@ Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine S \end{center} \end{figure} -\subsection{Winkelsumme} +\subsection{Winkelsumme und Flächeninhalt} \begin{figure} \begin{center} @@ -64,9 +52,9 @@ Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine S Die Winkel eines Kugeldreiecks sind die, welche die Halbtangenten in den Eckpunkten einschliessen. Für die Summe der Innenwinkel gilt \begin{align} - \alpha+\beta+\gamma &= \frac{F}{r^2} + \pi \ \text{und} \ \alpha+\beta+\gamma > \pi, \nonumber + \alpha+\beta+\gamma &= \frac{F}{r^2} + \pi \quad \text{und} \quad \alpha+\beta+\gamma > \pi, \nonumber \end{align} -wobei F der Flächeninhalt des Kugeldreiecks ist. +wobei $F$ der Flächeninhalt des Kugeldreiecks ist. \subsubsection{Sphärischer Exzess} Der sphärische Exzess \begin{align} @@ -77,32 +65,69 @@ beschreibt die Abweichung der Innenwinkelsumme von $\pi$ und ist proportional zu \subsubsection{Flächeninnhalt} Mithilfe des Radius $r$ und dem sphärischen Exzess $\epsilon$ gilt für den Flächeninhalt \[ F=\frac{\pi \cdot r^2}{\frac{\pi}{2}} \cdot \epsilon\]. -\subsection{Sphärischer Sinussatz} -In jedem Dreieck ist das Verhältnis des Sinus einer Seite zum Sinus des Gegenwinkels konstant. -Das bedeutet, dass +\subsection{Seiten und Winkelberechnung} +Es gibt in der sphärischen Trigonometrie eigentlich gar keinen Satz des Pythagoras, wie man ihn aus der zweidimensionalen Geometrie kennt. +Es gibt aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks, nicht aber für das rechtseitige Kugeldreieck, in eine Beziehung bringt und zum jetzigen Punkt noch unklar ist, weshalb dieser Satz so aussieht. +Die Approximation folgt noch. +Es gilt nämlich: +\begin{align} + \cos c = \cos a \cdot \cos b \quad \text{wenn} \nonumber & + \quad \alpha = \frac{\pi}{2} \nonumber +\end{align} + +\subsubsection{Approximation von kleinen Dreiecken} +Die Sätze in der ebenen Trigonometrie sind eigentlich Approximationen der sphärischen Trigonometrie. +So ist der Sinussatz in der Ebene nur eine Annäherung des sphärischen Sinussatzes. Das Gleiche gilt für den Kosinussatz und dem Satz des Pythagoras. +So kann mit dem Taylorpolynom 2. Grades den Sinus und den Kosinus vom Sphärischen in die Ebene approximieren: +\begin{align} + \sin(a) &\approx a \nonumber \intertext{und} + \cos(a)&\approx 1-\frac{a^2}{2}.\nonumber +\end{align} +Es gibt ebenfalls folgende Approximierung der Seiten von der Sphäre in die Ebene: +\begin{align} + a &\approx \sin(a) \nonumber \intertext{und} + a^2 &\approx 1-\cos(a). \nonumber +\end{align} +Die Korrespondenzen zwischen der ebenen- und sphärischen Trigonometrie werden in den kommenden Abschnitten erläutert. + +\subsubsection{Sphärischer Satz des Pythagoras} +Die Korrespondenz \[ a^2 \approx 1-cos(a)\] liefert unter Anderem einen entsprechenden Satz des Pythagoras, nämlich + +\begin{align} + \cos(a)\cdot \cos(b) &= \cos(c) \\ + \bigg[1-\frac{a^2}{2}\bigg] \cdot \bigg[1-\frac{b^2}{2}\bigg] &= 1-\frac{c^2}{2} \\ + \xcancel{1}- \frac{a^2}{2} - \frac{b^2}{2} + \xcancel{\frac{a^2b^2}{4}}&= \xcancel{1}- \frac{c^2}{2} \intertext{Höhere Potenzen vernachlässigen} + -a^2-b^2 &=-c^2\\ + a^2+b^2&=c^2 +\end{align} + +\subsubsection{Sphärischer Sinussatz} +Den sphärischen Sinussatz \begin{align} \frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} \nonumber \end{align} -auch beim Kugeldreieck gilt. +kann man ebenfalls mit der Korrespondenz \[a \approx \sin(a) \] zum entsprechenden ebenen Sinussatz \[\frac{a}{\sin (\alpha)} =\frac{b}{\sin (\beta)} = \frac{c}{\sin (\gamma)}\] approximieren. -\subsection{Sphärische Kosinussätze} -Auch in der sphärischen Trigonometrie gibt es den Seitenkosinussatz + +\subsubsection{Sphärische Kosinussätze} +In der sphärischen Trigonometrie gibt es den Seitenkosinussatz \begin{align} \cos \ a = \cos b \cdot \cos c + \sin b \cdot \sin c \cdot \cos \alpha \nonumber \end{align} %Seitenkosinussatz und den Winkelkosinussatz \begin{align} - \cos \gamma = -\cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \cdot \cos c. \nonumber -\end{align} + \cos \gamma = -\cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \cdot \cos c, \nonumber +\end{align} der nur in der sphärischen Trigonometrie vorhanden ist. -\subsection{Sphärischer Satz des Pythagoras für das rechtwinklige Kugeldreieck} -Es gibt in der sphärischen Trigonometrie eigentlich garkeinen "Satz des Pythagoras", wie man ihn aus der zweidimensionalen Geometrie kennt. -In der sphärischen Trigonometrie gibt es aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks, nicht aber für das rechtseitige Kugeldreieck, in eine Beziehung bringt. -Es gilt nämlich: +Analog gibt es auch beim Seitenkosinussatz eine Korrespondenz zu \[ a^2 \leftrightarrow 1-\cos(a),\] die den ebenen Kosinussatz herleiten lässt, nämlich \begin{align} - \cos c = \cos a \cdot \cos b \ \text{wenn} \nonumber & - \alpha = \frac{\pi}{2} \nonumber + \cos(a)&= \cos(b)\cdot \cos(c) + \sin(b) \cdot \sin(c)\cdot \cos(\alpha) \\ + 1-\frac{a^2}{2} &= \bigg[1-\frac{b^2}{2}\bigg]\bigg[1-\frac{c^2}{2}\bigg]+bc\cdot\cos(\alpha) \\ + \xcancel{1}-\frac{a^2}{2} &= \xcancel{1}-\frac{b^2}{2}-\frac{c^2}{2} \xcancel{+\frac{b^2c^2}{4}}+bc \cdot \cos(\alpha)\intertext{Höhere Potenzen vernachlässigen} + a^2&=b^2+c^2-2bc \cdot \cos(\alpha) \end{align} + + \ No newline at end of file -- cgit v1.2.1 From 76a5de291c288aa6e439fb97b0172dcfb5c9f1fe Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 25 May 2022 10:30:01 +0200 Subject: rationale Funktionen --- buch/chapters/060-integral/rational.tex | 169 ++++++++++++++++++++++++++++++++ 1 file changed, 169 insertions(+) (limited to 'buch') diff --git a/buch/chapters/060-integral/rational.tex b/buch/chapters/060-integral/rational.tex index 19f2ad9..989e65b 100644 --- a/buch/chapters/060-integral/rational.tex +++ b/buch/chapters/060-integral/rational.tex @@ -5,4 +5,173 @@ % \subsection{Rationale Funktionen und Funktionenkörper \label{buch:integral:subsection:rational}} +Welche Funktionen sollen als Antwort auf die Frage nach einer Stammfunktion +akzeptiert werden? +Polynome in der unabhängigen Variablen $x$ sollten sicher dazu gehören, +also alles, was man mit Hilfe der Multiplikation, Addition und Subtraktion +aus Koeffizienten zum Beispiel in den rationalen Zahlen $\mathbb{Q}$ und +der unabhängigen Variablen aufbauen kann. +Doch welche weiteren Operationen sollen zugelassen werden und was lässt +sich über die entstehende Funktionenmenge aussagen? + +\subsubsection{Körper} +Die kleinste Zahlenmenge, in der alle arithmetischen Operationen soweit +sinnvoll durchgeführt werden können, ist die Menge $\mathbb{Q}$ der +rationalen Zahlen. +Etwas formaler ist eine solche Menge, in der die Arithmetik uneingeschränkt +ausgeführt werden kann, ein Körper gemäss der folgenden Definition. +\index{Korper@Körper}% + +\begin{definition} +\label{buch:integral:definition:koerper} +Eine {\em Körper} ist eine Menge $K$ mit zwei Verknüpfungen $+$, die Addition, +und $\cdot$, die Multiplikation, +welche die folgenden Eigenschaften haben. +\begin{center} +\renewcommand{\tabcolsep}{0pt} +\begin{tabular}{p{68mm}p{4mm}p{68mm}} +%Eigenschaften der +Addition: +\begin{enumerate}[{\bf A}.1)] +\item assoziativ: $(a+b)+c=a+(b+c)$ +für alle $a,b,c\in K$ +\item kommutativ: $a+b=b+a$ +für alle $a,b\in K$ +\item Neutrales Element der Addition: es gibt ein Element $0\in K$ mit +der Eigenschaft $a+0=a$ für alle $a\in K$ +\item Additiv inverses Element: zu jedem Element $a\in K$ gibt es das Element +$-a$ mit der Eigenschaft $a+(-a)=0$. +\end{enumerate} +&&% +%Eigenschaften der +Multiplikation: +\begin{enumerate}[{\bf M}.1)] +\item assoziativ: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ +für alle $a,b,c\in K$ +\index{Assoziativgesetz}% +\index{assoziativ}% +\item kommutativ: $a\cdot b=b\cdot a$ +für alle $a,b\in K$ +\index{Kommutativgesetz}% +\index{kommutativ}% +\item Neutrales Element der Multiplikation: es gibt ein Element $1\in K$ mit +der Eigenschaft $a\cdot 1 =a$ für alle $a\in K$ +\index{neutrales Element}% +\item Multiplikativ inverses Element: zu jedem Element +\index{inverses Element}% +$a\in K^*=K\setminus\{0\}$ +gibt es das Element $a^{-1}$ mit der Eigenschaft $a\cdot a^{-1}=1$. +Die Menge $K^*$ heisst auch die {\em Einheitengruppe} oder die +{\em Gruppe der invertierbaren Elemente} des Körpers. +\index{Einheitengruppe}% +\index{Gruppe der invertierbaren Elemente}% +\end{enumerate} +\end{tabular} +\end{center} +\vspace{-10pt} +Ausserdem gilt das Distributivgesetz: für alle $a,b,c\in K$ gilt +$a\cdot(b+c)=a\cdot b + a\cdot c$. +\index{Disitributivgesetz}% +\end{definition} + +Das Assoziativgesetz {\bf A}.1 besagt, dass Summen mit beliebig +vielen Termen ohne Klammern geschrieben werden kann, weil es nicht +darauf ankommt, in welcher Reihenfolge die Additionen ausgeführt werden. +Ebenso für das Assoziativgesetz {\bf M}.1 der Multiplikation. +Die Kommutativgesetze {\bf A}.2 und {\bf M}.2 implizieren, dass man +nicht auf die Reihenfolge der Summanden oder Faktoren achten muss. +Das Distributivgesetz schliesslich besagt, dass man Produkte ausmultiplizieren +oder gemeinsame Faktoren ausklammern kann, wie man es in der Schule +gelernt hat. + +Die rellen Zahlen $\mathbb{R}$ und die komplexen Zahlen $\mathbb{C}$ +bilden ebenfalls einen Körper, die von den rationalen Zahlen geerbten +Eigenschaften der Verknüpfungen setzen sich auf $\mathbb{R}$ und +$\mathbb{C}$ fort. +Es lassen sich allerdings auch Zahlkörper zwischen $\mathbb{Q}$ und +$\mathbb{R}$ konstruieren, wie das folgende Beispiel zeigt. + +\begin{beispiel} +Die Menge +\[ +\mathbb{Q}(\!\sqrt{2}) += +\{ +a+b\sqrt{2} +\;|\; +a,b\in \mathbb{Q} +\} +\] +ist eine Teilmenge von $\mathbb{R}$. +Die Rechenoperationen haben alle verlangten Eigenschaften, wenn gezeigt +werden kann, dass Produkte und Quotienten von Zahlen in $\mathbb{Q}(\!\sqrt{2})$ +wieder in $\mathbb{Q}(\!\sqrt{2})$ sind. +Dazu rechnet man +\begin{align*} +(a+b\sqrt{2}) +(c+d\sqrt{2}) +&= +ac + 2bd + (ad+bc)\sqrt{2} \in \mathbb{Q}(\!\sqrt{2}) +\intertext{und} +\frac{a+b\sqrt{2}}{c+d\sqrt{2}} +&= +\frac{a+b\sqrt{2}}{c+d\sqrt{2}} +\cdot +\frac{c-d\sqrt{2}}{c-d\sqrt{2}} += +\frac{ac-2bd +(-ad+bc)\sqrt{2}}{c^2-2d^2} +\\ +&= +\underbrace{\frac{ac-2bd}{c^2-2d^2}}_{\displaystyle\in\mathbb{Q}} ++ +\underbrace{\frac{-ad+bc}{c^2-2d^2}}_{\displaystyle\in\mathbb{Q}} +\sqrt{2} +\in \mathbb{Q}(\!