From 85e7d741f78ca0874b42db5cfbd18f4c28a933b3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Patrik=20M=C3=BCller?= Date: Thu, 2 Jun 2022 15:23:21 +0200 Subject: Add presentation --- .../laguerre/presentation/sections/laguerre.tex | 88 ++++++++++++++++++++++ 1 file changed, 88 insertions(+) create mode 100644 buch/papers/laguerre/presentation/sections/laguerre.tex (limited to 'buch/papers/laguerre/presentation/sections/laguerre.tex') diff --git a/buch/papers/laguerre/presentation/sections/laguerre.tex b/buch/papers/laguerre/presentation/sections/laguerre.tex new file mode 100644 index 0000000..cba9ffb --- /dev/null +++ b/buch/papers/laguerre/presentation/sections/laguerre.tex @@ -0,0 +1,88 @@ +\section{Laguerre-Polynome} + +\begin{frame}{Laguerre-Differentialgleichung} + +\begin{itemize} +\item Benannt nach Edmond Nicolas Laguerre (1834-1886) +\item Aus Artikel von 1879, +in dem er $\int_0^\infty \exp(-x)/x \, dx$ analysierte +\end{itemize} + +\begin{align*} +x y''(x) + (1 - x) y'(x) + n y(x) + & = +0 +, \quad +n \in \mathbb{N}_0 +, \quad +x \in \mathbb{R} +\end{align*} + +\end{frame} + +\begin{frame}{Lösen der Differentialgleichung} + +\begin{align*} +x y''(x) + (1 - x) y'(x) + n y(x) + & = +0 +\\ +\end{align*} + +\uncover<2->{ +\centering +\begin{tikzpicture}[remember picture,overlay] +%% use here too +\path[draw=mainColor, very thick,->](0, 1.1) to +node[anchor=west]{Potenreihenansatz} (0, -0.8); +\end{tikzpicture} +} + +\begin{align*} +\uncover<3