\sqrt{2}). +\qedhere +\end{align*} +\end{beispiel} + + +\subsubsection{Rationalen Funktionen} +Die als Antworten auf die Frage nach einer Stammfunktion akzeptablen +Funktionen sollten alle rationalen Zahlen sowie die unabhängige +Variable $x$ enthalten. +Ausserdem sollte man beliebige arithmetische Operationen mit +diesen Ausdrücken durchführen können. +Mit Addition, Subtraktion und Multiplikation entstehen aus den +rationalen Zahlen und der unabhängigen Variablen die Polynome $\mathbb{Q}[x]$ +(siehe auch Abschnitt~\ref{buch:potenzen:section:polynome}). + + +\begin{definition} +Die Menge +\[ +\mathbb{Q}(x) += +\biggl\{ +\frac{p(x)}{q(x)} +\;\bigg|\; +p(x),q(x)\in\mathbb{Q}[x] +\wedge +q(x)\ne 0 +\biggr\}, +\] +bestehenden aus allen Quotienten von Polynome, deren Nenner nicht +das Nullpolynom ist, heisst der Körper der {\em rationalen Funktionen} +\index{rationale Funktion}% +mit Koeffizienten in $\mathbb{Q}$. +\end{definition} + +Die Definition erlaubt, dass der Nenner Nullstellen hat, die sich in +Polen der Funktion äussern. +Die Eigenschaften eines Körpers sind sicher erfüllt, wenn wir uns +nur davon überzeugen können, +dass die arithmetischen Operationen nicht aus dieser Funktionenmenge +herausführen. +Dazu muss man nur verstehen, dass die Operation des gleichnamig Machens +zweier Brüche auch für Nenner funktioniert, die Polynome sind, und die +Summe wzeier Brüche von Polynomen wieder in einen Bruch von Polynomen +umwandelt. + + -- cgit v1.2.1 From 8453542b493fe8396a406c5a195dc0a4125f638d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 25 May 2022 11:59:00 +0200 Subject: Koerpererweiterungen --- buch/chapters/060-integral/erweiterungen.tex | 128 ++++++++++++++++++++++++++- buch/chapters/060-integral/rational.tex | 1 + 2 files changed, 128 insertions(+), 1 deletion(-) (limited to 'buch') diff --git a/buch/chapters/060-integral/erweiterungen.tex b/buch/chapters/060-integral/erweiterungen.tex index f88f6e3..d5c7c72 100644 --- a/buch/chapters/060-integral/erweiterungen.tex +++ b/buch/chapters/060-integral/erweiterungen.tex @@ -5,8 +5,134 @@ % \subsection{Körpererweiterungen \label{buch:integral:subsection:koerpererweiterungen}} +Das Beispiel des Körpers $\mathbb{Q}(\!\sqrt{2})$ auf Seite +\pageref{buch:integral:beispiel:Qsqrt2} illustriert eine Möglichkeit, +einen kleinen Körper zu vergrössern. +Das Prinzip ist verallgemeinerungsfähig und soll in diesem Abschnitt +erarbeitet werden. + % % algebraische Zahl-Erweiterungen +\subsubsection{Algebraische Erweiterungen} +Der Körper $\mathbb{Q}(\!\sqrt{2})$ entsteht aus dem Körper $\mathbb{Q}$ +dadurch, dass man die Zahl $\sqrt{2}$ hinzufügt und alle erlaubten +arithmetischen Operationen zulässt. +Die Darstellung von Elementen von $\mathbb{Q}(\!\sqrt{2})$ als +$a+b\sqrt{2}$ ist möglich, weil die Zahl $\alpha=\sqrt{2}$ die +algebraische Relation +\[ +\alpha^2-2 = \sqrt{2}^2 -2 = 0 +\] +erfüllt. +Voraussetzung für diese Aussage ist, dass es die Zahl $\sqrt{2}$ in einem +geeigneten grösseren Körper gibt. +Die reellen oder komplexen Zahlen bilden einen solchen Körper. +Wir verallemeinern diese Situation wie folgt. + +\begin{definition} +Ist $K$ ein Körper, dann heisst ein Körper $L$ mit $K\subset L$ ein +{\em Erweiterungskörper} von $K$. +\index{Erweiterungskoerper@Erweiterungskörper} +\end{definition} + +\begin{definition} +\label{buch:integral:definition:algebraisch} +Sei $K\subset L$ eine Körpererweiterung. +Das Element $\alpha\in L$ heisst {\em algebraisch} über $K$, wenn es +ein Polynom $p(x)\in K[x]$ gibt derart, dass $\alpha$ eine Nullstelle +von $p(x)$ ist, also gibt mit $p(\alpha)=0$. +Das normierte Polynom $m(x)$ geringsten Grades, welches $m(\alpha)=0$ +erfüllt, heisst das {\em Minimalpolynom} von $\alpha$. +\index{Minimalpolynom}% +\end{definition} + +Man sagt auch $\alpha$ ist algebraisch vom Grad $n$, wenn das Minimalpolynom +den Grad $n$ hat. +Wenn $\alpha\ne 0$ algebraisch ist, dann ist auch $1/\alpha$ algebraisch, +wie das folgende Argument zeigt. +Für das Minimalpolynom $m(x)$ von $\alpha$, ist $m(\alpha)=0$. +Teilt man diese Gleichung durch $\alpha^n$ teilt, erhält man +\[ +m_0\frac{1}{\alpha^n} ++ +m_1\frac{1}{\alpha^{n-1}} ++ +\ldots ++ +m_{n-1}\frac{1}{\alpha} ++ +1 += +0, +\] +das Polynom +\[ +\hat{m}(x) += +m_0x^n + m_1x^{n-1} + \ldots m_{n-1} x + 1 +\in +K[x] +\] +hat also $\alpha^{-1}$ als Nullstelle. +Das Polynom $\hat{m}(x)$ beweist daher, dass $\alpha^{-1}$ algebraisch ist. + +Die Zahl $\sqrt{2}\in\mathbb{R}$ ist also algebraisch über $\mathbb{Q}$ +und jede andere Quadratwurzel von Elementen von $\mathbb{Q}$ ist +ebenfalls algebraisch über $\mathbb{Q}$. +Auch der Körper $\mathbb{Q}(\alpha)$ kann für jede andere Quadratwurzel +auf die genau gleiche Art wie für $\sqrt{2}$ konstruiert werden. + +\begin{definition} +\label{buch:integral:definition:algebraischeerweiterung} +Sei $K\subset L$ eine Körpererweiterung und $\alpha\in L$ ein algebraisches +Element mit Minimalpolynom $m(x)\in K[x]$. +Dann heisst die Menge +\begin{equation} +K(\alpha) += +\{ +a_0 + a_1\alpha + \ldots +a_n\alpha^n +\;|\; +a_i\in K +\} +\label{buch:integral:eqn:algelement} +\end{equation} +mit $n=\deg m(x) - 1$ der durch Adjunktion von $\alpha$ erhaltene +Erweiterungsköper. +\end{definition} + +Wieder muss nur überprüft werden, dass jedes Produkt oder jeder +Quotient von Ausdrücken der Form~\eqref{buch:integral:eqn:algelement} +wieder in diese Form gebracht werden kann. +Dazu sei +\[ +m(x) += +m_0+m_1x + m_2x^2 ++\ldots +m_{n-1}x^{n-1} + x^n +\] +das Minimalpolynom von $\alpha$. +Die Gleichung $m(\alpha)=0$ kann nach $\alpha^n$ aufgelöst werden und +liefert +\[ +\alpha^n = -m_0 - m_1\alpha - m_2\alpha^2 -\ldots -m_{n-1}\alpha^{n-1}. +\] +Damit kann jede Potenz von $\alpha$ mit einem Exponenten grösser als $n$ +in eine Linearkombination von Potenzen mit kleineren Exponenten +reduziert werden. +Ein Polynom in $\alpha$ kann also immer auf die +Form~\eqref{buch:integral:eqn:algelement} +gebracht werden. + +XXX Quotienten + % rationale Funktionen als Körpererweiterungen +\subsubsection{Rationale Funktionen als Körpererweiterung} + % Erweiterungen mit algebraischen Funktionen -% +\subsubsection{Algebraische Funktionen} + +% Transzendente Körpererweiterungen +\subsubsection{Transzendente Erweiterungen} + + diff --git a/buch/chapters/060-integral/rational.tex b/buch/chapters/060-integral/rational.tex index 989e65b..9cef3a7 100644 --- a/buch/chapters/060-integral/rational.tex +++ b/buch/chapters/060-integral/rational.tex @@ -92,6 +92,7 @@ Es lassen sich allerdings auch Zahlkörper zwischen $\mathbb{Q}$ und $\mathbb{R}$ konstruieren, wie das folgende Beispiel zeigt. \begin{beispiel} +\label{buch:integral:beispiel:Qsqrt2} Die Menge \[ \mathbb{Q}(\!\sqrt{2}) -- cgit v1.2.1 From 6e9f45ad084ca9341c2893bdfe1ddcb07ee8a45b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 25 May 2022 12:07:43 +0200 Subject: typo --- buch/chapters/060-integral/rational.tex | 6 +++--- buch/chapters/060-integral/sqrat.tex | 2 +- 2 files changed, 4 insertions(+), 4 deletions(-) (limited to 'buch') diff --git a/buch/chapters/060-integral/rational.tex b/buch/chapters/060-integral/rational.tex index 9cef3a7..4cd7d7f 100644 --- a/buch/chapters/060-integral/rational.tex +++ b/buch/chapters/060-integral/rational.tex @@ -61,17 +61,17 @@ der Eigenschaft $a\cdot 1 =a$ für alle $a\in K$ \index{inverses Element}% $a\in K^*=K\setminus\{0\}$ gibt es das Element $a^{-1}$ mit der Eigenschaft $a\cdot a^{-1}=1$. -Die Menge $K^*$ heisst auch die {\em Einheitengruppe} oder die -{\em Gruppe der invertierbaren Elemente} des Körpers. \index{Einheitengruppe}% \index{Gruppe der invertierbaren Elemente}% \end{enumerate} \end{tabular} \end{center} -\vspace{-10pt} +\vspace{-22pt} Ausserdem gilt das Distributivgesetz: für alle $a,b,c\in K$ gilt $a\cdot(b+c)=a\cdot b + a\cdot c$. \index{Disitributivgesetz}% +Die Menge $K^*$ heisst auch die {\em Einheitengruppe} oder die +{\em Gruppe der invertierbaren Elemente} des Körpers. \end{definition} Das Assoziativgesetz {\bf A}.1 besagt, dass Summen mit beliebig diff --git a/buch/chapters/060-integral/sqrat.tex b/buch/chapters/060-integral/sqrat.tex index 20f1ef7..f6838e5 100644 --- a/buch/chapters/060-integral/sqrat.tex +++ b/buch/chapters/060-integral/sqrat.tex @@ -332,7 +332,7 @@ Letzteres wird im nächsten Abschnitt berechnet. % \subsubsection{Das Integral von $1/y$} Eine Stammfunktion von $1/y$ kann mit etwas Geschick bekannten -Interationstechnikgen gefunden werden. +Interationstechniken gefunden werden. Durch Ableitung der Funktion \[ F -- cgit v1.2.1 From 4197abc20216c15f11660d63549eb8b765f1c892 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 25 May 2022 12:08:44 +0200 Subject: typos --- buch/chapters/060-integral/sqrat.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) (limited to 'buch') diff --git a/buch/chapters/060-integral/sqrat.tex b/buch/chapters/060-integral/sqrat.tex index f6838e5..ceb8650 100644 --- a/buch/chapters/060-integral/sqrat.tex +++ b/buch/chapters/060-integral/sqrat.tex @@ -337,7 +337,7 @@ Durch Ableitung der Funktion \[ F = -\frac{1}{\sqrt{a}}\log\biggl(x+\frac{b}{2a}+\frac{y}{\sqrt{y}}\biggr) +\frac{1}{\sqrt{a}}\log\biggl(x+\frac{b}{2a}+\frac{y}{\sqrt{a}}\biggr) \] kann man nachprüfen, dass $F$ eine Stammfunktion von $1/y$ ist, also @@ -345,7 +345,7 @@ also \int \frac{1}{y} = -\frac{1}{\sqrt{a}}\log\biggl(x+\frac{b}{2a}+\frac{y}{\sqrt{y}}\biggr). +\frac{1}{\sqrt{a}}\log\biggl(x+\frac{b}{2a}+\frac{y}{\sqrt{a}}\biggr). \end{equation} % @@ -458,7 +458,7 @@ Form = v_0 + C -\log\biggl(x+\frac{b}{2a}+\frac{y}{\sqrt{y}}\biggr) +\log\biggl(x+\frac{b}{2a}+\frac{y}{\sqrt{a}}\biggr) + \sum_{i=1}^n c_i \log v_i, -- cgit v1.2.1 From 03881a82e1a30cfaea1709f4f3f50c5cd9dfd0ea Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 25 May 2022 17:40:27 +0200 Subject: algebraische Erweiterungen --- buch/chapters/060-integral/erweiterungen.tex | 109 ++++++++++++++++++++++++++- buch/chapters/060-integral/rational.tex | 2 +- 2 files changed, 106 insertions(+), 5 deletions(-) (limited to 'buch') diff --git a/buch/chapters/060-integral/erweiterungen.tex b/buch/chapters/060-integral/erweiterungen.tex index d5c7c72..7039cc0 100644 --- a/buch/chapters/060-integral/erweiterungen.tex +++ b/buch/chapters/060-integral/erweiterungen.tex @@ -27,7 +27,7 @@ erfüllt. Voraussetzung für diese Aussage ist, dass es die Zahl $\sqrt{2}$ in einem geeigneten grösseren Körper gibt. Die reellen oder komplexen Zahlen bilden einen solchen Körper. -Wir verallemeinern diese Situation wie folgt. +Wir verallgemeinern diese Situation wie folgt. \begin{definition} Ist $K$ ein Körper, dann heisst ein Körper $L$ mit $K\subset L$ ein @@ -124,15 +124,116 @@ Ein Polynom in $\alpha$ kann also immer auf die Form~\eqref{buch:integral:eqn:algelement} gebracht werden. -XXX Quotienten +Es muss aber noch gezeigt werden, dass auch der Kehrwert eines Elements +der Form~\eqref{buch:integral:eqn:algelement} in dieser Form geschrieben +werden kann. +Sei also $a(\alpha)$ so ein Element, dann sind die beiden Polynome +$a(x)$ und $m(x)$ teilerfremd, der grösste gemeinsame Teiler ist $1$. +Mit dem erweiterten euklidischen Algorithmus kann man zwei Polynome +$s(x)$ und $t(x)$ finden derart, dass $s(x)a(x)+t(x)m(x)=1$. +Setzt man $\alpha$ für $x$ ein, verschwindet das Minimalpolynom und +es bleibt +\[ +s(\alpha)a(\alpha) = 1 +\qquad\Rightarrow\qquad +s(\alpha) = \frac{1}{a(\alpha)}. +\] +Damit ist $s(\alpha)$ eine Darstellung von $1/a(\alpha)$ in der +Form~\eqref{buch:integral:eqn:algelement}. + +% Transzendente Körpererweiterungen +\subsubsection{Transzendente Erweiterungen} +Nicht alle Zahlen in $\mathbb{R}$ sind algebraisch. +Lindemann bewies 1882 einen allgemeinen Satz, aus dem folgt, +dass $\pi$ und $e$ nicht algebraisch sind, es gibt also +kein Polynom mit rationalen Koeffizienten, welches $\pi$ +oder $e$ als Nullstelle hat. + +\begin{definition} +Eine Zahl $\alpha\in L$ in einer Körpererweiterung $K\subset L$ +heisst {\em transzendent}, wenn $\alpha$ nicht algebraisch ist, +wenn es also kein Polynom in $K[x]$ gibt, welches $\alpha$ als +Nullstelle hat. +\end{definition} + +Die Zahlen $\pi$ und $e$ sind also transzendent. +Eine andere Art, diese Eigenschaft zu beschreiben ist zu sagen, +dass die Potenzen +\[ +1=\pi^0, \pi, \pi^2,\pi^3,\dots +\] +linear unabhängig sind. +Gäbe es nämlich eine lineare Abhängigkeit, dann gäbe es Koeffizienten +$l_i$ derart, dass +\[ +l_0 + l_1\pi^1 + l_2\pi^2 + \ldots + l_{n-1}\pi^{n-1} + l_{n}\pi^n = l(\pi)=0, +\] +und damit wäre dann ein Polynom gefunden, welches $\pi$ als Nullstelle hat. + +Selbstverstländlich kann man zu einem transzendenten Element $\alpha$ +immer noch einen Körper konstruieren, der alle Zahlen enthält, welche man +mit den arithmetischen Operationen aus $\alpha$ bilden kann. +Man kann ihn schreiben als +\[ +K(\alpha) += +\biggl\{ +\frac{p(\alpha)}{q(\alpha)} +\;\bigg|\; +p(x),q(x)\in K[x] \wedge p(x)\ne 0 +\biggr\}, +\] +aber die Vereinfachungen zur +Form~\eqref{buch:integral:eqn:algelement}, die bei einem algebraischen +Element $\alpha$ möglich waren, können jetzt nicht mehr durchgeführt +werden. +$K\subset K(\alpha)$ ist zwar immer noch eine Körpererweiterung, aber +$K(\alpha)$ ist nicht mehr ein endlichdimensionaler Vektorraum. +Die Körpererweiterung $K\subset K(\alpha)$ heisst {\em transzendent}. % rationale Funktionen als Körpererweiterungen \subsubsection{Rationale Funktionen als Körpererweiterung} +Die unabhängige Variable wird bei Rechnen so behandelt, dass die +Potenzen alle linear unabhängig sind. +Dies ist die Grundlage für den Koeffizientenvergleich. +Der Körper der rationalen Funktion $K(x)$ +ist also eine transzendente Körpererweiterung von $K$. % Erweiterungen mit algebraischen Funktionen \subsubsection{Algebraische Funktionen} +Für das Integrationsproblem möchten wir nicht nur rationale Funktionen +verwenden können, sondern auch Wurzelfunktionen. +Wir möchten also zum Beispiel auch mit der Funktion $\sqrt{ax^2+bx+c}$ +und allem, was man mit arithmetischen Operationen daraus machen kann, +arbeiten können. +Eine Körpererweiterung, die $\sqrt{ax^2+bx+c}$ enthält, enthält auch +alles, was man daraus bilden kann. +Doch wie bekommen wir die Funktion $\sqrt{ax^2+bx+c}$ in den Körper? -% Transzendente Körpererweiterungen -\subsubsection{Transzendente Erweiterungen} +Die Art und Weise, wie man Wurzeln in der Schule kennenlernt ist als +eine neue Operation, die zu einer Zahl die Quadratwurzel liefert. +Diese Idee, den Körper mit einer weiteren Funktion anzureichern, +führt über nicht auf eine nützliche neue algebraische Struktur. +Wir dürfen daher $\sqrt{ax^2+bx+c}$ nicht als die Zusammensetzung +einer einzelnen neuen Funktion $\sqrt{\phantom{A}}$ mit +einem Polynom betrachten. + +Die Wurzel $\sqrt{ax^2+bx+c}$ ist aber auch die Nullstelle des Polynoms +\[ +p(z) += +z^2 - [ax^2+bx+c] +\in +K(x)[z] +\] +mit Koeffizienten in $K(x)$. +Die eckigen Klammern sollen helfen, die Koeffizienten in $K(x)$ +zu erkennen. +Die Funktion $\sqrt{ax^2+bx+c}$ ist also algebraisch über $K(x)$. +Einen Funktionenkörper, der die Funktion enthält, kann man also erhalten, +indem man den Körper $K(x)$ um das über $K(x)$ algebraische Element +$y=\sqrt{ax^2+bx+c}$ zu $K(x,y)=K(x,\sqrt{ax^2+bx+c}$ erweitert. +Wurzelfunktion werden daher nicht als Zusammensetzungen, sondern als +algebraische Erweiterungen eines Funktionenkörpers betrachtet. diff --git a/buch/chapters/060-integral/rational.tex b/buch/chapters/060-integral/rational.tex index 4cd7d7f..ae64c34 100644 --- a/buch/chapters/060-integral/rational.tex +++ b/buch/chapters/060-integral/rational.tex @@ -157,7 +157,7 @@ p(x),q(x)\in\mathbb{Q}[x] q(x)\ne 0 \biggr\}, \] -bestehenden aus allen Quotienten von Polynome, deren Nenner nicht +bestehend aus allen Quotienten von Polynome, deren Nenner nicht das Nullpolynom ist, heisst der Körper der {\em rationalen Funktionen} \index{rationale Funktion}% mit Koeffizienten in $\mathbb{Q}$. -- cgit v1.2.1 From 9a90404d081513254925c76b2fbaabb1a1c62982 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 25 May 2022 20:15:57 +0200 Subject: differenialkoerper --- buch/chapters/060-integral/diffke.tex | 33 ++++++++++++++++++++++++++++++++- 1 file changed, 32 insertions(+), 1 deletion(-) (limited to 'buch') diff --git a/buch/chapters/060-integral/diffke.tex b/buch/chapters/060-integral/diffke.tex index 53b46ad..a943fa3 100644 --- a/buch/chapters/060-integral/diffke.tex +++ b/buch/chapters/060-integral/diffke.tex @@ -5,16 +5,47 @@ % \subsection{Differentialkörper und ihre Erweiterungen \label{buch:integral:subsection:diffke}} +Die Ableitung wird in den Grundvorlesungen der Analysis jeweils +als ein Grenzprozess eingeführt. +Die praktische Berechnung von Ableitungen verwendet aber praktisch +nie diese Definition, sondern fast ausschliesslich die rein algebraischen +Ableitungsregeln. +So wie die Wurzelfunktionen im letzten Abschnitt als algebraische +Körpererweiterungen erkannt wurden, muss jetzt auch für die Ableitung +eine rein algebraische Definition gefunden werden. +Die entstehende Struktur ist der Differentialkörper, der in diesem +Abschnitt definiert werden soll. + +% +% Derivation % \subsubsection{Derivation} -% Ableitungsaxiome +\begin{definition} +Sei $\mathscr{F}$ ein Funktionenkörper. +Eine {\em Derivation} ist eine lineare Abbildung +$D\colon \mathscr{F}\to\mathscr{F}$ +mit der Eigenschaft +\[ +D(fg) = (Df)g+f(Dg). +\] +\end{definition} + +% +% Ableitungsregeln +% \subsubsection{Ableitungsregeln} % Ableitungsregeln +% +% Konstantenkörper +% \subsubsection{Konstantenkörper} % Konstantenkörper +% +% Logarithmus und Exponantialfunktion +% \subsubsection{Logarithmus und Exponentialfunktion} % Logarithmus und Exponentialfunktion -- cgit v1.2.1 From f24e5bd9fda39e2f8bbfb0946aac2ee7dcda547d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 26 May 2022 08:35:55 +0200 Subject: new stuff --- buch/chapters/060-integral/diffke.tex | 96 +++++++++++++++++++- buch/chapters/060-integral/elementar.tex | 8 ++ buch/chapters/060-integral/erweiterungen.tex | 10 +++ buch/chapters/060-integral/logexp.tex | 127 +++++++++++++++++++++++++-- buch/chapters/060-integral/rational.tex | 2 +- buch/chapters/060-integral/sqrat.tex | 7 +- 6 files changed, 237 insertions(+), 13 deletions(-) (limited to 'buch') diff --git a/buch/chapters/060-integral/diffke.tex b/buch/chapters/060-integral/diffke.tex index a943fa3..02e90f6 100644 --- a/buch/chapters/060-integral/diffke.tex +++ b/buch/chapters/060-integral/diffke.tex @@ -20,32 +20,120 @@ Abschnitt definiert werden soll. % Derivation % \subsubsection{Derivation} +Für die praktische Berechnung der Ableitung einer Funktion verwendet +man in erster Linie die bekannten Rechenregeln. +Dazu gehören für zwei Funktionen $f$ und $g$ +\begin{itemize} +\item Linearität: $(\alpha f+\beta g)' = \alpha f' + \beta g'$ für +Konstanten $\alpha$, $\beta$. +\item Produktregel: $(fg)'=f'g+fg'$. +\index{Produktregel}% +\item Quotientenregel: $(f/g)' = (f'g-fg')/g^2$. +\index{Quotientenregel}% +\end{itemize} +Die ebenfalls häufig verwendete Kettenregel $(f\circ g)' = (f'\circ g) g'$ +\index{Kettenregel}% +für zusammengesetzte Funktionen wird später kaum benötigt, da wir +Verkettungen durch Körpererweiterungen ersetzen wollen. +Die Ableitung hat somit die rein algebraischen Eigenschaften +einer Derivation gemäss folgender Definition. \begin{definition} -Sei $\mathscr{F}$ ein Funktionenkörper. +Sei $\mathscr{F}$ ein Körper. Eine {\em Derivation} ist eine lineare Abbildung +\index{Derivation}% $D\colon \mathscr{F}\to\mathscr{F}$ mit der Eigenschaft \[ D(fg) = (Df)g+f(Dg). \] +Ein {\em Differentialkörper} ist ein Körper mit einer Derivation. +\index{Differentialkoerper@Differentialkörper}% \end{definition} +Die Ableitung in einem Funktionenkörper ist eine Derivation, +die sich zusätzlich dadurch auszeichnet, dass $Dx=x'=1$. +Sie wird weiterhin mit dem Strich bezeichnet. + % % Ableitungsregeln % \subsubsection{Ableitungsregeln} -% Ableitungsregeln +Die Definition einer Derivation macht keine Aussagen über Quotienten, +diese kann man aber aus den Eigenschaften einer Derivation sofort +ableiten. +Wir schreiben $q=f/g$ für $f,g\in\mathscr{F}$, dann ist $f=qg$. +Nach der Kettenregel gilt +\( +f'=q'g+qg' +\). +Substituiert man darin $q=f/g$ und löst nach $q'$ auf, erhält man +\[ +f'=q'g+\frac{fg'}{g} +\qquad\Rightarrow\qquad +q'=\frac1{g}\biggl(f'-\frac{fg'}{g}\biggr) += +\frac{f'g-fg'}{g^2}. +\] + % % Konstantenkörper % \subsubsection{Konstantenkörper} -% Konstantenkörper +Die Ableitung einer Konstanten verschwindet. +Beim Hinzufügen von Funktionen zu einem Funktionenkörper können weitere +Konstanten hinzukommen, ohne dass dies auf den ersten Blick sichtbar wird. +Zum Beispiel enthält $\mathbb{Q}(x,\!\sqrt{x+\pi})$ wegen +$(\!\sqrt{x+\pi})^2-x=\pi$ auch die Konstante $\pi$. +Eine Derivation ermöglicht dank des nachfolgenden Satzes auch, +solche Konstanten zu erkennen. + +\begin{satz} +Sei $\mathscr{F}$ ein Körper und $D$ eine Derivation in $\mathscr{F}$. +Dann ist die Menge $C=\{a\in\mathscr{F}\;|\;Da=0\}$ ein Körper. +\end{satz} + +\begin{proof}[Beweis] +Es muss gezeigt werden, dass Summe und Produkt von Element von $C$ +wieder in $C$ liegen. +Wenn $Da=Db=0$, dann ist $D(a+b)=Da+Db=0$, also ist $a+b\in C$. +Für das Produkt gilt $D(ab)=(Da)b+a(Db)=0b+a0=0$, also ist auch +$ab\in C$. +\end{proof} + +Die Menge $C$ heisst der {\em Konstantenkörper} von $\mathscr{F}$. +\index{Konstantenkörper}% % % Logarithmus und Exponantialfunktion % \subsubsection{Logarithmus und Exponentialfunktion} -% Logarithmus und Exponentialfunktion +Die Exponentialfunktion und der Logarithmus sind nicht algebraisch +über $\mathbb{Q}(x)$, sie lassen sich nicht durch eine algebraische +Gleichung charakterisieren. +Sie zeichnen sich aber durch besondere Ableitungseigenschaften aus. +Die Theorie der gewöhnlichen Differentialgleichungen garantiert, +dass eine Funktion durch eine Differentialgleichung und Anfangsbedingungen +festgelegt ist. +Für die Exponentialfunktion und der Logarithmus haben die +Ableitungseigenschaften +\[ +\exp'(x) = \exp(x) +\qquad\text{und}\qquad +x \log'(x) = 1. +\] +\index{Exponentialfunktion}% +\index{Logarithmus}% +In der algebraischen Beschreibung eines Funktionenkörpers gibt es +das Konzept des Wertes einer Funktion an einer bestimmten Stelle nicht. +Somit können keine Anfangsbedingungen vorgegeben werden. +Da die Gleichungen linear sind, sind Vielfache einer Lösung wieder +Lösungen. +Insbesondere ist mit $\exp(x)$ auch $a\exp(x)$ eine Lösung und mit +$\log(x)$ auch $a\log(x)$ für alle Konstanten $a$. + +Die Eigenschaft, dass die Exponentialfunktion die Umkehrfunktion +des Logarithmus ist, lässt sich mit den Mitteln eines Differentialkörpers +nicht ausdrücken. diff --git a/buch/chapters/060-integral/elementar.tex b/buch/chapters/060-integral/elementar.tex index 2962178..854a875 100644 --- a/buch/chapters/060-integral/elementar.tex +++ b/buch/chapters/060-integral/elementar.tex @@ -5,3 +5,11 @@ % \subsection{Elementare Funktionen \label{buch:integral:subsection:elementar}} +Etwas allgemeiner kann man sagen, dass in den +Beispielen~\eqref{buch:integration:risch:eqn:integralbeispiel2} +algebraische Erweiterungen von $\mathbb{Q}(x)$ und Erweiterungen +um Logarithmen oder Exponentialfunktionen vorgekommen sind. +Die Stammfunktionen verwenden dieselben Funktionen oder höchstens +Erweiterungen um Logarithmen von Funktionen, die man schon im +Integranden gesehen hat. + diff --git a/buch/chapters/060-integral/erweiterungen.tex b/buch/chapters/060-integral/erweiterungen.tex index 7039cc0..a999ebb 100644 --- a/buch/chapters/060-integral/erweiterungen.tex +++ b/buch/chapters/060-integral/erweiterungen.tex @@ -141,6 +141,16 @@ s(\alpha) = \frac{1}{a(\alpha)}. Damit ist $s(\alpha)$ eine Darstellung von $1/a(\alpha)$ in der Form~\eqref{buch:integral:eqn:algelement}. +% +% Komplexe Zahlen +% +\subsubsection{Komplexe Zahlen} +Die imaginäre Einheit $i$ hat die Eigenschaft, dass $i^2=-1$, insbesondere +ist sie Nullstelle des Polynoms $m(x)=x^2+1\in\mathbb{Q}[x]$. +Die Menge $\mathbb{Q}(i)$ ist daher eine algebraische Körpererweiterung +von $\mathbb{Q}$ bestehend aus den komplexen Zahlen mit rationalem +Real- und Imaginärteil. + % Transzendente Körpererweiterungen \subsubsection{Transzendente Erweiterungen} Nicht alle Zahlen in $\mathbb{R}$ sind algebraisch. diff --git a/buch/chapters/060-integral/logexp.tex b/buch/chapters/060-integral/logexp.tex index 7cbb906..2bfe0e1 100644 --- a/buch/chapters/060-integral/logexp.tex +++ b/buch/chapters/060-integral/logexp.tex @@ -13,15 +13,132 @@ $\log(x-\alpha)$ hinzuzufügen. Es können jedoch noch ganz andere neue Funktionen auftreten, wie die folgende Zusammenstellung einiger Stammfunktionen zeigt: -\begin{align*} +\begin{equation} +\begin{aligned} \int\frac{dx}{1+x^2} &= -\arctan x +\arctan x, \\ -\end{align*} - - +\int \cos x\,dx +&= +\sin x, +\\ +\int\frac{dx}{\sqrt{1-x^2}} +&= +\arcsin x, +\\ +\int +\operatorname{arcosh} x\,dx +&= +x \operatorname{arcosh} x - \sqrt{x^2-1}. +\end{aligned} +\label{buch:integration:risch:allgform} +\end{equation} +In der Stammfunktion treten Funktionen auf, die auf den ersten +Blick nichts mit den Funktionen im Integranden zu tun haben. +Die trigonometrischen und hyperbolichen Funktionen +in~\eqref{buch:integration:risch:allgform} +lassen sich alle durch Exponentialfunktionen ausdrücken. +So gilt +\begin{equation} +\begin{aligned} +\sin x &= \frac{1}{2i}\bigl( e^{ix} - e^{-ix}\bigr), +& +&\qquad& +\cos x &= \frac{1}{2}\bigl( e^{ix} + e^{-ix}\bigr), +\\ +\sinh x &= \frac12\bigl( e^x - e^{-x} \bigr), +& +&\qquad& +\cosh x &= \frac12\bigl( e^x + e^{-x} \bigr). +\end{aligned} +\label{buch:integral:risch:trighypinv} +\end{equation} +Nach Multiplikation mit $e^{ix}$ bzw.~$e^{x}$ entsteht eine +quadratische Gleichung in $e^{ix}$ bzw.~$e^{x}$. +Die Lösungsformel für quadratische Gleichungen erlaubt daher, $e^{ix}$ +bzw.~$e^{x}$ zu finden und damit auch die Umkehrfunktionen. +Die Rechnung ergibt +\begin{equation} +\begin{aligned} +\arcsin y +&= +\frac{1}{i}\log\bigl( +iy\pm\sqrt{1-y^2} +\bigr) +& +&\qquad& +\arccos y +&= +\log\bigl( +y\pm \sqrt{y^2-1} +\bigr) +\\ +\operatorname{arsinh}y +&= +\log\bigl( +y \pm \sqrt{1+y^2} +\bigr) +& +&\qquad& +\operatorname{arcosh} y +&= +\log\bigl( +y\pm \sqrt{y^2-1} +\bigr) +\end{aligned} +\label{buch:integral:risch:trighypinv} +\end{equation} +Alle Funktionen, die man aus dem elementaren Analysisunterricht +kennt, können also mit Hilfe von Exponentialfunktionen und Logarithmen +geschrieben werden. +Man nennt dies die $\log$-$\exp$-Notation der trigonometrischen +und hyperbolischen Funktionen. +\index{logexpnotation@$\log$-$\exp$-Notation}% +Wendet man die Substitutionen +\eqref{buch:integral:risch:trighyp} +und +\eqref{buch:integral:risch:trighypinv} +auf die Integrale +\eqref{buch:integration:risch:allgform} +an, entstehen die Beziehungen +\begin{equation} +\begin{aligned} +\int\frac{1}{1+x^2} +&= +\frac12i\bigl( +\log(1-ix) - \log(1+ix) +\bigr) +\\ +\int\bigl( +{\textstyle\frac12} +e^{ix} ++ +{\textstyle\frac12} +e^{-ix} +\bigr) +&= +-{\textstyle\frac12}ie^{ix} ++{\textstyle\frac12}ie^{-ix} +\\ +\int +\frac{1}{\sqrt{1-x^2}} +&= +-i\log\bigl(ix+\sqrt{1-x^2}) +\\ +\int \log\bigl(x+\sqrt{x^2-1}\bigr) +&= +x\log\bigl(x+\sqrt{x^2-1}\bigr) - \sqrt{x^2-1}. +\end{aligned} +\label{buch:integration:risch:eqn:integralbeispiel2} +\end{equation} +Die in den Stammfuntionen auftretenden Funktionen treten entweder +schon im Integranden auf oder sind Logarithmen von solchen +Funktionen. +Zum Beispiel hat der Nenner im ersten Integral die Faktorisierung +$1+x^2=(1+ix)(1-ix)$, in der Stammfunktion findet man die Logarithmen +der Faktoren. diff --git a/buch/chapters/060-integral/rational.tex b/buch/chapters/060-integral/rational.tex index ae64c34..7b24e9f 100644 --- a/buch/chapters/060-integral/rational.tex +++ b/buch/chapters/060-integral/rational.tex @@ -157,7 +157,7 @@ p(x),q(x)\in\mathbb{Q}[x] q(x)\ne 0 \biggr\}, \] -bestehend aus allen Quotienten von Polynome, deren Nenner nicht +bestehend aus allen Quotienten von Polynomen, deren Nenner nicht das Nullpolynom ist, heisst der Körper der {\em rationalen Funktionen} \index{rationale Funktion}% mit Koeffizienten in $\mathbb{Q}$. diff --git a/buch/chapters/060-integral/sqrat.tex b/buch/chapters/060-integral/sqrat.tex index ceb8650..787cfc9 100644 --- a/buch/chapters/060-integral/sqrat.tex +++ b/buch/chapters/060-integral/sqrat.tex @@ -331,8 +331,9 @@ Letzteres wird im nächsten Abschnitt berechnet. % Das Integral von $1/y$ % \subsubsection{Das Integral von $1/y$} -Eine Stammfunktion von $1/y$ kann mit etwas Geschick bekannten -Interationstechniken gefunden werden. +Eine Stammfunktion von $1/y$ kann mit etwas Geschick mit den +Interationstechniken gefunden werden, die man in einem Analysis-Kurs +lernt. Durch Ableitung der Funktion \[ F @@ -471,7 +472,7 @@ die bei der Berechnung der Integrale \eqref{buch:integral:sqrat:eqn:2teart} auftreten. Insbesondere liefert die Rechnung eine Körpererweiterung von $\mathcal{K}(x,y)$ um die logarithmische Funktionen -$\log(x+b/2a+y/\sqrt{y})$ und $\log v_i$, in der $R(x,y)$ eine +$\log(x+b/2a+y/\!\sqrt{y})$ und $\log v_i$, in der $R(x,y)$ eine Stammfunktion hat. -- cgit v1.2.1 From 50ecdafd467b0ec21be5b3bffce1d4c5acbb4fe6 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 26 May 2022 08:36:59 +0200 Subject: add missing file --- buch/SeminarSpezielleFunktionen.tex | 7 +++++++ 1 file changed, 7 insertions(+) create mode 100644 buch/SeminarSpezielleFunktionen.tex (limited to 'buch') diff --git a/buch/SeminarSpezielleFunktionen.tex b/buch/SeminarSpezielleFunktionen.tex new file mode 100644 index 0000000..4ee1900 --- /dev/null +++ b/buch/SeminarSpezielleFunktionen.tex @@ -0,0 +1,7 @@ +% +% buch.tex -- Buch zum mathematischen Seminar Spezielle Funktionen +% +% (c) 2022 Prof. Dr. Andreas Mueller, OST Ostschweizer Fachhochschule +% +\def\IncludeBookCover{1} +\input{common/content.tex} -- cgit v1.2.1 From 14b48dfeb636fe25b0745a2ab617cc5d307c06e6 Mon Sep 17 00:00:00 2001 From: runterer Date: Thu, 26 May 2022 20:38:30 +0200 Subject: =?UTF-8?q?tikz=20und=20eulerprodukt=20hinzugef=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- buch/papers/zeta/analytic_continuation.tex | 23 ++++++- buch/papers/zeta/continuation_overview.tikz.tex | 7 +- buch/papers/zeta/euler_product.tex | 85 +++++++++++++++++++++++++ buch/papers/zeta/main.tex | 1 + buch/papers/zeta/zeta_gamma.tex | 7 +- 5 files changed, 114 insertions(+), 9 deletions(-) create mode 100644 buch/papers/zeta/euler_product.tex (limited to 'buch') diff --git a/buch/papers/zeta/analytic_continuation.tex b/buch/papers/zeta/analytic_continuation.tex index 5e09e42..408a1f7 100644 --- a/buch/papers/zeta/analytic_continuation.tex +++ b/buch/papers/zeta/analytic_continuation.tex @@ -1,7 +1,26 @@ \section{Analytische Fortsetzung} \label{zeta:section:analytische_fortsetzung} \rhead{Analytische Fortsetzung} -%TODO missing Text +Die analytische Fortsetzung der Riemannschen Zetafunktion ist äusserst interessant. +Sie ermöglicht die Berechnung von $\zeta(-1)$ und weiterer spannender Werte. +So liegen zum Beispiel unendlich viele Nullstellen der Zetafunktion bei $\Re(s) = 0.5$. +Diese sind relevant für die Primzahlverteilung und sind Gegenstand der Riemannschen Vermutung. + +Es werden zwei verschiedene Fortsetzungen benötigt. +Die erste erweitert die Zetafunktion auf $\Re(s) > 0$. +Die zweite verwendet eine Spiegelung an der $\Re(s) = 0.5$ Linie und erschliesst damit die ganze komplexe Ebene. +Eine grafische Darstellung dieses Plans ist in Abbildung \ref{zeta:fig:continuation_overview} zu sehen. +\begin{figure} + \centering + \input{papers/zeta/continuation_overview.tikz.tex} + \caption{ + Die verschiedenen Abschnitte der Riemannschen Zetafunktion. + Die originale Definition von \eqref{zeta:equation1} ist im grünen Bereich gültig. + Für den blauen Bereich gilt \eqref{zeta:equation:fortsetzung1}. + Um den roten Bereich zu bekommen verwendet die Funktionalgleichung \eqref{zeta:equation:functional} eine Spiegelung an $\Re(s) = 0.5$. + } + \label{zeta:fig:continuation_overview} +\end{figure} \subsection{Fortsetzung auf $\Re(s) > 0$} \label{zeta:subsection:auf_bereich_ge_0} Zuerst definieren die Dirichletsche Etafunktion als @@ -42,7 +61,7 @@ Durch Subtraktion der beiden Gleichungen \eqref{zeta:align1} minus \eqref{zeta:a &= \eta(s). \end{align} Dies ist die Fortsetzung auf den noch unbekannten Bereich $0 < \Re(s) < 1$ -\begin{equation} +\begin{equation} \label{zeta:equation:fortsetzung1} \zeta(s) := \left(1 - \frac{1}{2^{s-1}} \right)^{-1} \eta(s). diff --git a/buch/papers/zeta/continuation_overview.tikz.tex b/buch/papers/zeta/continuation_overview.tikz.tex index 03224ff..836ab1d 100644 --- a/buch/papers/zeta/continuation_overview.tikz.tex +++ b/buch/papers/zeta/continuation_overview.tikz.tex @@ -1,12 +1,13 @@ \begin{tikzpicture}[>=stealth', auto, node distance=0.9cm, scale=2, dot/.style={fill, circle, inner sep=0, minimum size=0.1cm}] - \draw[->] (-2,0) -- (-1,0) node[dot]{} node[anchor=north]{$-1$} -- (0,0) node[anchor=north west]{$0$} -- (1,0) node[anchor=north west]{$1$} -- (2,0) node[anchor=west]{Re$(s)$}; + \draw[->] (-2,0) -- (-1,0) node[dot]{} node[anchor=north]{$-1$} -- (0,0) node[anchor=north west]{$0$} -- (0.5,0) node[anchor=north west]{$0.5$}-- (1,0) node[anchor=north west]{$1$} -- (2,0) node[anchor=west]{$\Re(s)$}; - \draw[->] (0,-1.2) -- (0,1.2) node[anchor=south]{Im$(s)$}; + \draw[->] (0,-1.2) -- (0,1.2) node[anchor=south]{$\Im(s)$}; \begin{scope}[yscale=0.1] \draw[] (1,-1) -- (1,1); \end{scope} + \draw[dotted] (0.5,-1) -- (0.5,1); \begin{scope}[] \fill[opacity=0.2, red] (-1.8,1) rectangle (0, -1); @@ -14,4 +15,4 @@ \fill[opacity=0.2, green] (1,1) rectangle (1.8, -1); \end{scope} -\end{tikzpicture} \ No newline at end of file +\end{tikzpicture} diff --git a/buch/papers/zeta/euler_product.tex b/buch/papers/zeta/euler_product.tex new file mode 100644 index 0000000..a6ed512 --- /dev/null +++ b/buch/papers/zeta/euler_product.tex @@ -0,0 +1,85 @@ +\section{Eulerprodukt} \label{zeta:section:eulerprodukt} +\rhead{Eulerprodukt} + +Das Eulerprodukt stellt die Verbindung der Zetafunktion und der Primzahlen her. +Diese Verbindung ist sehr wichtig, da durch sie eine Aussage zur Primzahlverteilung gemacht werden kann. +Die Verteilung der Primzahlen ist Gegenstand der Riemannschen Vermutung, welche eines der grössten ungelösten Probleme der Mathematik ist. + +\begin{satz} + Für alle Zahlen $s$ mit $\Re(s) > 1$ ist die Zetafunktion identisch mit dem unendlichen Eulerprodukt + \begin{equation}\label{zeta:eq:eulerprodukt} + \zeta(s) + = + \sum_{n=1}^\infty + \frac{1}{n^s} + = + \prod_{p \in P} + \frac{1}{1-p^{-s}} + \end{equation} + wobei $P$ die Menge aller Primzahlen darstellt. +\end{satz} + +\begin{proof}[Beweis] + Der Beweis startet mit dem Eulerprodukt und stellt dieses so um, dass die Zetafunktion erscheint. + Als erstes ersetzen wir die Faktoren durch geometrische Reihen + \begin{equation} + \prod_{i=1}^{\infty} + \frac{1}{1-p^{-s}} + = + \prod_{p \in P} + \sum_{k_i=0}^{\infty} + \left( + \frac{1}{p_i^s} + \right)^{k_i} + = + \prod_{p \in P} + \sum_{k_i=0}^{\infty} + \frac{1}{p_i^{s k_i}}, + \end{equation} + dabei iteriert der Index $i$ über alle Primzahlen $p_i$. + Durch Ausschreiben der Multiplikation und Ausklammern der Summen erhalten wir + \begin{align} + \prod_{p \in P} + \sum_{k_i=0}^{\infty} + \frac{1}{p_i^{s k_i}} + &= + \sum_{k_1=0}^{\infty} + \frac{1}{p_1^{s k_1}} + \sum_{k_2=0}^{\infty} + \frac{1}{p_2^{s k_2}} + \ldots + \nonumber \\ + &= + \sum_{k_1=0}^{\infty} + \sum_{k_2=0}^{\infty} + \ldots + \left( + \frac{1}{p_1^{k_1}} + \frac{1}{p_2^{k_2}} + \ldots + \right)^s. + \label{zeta:equation:eulerprodukt2} + \end{align} + Der Fundamentalsatz der Arithmetik (Primfaktorzerlegung) besagt, dass jede beliebige Zahl $n \in \mathbb{N}$ durch eine eindeutige Primfaktorzerlegung beschrieben werden kann + \begin{equation} + n = \prod_i p_i^{k_i} \quad \forall \quad n \in \mathbb{N}. + \end{equation} + Jeder Summand der Summen in \eqref{zeta:equation:eulerprodukt2} ist somit eine Zahl $n$. + Da die Summen alle möglichen Kombinationen von Exponenten und Primzahlen in \eqref{zeta:equation:eulerprodukt2} enthält haben wir + \begin{equation} + \sum_{k_1=0}^{\infty} + \sum_{k_2=0}^{\infty} + \ldots + \left( + \frac{1}{p_1^{k_1}} + \frac{1}{p_2^{k_2}} + \ldots + \right)^s + = + \sum_{n=1}^\infty + \frac{1}{n^s} + = + \zeta(s) + \end{equation} +\end{proof} + diff --git a/buch/papers/zeta/main.tex b/buch/papers/zeta/main.tex index e0ea8e1..caddace 100644 --- a/buch/papers/zeta/main.tex +++ b/buch/papers/zeta/main.tex @@ -11,6 +11,7 @@ %TODO Einleitung \input{papers/zeta/einleitung.tex} +\input{papers/zeta/euler_product.tex} \input{papers/zeta/zeta_gamma.tex} \input{papers/zeta/analytic_continuation.tex} diff --git a/buch/papers/zeta/zeta_gamma.tex b/buch/papers/zeta/zeta_gamma.tex index 49fea74..db41676 100644 --- a/buch/papers/zeta/zeta_gamma.tex +++ b/buch/papers/zeta/zeta_gamma.tex @@ -2,9 +2,8 @@ \rhead{Zusammenhang mit der Gammafunktion} In diesem Abschnitt wird gezeigt, wie sich die Zetafunktion durch die Gammafunktion $\Gamma(s)$ ausdrücken lässt. -Dieser Zusammenhang der Art $\zeta(s) = f(\Gamma(s))$ wird später für die Herleitung der analytischen Fortsetzung gebraucht. +Dieser Zusammenhang der Art $\zeta(s) = f(\Gamma(s))$ ist nicht nur interessant, er wird später auch für die Herleitung der analytischen Fortsetzung gebraucht. -%TODO ref Gamma Wir erinnern uns an die Definition der Gammafunktion in \eqref{buch:rekursion:gamma:integralbeweis} \begin{equation*} \Gamma(s) @@ -51,12 +50,12 @@ Die Summe über $e^{-nu}$ können wir als geometrische Reihe schreiben und erhal &= \frac{1}{e^u - 1}. \end{align} -Wenn wir dieses Resultat einsetzen in \eqref{zeta:equation:zeta_gamma1} und durch $\Gamma(s)$ teilen, erhalten wir %TODO formulieren als Satz +Wenn wir dieses Resultat einsetzen in \eqref{zeta:equation:zeta_gamma1} und durch $\Gamma(s)$ teilen, erhalten wir den gewünschten Zusammenhang \begin{equation}\label{zeta:equation:zeta_gamma_final} \zeta(s) = \frac{1}{\Gamma(s)} \int_0^{\infty} \frac{u^{s-1}}{e^u -1} - du. + du \qed \end{equation} -- cgit v1.2.1 From 7459c95431d89576126a6a0007238592a4f5f033 Mon Sep 17 00:00:00 2001 From: runterer Date: Fri, 27 May 2022 20:10:13 +0200 Subject: Minor improvements --- buch/papers/zeta/analytic_continuation.tex | 26 ++++++++++++++------------ 1 file changed, 14 insertions(+), 12 deletions(-) (limited to 'buch') diff --git a/buch/papers/zeta/analytic_continuation.tex b/buch/papers/zeta/analytic_continuation.tex index 408a1f7..40424e0 100644 --- a/buch/papers/zeta/analytic_continuation.tex +++ b/buch/papers/zeta/analytic_continuation.tex @@ -14,7 +14,7 @@ Eine grafische Darstellung dieses Plans ist in Abbildung \ref{zeta:fig:continuat \centering \input{papers/zeta/continuation_overview.tikz.tex} \caption{ - Die verschiedenen Abschnitte der Riemannschen Zetafunktion. + Die verschiedenen Abschnitte der Riemannschen Zetafunktion. Die originale Definition von \eqref{zeta:equation1} ist im grünen Bereich gültig. Für den blauen Bereich gilt \eqref{zeta:equation:fortsetzung1}. Um den roten Bereich zu bekommen verwendet die Funktionalgleichung \eqref{zeta:equation:functional} eine Spiegelung an $\Re(s) = 0.5$. @@ -76,33 +76,35 @@ Wir beginnen damit, die Gammafunktion für den halben Funktionswert zu berechnen \int_0^{\infty} t^{\frac{s}{2}-1} e^{-t} dt. \end{equation} Nun substituieren wir $t$ mit $t = \pi n^2 x$ und $dt=\pi n^2 dx$ und erhalten -\begin{align} +\begin{equation} \Gamma \left( \frac{s}{2} \right) - &= + = \int_0^{\infty} (\pi n^2)^{\frac{s}{2}} x^{\frac{s}{2}-1} e^{-\pi n^2 x} - \,dx - && \text{Division durch } (\pi n^2)^{\frac{s}{2}} - \\ + \,dx. +\end{equation} +Analog zum Abschnitt \ref{zeta:section:zusammenhang_mit_gammafunktion} teilen wir durch $(\pi n^2)^{\frac{s}{2}}$ +\begin{equation} \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}} n^s} - &= + = \int_0^{\infty} x^{\frac{s}{2}-1} e^{-\pi n^2 x} - \,dx - && \text{Zeta durch Summenbildung } \sum_{n=1}^{\infty} - \\ + \,dx, +\end{equation} +und finden Zeta durch die Summenbildung $\sum_{n=1}^{\infty}$ +\begin{equation} \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} \zeta(s) - &= + = \int_0^{\infty} x^{\frac{s}{2}-1} \sum_{n=1}^{\infty} e^{-\pi n^2 x} \,dx. \label{zeta:equation:integral1} -\end{align} +\end{equation} Die Summe kürzen wir ab als $\psi(x) = \sum_{n=1}^{\infty} e^{-\pi n^2 x}$. %TODO Wieso folgendes -> aus Fourier Signal Es gilt -- cgit v1.2.1 From 42a5955183a1bc0678158c61fd6189c39d305697 Mon Sep 17 00:00:00 2001 From: runterer Date: Fri, 27 May 2022 23:29:56 +0200 Subject: added poissonsche summenformel --- buch/papers/zeta/analytic_continuation.tex | 176 ++++++++++++++++++++++++++++- 1 file changed, 171 insertions(+), 5 deletions(-) (limited to 'buch') diff --git a/buch/papers/zeta/analytic_continuation.tex b/buch/papers/zeta/analytic_continuation.tex index 40424e0..0ccc116 100644 --- a/buch/papers/zeta/analytic_continuation.tex +++ b/buch/papers/zeta/analytic_continuation.tex @@ -106,15 +106,65 @@ und finden Zeta durch die Summenbildung $\sum_{n=1}^{\infty}$ \,dx. \label{zeta:equation:integral1} \end{equation} Die Summe kürzen wir ab als $\psi(x) = \sum_{n=1}^{\infty} e^{-\pi n^2 x}$. -%TODO Wieso folgendes -> aus Fourier Signal -Es gilt +Im Abschnitt \ref{zeta:subsec:poisson_summation} wird die poissonsche Summenformel $\sum f(n) = \sum F(n)$ bewiesen. +In unserem Problem ist $f(n) = e^{-\pi n^2 x}$ und die zugehörige Fouriertransformation $F(n)$ ist +\begin{equation} + F(n) + = + \mathcal{F} + ( + e^{-\pi n^2 x} + ) + = + \frac{1}{\sqrt{x}} + e^{\frac{-n^2 \pi}{x}}. +\end{equation} +Dadurch ergibt sich \begin{equation}\label{zeta:equation:psi} - \psi(x) + \sum_{n=-\infty}^{\infty} + e^{-\pi n^2 x} = + \frac{1}{\sqrt{x}} + \sum_{n=-\infty}^{\infty} + e^{\frac{-n^2 \pi}{x}}, +\end{equation} +wobei wir die Summen so verändern müssen, dass sie bei $n=1$ beginnen und wir $\psi(x)$ erhalten als +\begin{align} + 2 + \sum_{n=1}^{\infty} + e^{-\pi n^2 x} + + + 1 + &= + \frac{1}{\sqrt{x}} + \left( + 2 + \sum_{n=1}^{\infty} + e^{\frac{-n^2 \pi}{x}} + + + 1 + \right) + \\ + 2 + \psi(x) + + + 1 + &= + \frac{1}{\sqrt{x}} + \left( + 2 + \psi\left(\frac{1}{x}\right) + + + 1 + \right) + \\ + \psi(x) + &= - \frac{1}{2} + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}} - + \frac{1}{2 \sqrt{x}}. -\end{equation} + + \frac{1}{2 \sqrt{x}}.\label{zeta:equation:psi} +\end{align} +Diese Gleichung wird später wichtig werden. Zunächst teilen wir nun das Integral aus \eqref{zeta:equation:integral1} auf als \begin{equation}\label{zeta:equation:integral2} @@ -309,3 +359,119 @@ Somit haben wir die analytische Fortsetzung gefunden als \zeta(1-s). \end{equation} %TODO Definitionen und Gleichungen klarer unterscheiden + +\subsection{Poissonsche Summenformel} \label{zeta:subsec:poisson_summation} + +Der Beweis für Gleichung \ref{zeta:equation:psi} folgt direkt durch die poissonsche Summenformel. +Um diese zu beweisen, berechnen wir zunächst die Fourierreihe der Dirac Delta Funktion. + +\begin{lemma} + Die Fourierreihe der periodischen Dirac Delta Funktion $\sum \delta(x - 2\pi k)$ ist + \begin{equation} \label{zeta:equation:fourier_dirac} + \sum_{k=-\infty}^{\infty} + \delta(x - 2\pi k) + = + \frac{1}{2\pi} + \sum_{n=-\infty}^{\infty} + e^{i n x}. + \end{equation} +\end{lemma} + +\begin{proof}[Beweis] + Eine Fourierreihe einer beliebigen periodischen Funktion $f(x)$ berechnet sich als + \begin{align} + f(x) + &= + \sum_{n=-\infty}^{\infty} + c_n + e^{i n x} \\ + c_n + &= + \frac{1}{2\pi} + \int_{-\pi}^{\pi} + f(x) + e^{-i n x} + \, dx. + \end{align} + Wenn $f(x)=\delta(x)$ eingesetz wird ergeben sich konstante Koeffizienten + \begin{equation} + c_n + = + \frac{1}{2\pi} + \int_{-\pi}^{\pi} + \delta(x) + e^{-i n x} + \, dx + = + \frac{1}{2\pi}, + \end{equation} + womit die sehr einfache Fourierreihe der Dirac Delta Funktion berechnet wäre. +\end{proof} + +\begin{satz}[Poissonsche Summernformel] + Die Summe einer Funktion $f(n)$ über alle ganzen Zahlen $n$ ist äquivalent zur Summe ihrer Fouriertransformation $F(k)$ über alle ganzen Zahlen $k$ + \begin{equation} + \sum_{n=-\infty}^{\infty} + f(n) + = + \sum_{k=-\infty}^{\infty} + F(k). + \end{equation} +\end{satz} + +\begin{proof}[Beweis] + Wir schreiben die Summe über die Fouriertransformation aus + \begin{align} + \sum_{k=-\infty}^{\infty} + F(k) + &= + \sum_{k=-\infty}^{\infty} + \int_{-\infty}^{\infty} + f(x) + e^{-i 2\pi x k} + \, dx + \\ + &= + \int_{-\infty}^{\infty} + f(x) + \underbrace{ + \sum_{k=-\infty}^{\infty} + e^{-i 2\pi x k} + }_{\text{\eqref{zeta:equation:fourier_dirac}}} + \, dx, + \end{align} + und verwenden die Fouriertransformation der Dirac Funktion aus \eqref{zeta:equation:fourier_dirac} + \begin{align} + \sum_{k=-\infty}^{\infty} + e^{-i 2\pi x k} + &= + 2 \pi + \sum_{k=-\infty}^{\infty} + \delta(-2\pi x - 2\pi k) + \\ + &= + \frac{2 \pi}{2 \pi} + \sum_{k=-\infty}^{\infty} + \delta(x + k). + \end{align} + Wenn wir dies einsetzen und erhalten wir den gesuchten Beweis für die poissonsche Summenformel + \begin{equation} + \sum_{k=-\infty}^{\infty} + F(k) + = + \int_{-\infty}^{\infty} + f(x) + \sum_{k=-\infty}^{\infty} + \delta(x + k) + \, dx + = + \sum_{k=-\infty}^{\infty} + \int_{-\infty}^{\infty} + f(x) + \delta(x + k) + \, dx + = + \sum_{k=-\infty}^{\infty} + f(k). + \end{equation} +\end{proof} -- cgit v1.2.1 From df8e535423f408f789f0cb624df7a4980572bc4d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 28 May 2022 14:57:18 +0200 Subject: more onm integration and lemniscate --- buch/chapters/060-integral/differentialkoerper.tex | 5 + buch/chapters/060-integral/diffke.tex | 106 ++++++++++- buch/chapters/060-integral/elementar.tex | 199 +++++++++++++++++++++ buch/chapters/060-integral/erweiterungen.tex | 98 +++++++++- buch/chapters/060-integral/logexp.tex | 20 ++- buch/chapters/060-integral/rational.tex | 27 ++- buch/chapters/060-integral/risch.tex | 12 ++ buch/chapters/110-elliptisch/lemniskate.tex | 24 ++- buch/chapters/references.bib | 7 + 9 files changed, 475 insertions(+), 23 deletions(-) (limited to 'buch') diff --git a/buch/chapters/060-integral/differentialkoerper.tex b/buch/chapters/060-integral/differentialkoerper.tex index a071ae2..a112e33 100644 --- a/buch/chapters/060-integral/differentialkoerper.tex +++ b/buch/chapters/060-integral/differentialkoerper.tex @@ -15,6 +15,11 @@ Doch woher weiss man, dass es keine solche Funktion gibt, und was heisst überhaupt ``Stammfunktion in geschlossener Form''? In diesem Abschnitt wird daher ein algebraischer Rahmen entwickelt, in dem diese Frage sinnvoll gestellt werden kann. +Das ultimative Ziel, welches aber erst in +Abschnitt~\ref{buch:integral:section:risch} in Angriff genommen +wird, ist ein Computer-Algorithmus, der Integrale in geschlossener +Form findet oder beweist, dass dies für einen gegebenen Integranden +nicht möglich ist. \input{chapters/060-integral/rational.tex} \input{chapters/060-integral/erweiterungen.tex} diff --git a/buch/chapters/060-integral/diffke.tex b/buch/chapters/060-integral/diffke.tex index 02e90f6..61badc8 100644 --- a/buch/chapters/060-integral/diffke.tex +++ b/buch/chapters/060-integral/diffke.tex @@ -105,6 +105,94 @@ $ab\in C$. Die Menge $C$ heisst der {\em Konstantenkörper} von $\mathscr{F}$. \index{Konstantenkörper}% +% +% Ableitung algebraischer Elemente +% +\subsubsection{Ableitung und algebraische Körpererweiterungen} +Die Rechenregeln in einem Differentialkörper $\mathscr{F}$ legen auch die +Ableitung eines algebraischen Elements fest. +Sei $m(z)=m_0+m_1z+\ldots+m_{n-1}z^{n-1}+z^n$ das Minimalpolynom eines +über $\mathscr{F}$ algebraischen Elements $f$. +Aus $m(f)=0$ folgt durch Ableiten +\[ +0 += +m(f)' += +m_0' ++ +(m_1'f+m_1f') ++ +(m_2'f + m_12f'f) ++ +\ldots ++ +(m_{n-1}'f^{n-1} + m_{n-1} (n-1)f'f^{n-2}) ++ +nf'f^{n-1}. +\] +Zusammenfassen der Ableitung $f'$ auf der linken Seite liefert die +Gleichung +\[ +f'( +m_1+2m_2f+\ldots+(n-1)m_{n-1}f^{n-2}+nf^{n-1} +) += +m_0' + m_1'f + m_2'f^2 + \ldots + m_{n-1}'f^{n-1} + f^n, +\] +aus der +\[ +f' += +\frac{ +m_0' + m_1'f + m_2'f^2 + \ldots + m_{n-1}'f^{n-1} + f^n +}{ +m_1+2m_2f+\ldots+(n-1)m_{n-1}f^{n-2}+nf^{n-1} +} +\] +als Element von $\mathscr{F}(g)$ berechnet werden kann. +Die Ableitungsoperation lässt sich somit auf die Körpererweiterung +$\mathscr{F}(f)$ fortsetzen. + +\begin{beispiel} +Das über $\mathbb{Q}(x)$ algebraische Element $y=\sqrt{ax^2+bx+c}$ +hat das Minimalpolynom +\[ +m(z) += +z^2 - [ax^2+bx+c] +\in +\mathbb{Q}(x)[z] +\] +mit Koeffizienten $m_0 = ax^2+bx+c,$ $m_1=0$ und $m_2=1$. +Es hat die Ableitung +\[ +y' += +\frac{m_0'}{2m_2y} += +\frac{ +2ax+b +}{ +2y +} +\in +\mathbb{Q}(x,y) +\] +wegen $m_0'=2ax+b$. +\end{beispiel} + +\begin{definition} +Eine differentielle Körpererweiterung ist eine Körpererweiterung +$\mathscr{K}\subset\mathscr{F}$ von Differentialkörpern derart, dass +die Ableitungen $D_{\mathscr{K}}$ in $\mathscr{K}$ +und $D_{\mathscr{F}}$ in $\mathscr{F}$ übereinstimmen: +\( +D_{\mathscr{K}}g= D_{\mathscr{F}} g +\) +für alle $g\in\mathscr{K}$. +\end{definition} + % % Logarithmus und Exponantialfunktion % @@ -116,6 +204,7 @@ Sie zeichnen sich aber durch besondere Ableitungseigenschaften aus. Die Theorie der gewöhnlichen Differentialgleichungen garantiert, dass eine Funktion durch eine Differentialgleichung und Anfangsbedingungen festgelegt ist. +\label{buch:integral:expundlog} Für die Exponentialfunktion und der Logarithmus haben die Ableitungseigenschaften \[ @@ -128,10 +217,19 @@ x \log'(x) = 1. In der algebraischen Beschreibung eines Funktionenkörpers gibt es das Konzept des Wertes einer Funktion an einer bestimmten Stelle nicht. Somit können keine Anfangsbedingungen vorgegeben werden. -Da die Gleichungen linear sind, sind Vielfache einer Lösung wieder -Lösungen. -Insbesondere ist mit $\exp(x)$ auch $a\exp(x)$ eine Lösung und mit -$\log(x)$ auch $a\log(x)$ für alle Konstanten $a$. +Da die Gleichung für $\exp$ linear sind, sind Vielfache einer Lösung wieder +Lösungen, +insbesondere ist mit $\exp(x)$ auch $a\exp(x)$ eine Lösung. +Die Gleichung für $\log$ ist nicht linear, aber es ist +$\log'(x) = 1/x$, $\log$ ist eine Stammfunktion von $1/x$, die +nur bis auf eine Konstante bestimmt ist. +Tatsächlich gilt +\[ +x(\log(x)+a)' += +x\log(x) + xa' = x\log(x)=1, +\] +die Funktion $\log(x)+a$ ist also auch eine Lösung für den Logarithmus. Die Eigenschaft, dass die Exponentialfunktion die Umkehrfunktion des Logarithmus ist, lässt sich mit den Mitteln eines Differentialkörpers diff --git a/buch/chapters/060-integral/elementar.tex b/buch/chapters/060-integral/elementar.tex index 854a875..fd5f051 100644 --- a/buch/chapters/060-integral/elementar.tex +++ b/buch/chapters/060-integral/elementar.tex @@ -13,3 +13,202 @@ Die Stammfunktionen verwenden dieselben Funktionen oder höchstens Erweiterungen um Logarithmen von Funktionen, die man schon im Integranden gesehen hat. +% +% Exponentielle und logarithmische Funktione +% +\subsubsection{Exponentielle und logarithmische Funktionen} +In Abschnitt~\ref{buch:integral:subsection:diffke} haben wir +bereits die Exponentialfunktion $e^x$ und die Logarithmusfunktion +$\log x$ charakterisiert als eine Körpererweiterung durch +Elemente, die der Differentialgleichung +\[ +\exp' = \exp +\qquad\text{und}\qquad +\log' = \frac{1}{x} +\] +genügen. +Für die Stammfunktionen, die in +Abschnitt~\ref{buch:integral:subsection:logexp} +gefunden wurden, sind aber Logarithmusfunktionen nicht von +$x$ sondern von beliebigen über $\mathbb{Q}$ algebraischen Elementen +nötig. +Um zu verstehen, wie wir diese Funktion als Körpererweiterung erhalten +könnten, betrachten wir die Ableitung einer Exponentialfunktion +$\vartheta(x) = \exp(f(x))$ und eines +Logarithmus +$\psi(x) = \log(f(x))$, wie man sie mit der Kettenregel +berechnet hätte: +\begin{align*} +\vartheta'(x) +&=\exp(f(x)) \cdot f'(x) +& +\psi'(x) +&= +\frac{f'(x)}{f(x)} +\quad\Leftrightarrow\quad +f(x)\psi'(x) += +f'(x). +\end{align*} +Dies motiviert die folgende Definition + +\begin{definition} +\label{buch:integral:def:explog} +Sei $\mathscr{F}$ ein Differentialklörper und $f\in\mathscr{F}$. +Ein Exponentialfunktion von $f$ ist ein $\vartheta\in \mathscr{F}$mit +$\vartheta' = \vartheta f'$. +Ein Logarithmus von $f$ ist ein $\vartheta\in\mathscr{F}$ mit +$f\vartheta'=f'$. +\end{definition} + +Für $f=x$ mit $f'=1$ reduziert sich die +Definition~\ref{buch:integral:def:explog} +auf die Definition der Exponentialfunktion $\exp(x)$ und +Logarithmusfunktion $\log(x)$ auf Seite~\pageref{buch:integral:expundlog}. + + +% +% +% +\subsubsection{Transzendente Körpererweiterungen} +Die Wurzelfunktionen haben wir früher als algebraische Erweiterungen +eines Differentialkörpers erkannt. +Die logarithmischen und exponentiellen Elemente gemäss +Definition~\ref{buch:integral:def:explog} sind nicht algebraisch. + +\begin{definition} +\label{buch:integral:def:transzendent} +Sei $\mathscr{F}\subset\mathscr{G}$ eine Körpererweiterung und +$\vartheta\in\mathscr{G}$. +$\vartheta$ heisst {\em transzendent}, wenn $\vartheta$ nicht +algebraisch ist. +\end{definition} + +\begin{beispiel} +Die Funktion $f = e^x + e^{2x} + e^{x/2}$ ist sicher transzendent, +in diesem Beispiel zeigen wir, dass es mindestens drei verschiedene +Möglichkeiten gibt, eine Körpererweiterung von $\mathbb{Q}(x)$ zu +konstruieren, die $f$ enthält. + +Erste Möglichkeit: $f=\vartheta_1 + \vartheta_2 + \vartheta_3$ mit +$\vartheta_1=e^x$, +$\vartheta_2=e^{2x}$ +und +$\vartheta_3=e^{x/2}$. +Jedes der Elemente $\vartheta_i$ ist exponentiell über $\mathbb{Q}(x)$ und +$f$ ist in +\[ +\mathbb{Q}(x) +\subset +\mathbb{Q}(x,\vartheta_1) +\subset +\mathbb{Q}(x,\vartheta_1,\vartheta_2) +\subset +\mathbb{Q}(x,\vartheta_1,\vartheta_2,\vartheta_3) +\ni +f. +\] +Jede dieser Körpererweiterungen ist transzendent. + +Zweite Möglichkeit: $\vartheta_1=e^x$ ist exponentiell über +$\mathbb{Q}(x)$ und $\mathbb{Q}(x,\vartheta_1)$ enthält wegen +\[ +(\vartheta_1^2)' += +2\vartheta_1\vartheta_1' += +2\vartheta_1^2, +\] +somit ist $\vartheta_1^2=\vartheta_2$ eine Exponentialfunktion von $2x$ +über $\mathbb{Q}(x)$. +Das Element $\vartheta_3=e^{x/2}$ ist zwar auch exponentiell über +$\mathbb{Q}(x)$, es ist aber auch eine Nullstelle des Polynoms +$m(z)=z^2-[\vartheta_1]$. +Die Erweiterung +$\mathbb{Q}(x,\vartheta_1)\subset\mathbb{Q}(x,\vartheta_1,\vartheta_3)$ +ist eine algebraische Erweiterung, die +$f=\vartheta_1 + \vartheta_1^2+\vartheta_3$ enthält. + +Dritte Möglichkeit: $\vartheta_3=e^{x/2}$ ist exponentiell über +$\mathbb{Q}(x)$. +Die transzendente Körpererweiterung +\[ +\mathbb{Q}(x) \subset \mathbb{Q}(x,\vartheta_3) +\] +enthält das Element +$f=\vartheta_3^4+\vartheta_3^2 + \vartheta_3 $. +\end{beispiel} + +Das Beispiel zeigt, dass man nicht sagen kann, dass eine Funktion +ausschliesslich in einer algebraischen oder transzendenten Körpererweiterung +zu finden ist. +Vielmehr gibt es für die gleiche Funktion möglicherweise verschiedene +Körpererweiterungen, die alle die Funktion enthalten können. + +% +% Elementare Funktionen +% +\subsubsection{Elementare Funktionen} +Die Stammfunktionen~\eqref{buch:integration:risch:eqn:integralbeispiel2} +können aufgebaut werden, indem man dem Körper $\mathbb{Q}(x)$ schrittweise +sowohl algebraische wie auch transzendente Elemente hinzufügt, +wie in der folgenden Definition, die dies für abstrakte +Differentialkörpererweiterungen formuliert. + +\begin{definition} +Eine Körpererweiterung $\mathscr{F}\subset\mathscr{G}$ heisst +{\em transzendente elementare Erweiterung}, wenn +$\mathscr{G} = \mathscr{F}(\vartheta_1,\dots,\vartheta_n)$ und +jedes der Element $\vartheta_i$ transzendent und logarithmisch oder +exponentiell ist über +$\mathscr{F}_{i-1}=\mathscr{F}(\vartheta_1,\dots,\vartheta_{i-1})$. +Die Körpererweiterung $\mathscr{F}\subset\mathscr{G}$ heisst +{\em elementare Erweiterung}, wenn +$\mathscr{G} = \mathscr{F}(\vartheta_1,\dots,\vartheta_n)$ und +jedes Element $\vartheta_i$ ist entweder logarithmisch, exponentiell +oder algebraisch über $\mathscr{F}_{i-1}$. +\end{definition} + +Die Funktionen, die als akzeptable Stammfunktionen für das Integrationsproblem +in Betracht kommen, sind also jene, die in einer geeigneten elementaren +Erweiterung des von $\mathbb{Q}(x)$ liegen. +Ausserdem können auch noch weitere Konstanten nötig sein, sowohl +algebraische Zahlen wie auch Konstanten wie $\pi$ oder $e$. + +\begin{definition} +Sei $\mathscr{K}(x)$ der Differentialklörper der rationalen Funktionen +über dem Konstantenkörper $\mathscr{K}\supset\mathbb{Q}$, der in $\mathbb{C}$ +enthalten ist. +Ist $\mathscr{F}\supset \mathscr{K}(x)$ eine transzendente elementare +Erweiterung von $\mathscr{K}(x)$, dann heisst $\mathscr{F}$ +ein Körper von {\em transzendenten elementaren Funktionen}. +Ist $\mathscr{F}$ eine elementare Erweiterung von $\mathscr{K}(x)$, dann +heisst $\mathscr{F}$ ein Körper von {\em elementaren Funktionen}. +\end{definition} + +\subsubsection{Das Integrationsproblem} +Die elementaren Funktionen enthalten alle Funktionen, die sich mit +arithmetischen Operationen, Wurzeln, Exponentialfunktionen, Logarithmen und +damit auch mit trigonometrischen und hyperbolischen Funktionen und ihren +Umkehrfunktionen aus den rationalen Zahlen, der unabhängigen Variablen $x$ +und möglicherweise einigen zusätzlichen Konstanten aufbauen lassen. +Sei also $f$ eine Funktion in einem Körper von elementaren +Funktionen +\[ +\mathscr(F) += +\mathbb{Q}(\alpha_1,\dots,\alpha_l)(x,\vartheta_1,\dots,\vartheta_n). +\] +Eine elementare Stammfunktion ist eine Funktion $F=\int f$ in einer +elementaren Körpererweiterung +\[ +\mathscr{G} += +\mathbb{Q}(\alpha_1,\dots,\alpha_l,\dots,\alpha_{l+k}) +(x,\vartheta_1,\dots,\vartheta_n,\dots,\vartheta_{n+m}) +\] +mit $F'=f$. +Das Ziel ist, $F$ mit Hilfe eines Algorithmus zu bestimmen. + + + diff --git a/buch/chapters/060-integral/erweiterungen.tex b/buch/chapters/060-integral/erweiterungen.tex index a999ebb..9138f3e 100644 --- a/buch/chapters/060-integral/erweiterungen.tex +++ b/buch/chapters/060-integral/erweiterungen.tex @@ -97,8 +97,8 @@ a_i\in K \} \label{buch:integral:eqn:algelement} \end{equation} -mit $n=\deg m(x) - 1$ der durch Adjunktion von $\alpha$ erhaltene -Erweiterungsköper. +mit $n=\deg m(x) - 1$ der durch {\em Adjunktion} oder Hinzufügen +von $\alpha$ erhaltene Erweiterungsköper. \end{definition} Wieder muss nur überprüft werden, dass jedes Produkt oder jeder @@ -151,7 +151,9 @@ Die Menge $\mathbb{Q}(i)$ ist daher eine algebraische Körpererweiterung von $\mathbb{Q}$ bestehend aus den komplexen Zahlen mit rationalem Real- und Imaginärteil. +% % Transzendente Körpererweiterungen +% \subsubsection{Transzendente Erweiterungen} Nicht alle Zahlen in $\mathbb{R}$ sind algebraisch. Lindemann bewies 1882 einen allgemeinen Satz, aus dem folgt, @@ -201,7 +203,9 @@ $K\subset K(\alpha)$ ist zwar immer noch eine Körpererweiterung, aber $K(\alpha)$ ist nicht mehr ein endlichdimensionaler Vektorraum. Die Körpererweiterung $K\subset K(\alpha)$ heisst {\em transzendent}. +% % rationale Funktionen als Körpererweiterungen +% \subsubsection{Rationale Funktionen als Körpererweiterung} Die unabhängige Variable wird bei Rechnen so behandelt, dass die Potenzen alle linear unabhängig sind. @@ -209,7 +213,9 @@ Dies ist die Grundlage für den Koeffizientenvergleich. Der Körper der rationalen Funktion $K(x)$ ist also eine transzendente Körpererweiterung von $K$. +% % Erweiterungen mit algebraischen Funktionen +% \subsubsection{Algebraische Funktionen} Für das Integrationsproblem möchten wir nicht nur rationale Funktionen verwenden können, sondern auch Wurzelfunktionen. @@ -246,4 +252,92 @@ $y=\sqrt{ax^2+bx+c}$ zu $K(x,y)=K(x,\sqrt{ax^2+bx+c}$ erweitert. Wurzelfunktion werden daher nicht als Zusammensetzungen, sondern als algebraische Erweiterungen eines Funktionenkörpers betrachtet. +% +% Konjugation +% +\subsubsection{Konjugation} +Die komplexen Zahlen sind die algebraische Erweiterung der reellen Zahlen +um die Nullstelle $i$ des Polynoms $m(x)=x^2+1$. +Die Zahl $-i$ ist aber auch eine Nullstelle von $m(x)$, die mit algebraischen +Mitteln nicht von $i$ unterscheidbar ist. +Die komplexe Konjugation $a+bi\mapsto a-bi$ vertauscht die beiden +\index{Konjugation, komplexe}% +\index{komplexe Konjugation}% +Nullstellen des Minimalpolynoms. + +Ähnliches gilt für die Körpererweiterung $\mathbb{Q}(\!\sqrt{2})$. +$\sqrt{2}$ und $\sqrt{2}$ sind beide Nullstellen des Minimalpolynoms +$m(x)=x^2-2$, die mit algebraischen Mitteln nicht unterschiedbar sind. +Sie haben zwar verschiedene Vorzeichen, doch ohne eine Ordnungsrelation +können diese nicht unterschieden werden. +\index{Ordnungsrelation}% +Eine Ordnungsrelation zwischen rationalen Zahlen lässt sich zwar +definieren, aber die Zahl $\sqrt{2}$ ist nicht rational, es braucht +also eine zusätzliche Annahme, zum Beispiel die Identifikation von +$\sqrt{2}$ mit einer reellen Zahl in $\mathbb{R}$, wo der Vergleich +möglich ist. + +Auch in $\mathbb{Q}(\!\sqrt{2})$ ist die Konjugation +$a+b\sqrt{2}\mapsto a-b\sqrt{2}$ eine Selbstabbildung, die +die Körperoperationen respektiert. + +Das Polynom $m(x)=x^2-x-1$ hat die Nullstellen +\[ +\frac12 \pm\sqrt{\biggl(\frac12\biggr)^2+1} += +\frac{1\pm\sqrt{5}}{2} += +\left\{ +\bgroup +\renewcommand{\arraystretch}{2.20} +\renewcommand{\arraycolsep}{2pt} +\begin{array}{lcl} +\displaystyle +\frac{1+\sqrt{5}}{2} &=& \phantom{-}\varphi \\ +\displaystyle +\frac{1-\sqrt{5}}{2} &=& \displaystyle-\frac{1}{\varphi}. +\end{array} +\egroup +\right. +\] +Sie erfüllen die gleiche algebraische Relation $x^2=x+1$. +Sie sind sowohl im Vorzeichen wie auch im absoluten Betrag +verschieden, beides verlangt jedoch eine Ordnungsrelation als +Voraussetzung, die uns fehlt. +Aus beiden kann man mit rationalen Operationen $\sqrt{5}$ gewinnen, +denn +\[ +\sqrt{5} += +4\varphi-1 += +-4\biggl(-\frac{1}{\varphi}\biggr)^2-1 +\qquad\Rightarrow\qquad +\mathbb{Q}(\!\sqrt{5}) += +\mathbb{Q}(\varphi) += +\mathbb{Q}(-1/\varphi). +\] +Die Abbildung $a+b\varphi\mapsto a-b/\varphi$ ist eine Selbstabbildung +des Körpers $\mathbb{Q}(\!\sqrt{5})$, welche die beiden Nullstellen +vertauscht. + +Dieses Phänomen gilt für jede algebraische Erweiterung. +Die Nullstellen des Minimalpolynoms, welches die Erweiterung +definiert, sind grundsätzlich nicht unterscheidbar. +Mit der Adjunktion einer Nullstelle enthält der Erweiterungskörper +auch alle anderen. +Sind $\alpha_1$ und $\alpha_2$ zwei Nullstellen des Minimalpolynoms, +dann definiert die Abbildung $\alpha_1\mapsto\alpha_2$ eine Selbstabbildung, +die die Nullstellen permutiert. + +Die algebraische Körpererweiterung +$\mathbb{Q}(x)\subset \mathbb{Q}(x,\sqrt{ax^2+bx+c})$ +ist nicht unterscheidbar von +$\mathbb{Q}(x)\subset \mathbb{Q}(x,-\!\sqrt{ax^2+bx+c})$. +Für das Integrationsproblem bedeutet dies, dass alle Methoden so +formuliert werden müssen, dass die Wahl der Nullstellen auf die +Lösung keinen Einfluss haben. + diff --git a/buch/chapters/060-integral/logexp.tex b/buch/chapters/060-integral/logexp.tex index 2bfe0e1..e0efab2 100644 --- a/buch/chapters/060-integral/logexp.tex +++ b/buch/chapters/060-integral/logexp.tex @@ -3,7 +3,7 @@ % % (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschlue % -\subsection{Log-Exp-Notation für elementare Funktionen +\subsection{Log-Exp-Notation für trigonometrische und hyperbolische Funktionen \label{buch:integral:subsection:logexp}} Die Integration rationaler Funktionen hat bereits gezeigt, dass eine Stammfunktion nicht immer im Körper der rationalen Funktionen @@ -37,6 +37,7 @@ x \operatorname{arcosh} x - \sqrt{x^2-1}. In der Stammfunktion treten Funktionen auf, die auf den ersten Blick nichts mit den Funktionen im Integranden zu tun haben. +\subsubsection{Trigonometrische und hyperbolische Funktionen} Die trigonometrischen und hyperbolichen Funktionen in~\eqref{buch:integration:risch:allgform} lassen sich alle durch Exponentialfunktionen ausdrücken. @@ -53,7 +54,7 @@ So gilt &\qquad& \cosh x &= \frac12\bigl( e^x + e^{-x} \bigr). \end{aligned} -\label{buch:integral:risch:trighypinv} +\label{buch:integral:risch:trighyp} \end{equation} Nach Multiplikation mit $e^{ix}$ bzw.~$e^{x}$ entsteht eine quadratische Gleichung in $e^{ix}$ bzw.~$e^{x}$. @@ -66,27 +67,27 @@ Die Rechnung ergibt &= \frac{1}{i}\log\bigl( iy\pm\sqrt{1-y^2} -\bigr) +\bigr), & &\qquad& \arccos y &= \log\bigl( y\pm \sqrt{y^2-1} -\bigr) +\bigr), \\ \operatorname{arsinh}y &= \log\bigl( y \pm \sqrt{1+y^2} -\bigr) +\bigr), & &\qquad& \operatorname{arcosh} y &= \log\bigl( y\pm \sqrt{y^2-1} -\bigr) +\bigr). \end{aligned} \label{buch:integral:risch:trighypinv} \end{equation} @@ -97,6 +98,7 @@ Man nennt dies die $\log$-$\exp$-Notation der trigonometrischen und hyperbolischen Funktionen. \index{logexpnotation@$\log$-$\exp$-Notation}% +\subsubsection{$\log$-$\exp$-Notation} Wendet man die Substitutionen \eqref{buch:integral:risch:trighyp} und @@ -110,7 +112,7 @@ an, entstehen die Beziehungen &= \frac12i\bigl( \log(1-ix) - \log(1+ix) -\bigr) +\bigr), \\ \int\bigl( {\textstyle\frac12} @@ -121,12 +123,12 @@ e^{-ix} \bigr) &= -{\textstyle\frac12}ie^{ix} -+{\textstyle\frac12}ie^{-ix} ++{\textstyle\frac12}ie^{-ix}, \\ \int \frac{1}{\sqrt{1-x^2}} &= --i\log\bigl(ix+\sqrt{1-x^2}) +-i\log\bigl(ix+\sqrt{1-x^2}), \\ \int \log\bigl(x+\sqrt{x^2-1}\bigr) &= diff --git a/buch/chapters/060-integral/rational.tex b/buch/chapters/060-integral/rational.tex index 7b24e9f..0ca164d 100644 --- a/buch/chapters/060-integral/rational.tex +++ b/buch/chapters/060-integral/rational.tex @@ -132,7 +132,9 @@ ac + 2bd + (ad+bc)\sqrt{2} \in \mathbb{Q}(\!\sqrt{2}) \end{align*} \end{beispiel} - +% +% Rationale Funktionen +% \subsubsection{Rationalen Funktionen} Die als Antworten auf die Frage nach einer Stammfunktion akzeptablen Funktionen sollten alle rationalen Zahlen sowie die unabhängige @@ -174,5 +176,28 @@ zweier Brüche auch für Nenner funktioniert, die Polynome sind, und die Summe wzeier Brüche von Polynomen wieder in einen Bruch von Polynomen umwandelt. +% +% Warum rationale Zahlen? +% +\subsubsection{Warum die Beschränkung auf rationale Zahlen?} +Aus mathematischer Sicht gibt es gute Gründe, Analysis im Körper $\mathbb{R}$ +oder $\mathbb{C}$ zu betreiben. +Da Ableitung und Integral als Grenzwerte definiert sind, stellt diese +Wahl des Körpers sicher, dass die Grenzwerte auch tatsächlich existieren. +Der Fundamentalsatz der Algebra garantiert, dass über $\mathbb{C}$ +jedes Polynome in Linearfaktoren zerlegt werden kann. + +Der Einfachheit der Analyse in $\mathbb{R}$ oder $\mathbb{C}$ steht +die Schwierigkeit gegenüber, beliebige Elemente von $\mathbb{R}$ in +einem Computer exakt darzustellen. +Für Brüche in $\mathbb{Q}$ gibt es eine solche Darstellung durch +Paare von Ganzzahlen, wie sie die GNU Multiprecision Arithmetic Library +\cite{buch:gmp} realisiert. +Irrationale Zahlen dagegen können nur exakt gehandhabt werden, wenn +man im wesentlichen symbolisch mit ihnen rechnet. +Die Grundlage dafür wird in +Abschnitt~\ref{buch:integral:subsection:koerpererweiterungen} +gelegt. + diff --git a/buch/chapters/060-integral/risch.tex b/buch/chapters/060-integral/risch.tex index 1ba746a..2080ce8 100644 --- a/buch/chapters/060-integral/risch.tex +++ b/buch/chapters/060-integral/risch.tex @@ -6,6 +6,18 @@ \section{Der Risch-Algorithmus \label{buch:integral:section:risch}} \rhead{Risch-Algorithmus} +Die Lösung des Integrationsproblem für $\mathbb{Q}(x)$ und für +$\mathbb{Q}(x,y)$ mit $y=\!\sqrt{ax^2+bx+c}$ hat gezeigt, dass +ein Differentialkörper genau die richtige Bühne für dieses Unterfangen +sein dürfte. +Die Stammfunktionen konnten in einem Erweiterungskörper gefunden +werden, der ein paar Logarithmen hinzugefügt worden sind. +Tatsächlich lässt sich in diesem Rahmen sogar ein Algorithmus +formulieren, der in einem noch zu definierenden Sinn ``elementare'' +Funktionen als Stammfunktionen finden kann oder beweisen kann, dass +eine solche nicht existiert. +Dieser Abschnitt soll einen Überblick darüber geben. + \input{chapters/060-integral/logexp.tex} \input{chapters/060-integral/elementar.tex} diff --git a/buch/chapters/110-elliptisch/lemniskate.tex b/buch/chapters/110-elliptisch/lemniskate.tex index 0df27a7..f750a82 100644 --- a/buch/chapters/110-elliptisch/lemniskate.tex +++ b/buch/chapters/110-elliptisch/lemniskate.tex @@ -20,7 +20,9 @@ elliptischen Funktionen hergestellt werden. \caption{Bogenlänge und Radius der Lemniskate von Bernoulli. \label{buch:elliptisch:fig:lemniskate}} \end{figure} -Die Lemniskate von Bernoulli ist die Kurve vierten Grades mit der Gleichung +Die {\em Lemniskate von Bernoulli} ist die Kurve vierten Grades +mit der Gleichung +\index{Lemniskate von Bernoulli}% \begin{equation} (X^2+Y^2)^2 = 2a^2(X^2-Y^2). \label{buch:elliptisch:eqn:lemniskate} @@ -161,13 +163,14 @@ Parameters $k$. Die Länge des rechten Blattes der Lemniskate wird mit $\varpi$ bezeichnet und hat den numerischen Wert -\[ +\begin{equation} \varpi = 2\int_0^1\sqrt{\frac{1}{1-t^4}}\,dt = 2.6220575542. -\] +\label{buch:elliptisch:eqn:varpi} +\end{equation} $\varpi$ ist auch als die {\em lemniskatische Konstante} bekannt. \index{lemniskatische Konstante}% Der Lemniskatenbogen zwischen dem Nullpunkt und $(1,0)$ hat die Länge @@ -179,7 +182,7 @@ $\varpi/2$. \subsection{Bogenlängenparametrisierung} Die Lemniskate mit der Gleichung \[ -(X^2+X^2)^2=2(X^2-X^2) +(X^2+Y^2)^2=2(X^2-Y^2) \] (der Fall $a=1$ in \eqref{buch:elliptisch:eqn:lemniskate}) kann mit Jacobischen elliptischen Funktionen @@ -332,7 +335,8 @@ Dies bedeutet, dass die Bogenlänge zwischen den Parameterwerten $0$ und $s$ = s, \] -der Parameter $t$ ist also ein Bogenlängenparameter. +der Parameter $t$ ist also ein Bogenlängenparameter, man darf also +$s=t$ schreiben. Die mit dem Faktor $1/\sqrt{2}$ skalierte Standard-Lemniskate mit der Gleichung @@ -355,10 +359,9 @@ y(t) \end{equation} \subsection{Der lemniskatische Sinus und Kosinus} -Der Sinus Berechnet die Gegenkathete zu einer gegebenen Bogenlänge des +Der Sinus berechnet die Gegenkathete zu einer gegebenen Bogenlänge des Kreises, er ist die Umkehrfunktion der Funktion, die der Gegenkathete die Bogenlänge zuordnet. - Daher ist es naheliegend, die Umkehrfunktion von $s(r)$ in \eqref{buch:elliptisch:eqn:lemniskatebogenlaenge} den {\em lemniskatischen Sinus} zu nennen mit der Bezeichnung @@ -368,6 +371,13 @@ Der Kosinus ist der Sinus des komplementären Winkels. Auch für die lemniskatische Bogenlänge $s(r)$ lässt sich eine komplementäre Bogenlänge definieren, nämlich die Bogenlänge zwischen dem Punkt $(x(r), y(r))$ und $(1,0)$. +Da die Bogenlänge zwischen $(0,0)$ und $(1,0)$ in +in \eqref{buch:elliptisch:eqn:varpi} bereits bereichnet wurde. +ist sie $\varpi/2-s$. +Der {\em lemniskatische Kosinus} ist daher +$\operatorname{cl}(s) = \operatorname{sl}(\varpi/2-s)$ +Graphen des lemniskatische Sinus und Kosinus sind in +Abbildung~\label{buch:elliptisch:figure:slcl} dargestellt. Da die Parametrisierung~\eqref{buch:elliptisch:lemniskate:bogenlaenge} eine Bogenlängenparametrisierung ist, darf man $t=s$ schreiben. diff --git a/buch/chapters/references.bib b/buch/chapters/references.bib index 17ef273..32a86ec 100644 --- a/buch/chapters/references.bib +++ b/buch/chapters/references.bib @@ -111,3 +111,10 @@ publisher = { Addison-Wesley } } +@online{buch:gmp, + title = {GNU Multiprecision Arithmetic Library}, + DAY = 26, + MONTH = 5, + YEAR = 2022, + url = {https://de.wikipedia.org/wiki/GNU_Multiple_Precision_Arithmetic_Library} +} -- cgit v1.2.1 From 2fad6877aa1883714a060e1204e6d4d3566541d9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 28 May 2022 19:02:25 +0200 Subject: add example --- buch/papers/nav/beispiel.txt | 24 ++++++++++++++++++++++++ 1 file changed, 24 insertions(+) create mode 100644 buch/papers/nav/beispiel.txt (limited to 'buch') diff --git a/buch/papers/nav/beispiel.txt b/buch/papers/nav/beispiel.txt new file mode 100644 index 0000000..c63525b --- /dev/null +++ b/buch/papers/nav/beispiel.txt @@ -0,0 +1,24 @@ +Datum: 28. 5. 2022 +Zeit: 15:29:49 UTC +Sternzeit: 7h 54m 26.593s + +Deneb + +RA 20h 42m 12.14s 10.703372h +DEC 45 21' 40.3" 45.361194 + +H 50g 15' 17.1" 50.254750h +Azi 59g 36' 02.0" 59.600555 + +Spica + +RA 13h 26m 23.44s 13.439844h +DEC -11g 16' 46.8" 11.279666 + +H 18g 27' 30.0" 18.458333 +Azi 240g 23' 52.5" 240.397916 + +Position: + +l = 140.228920 E +b = 35.734946 N -- cgit v1.2.1 From 082afe0e8250519008c73b947922be22afda3fd5 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 28 May 2022 19:14:50 +0200 Subject: beispiel korrektur --- buch/papers/nav/beispiel.txt | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'buch') diff --git a/buch/papers/nav/beispiel.txt b/buch/papers/nav/beispiel.txt index c63525b..853ae4e 100644 --- a/buch/papers/nav/beispiel.txt +++ b/buch/papers/nav/beispiel.txt @@ -20,5 +20,5 @@ Azi 240g 23' 52.5" 240.397916 Position: -l = 140.228920 E -b = 35.734946 N +l = 140 14' 00.01" E 140.233336 E +b = 35 43' 00.02" N 35.716672 N -- cgit v1.2.1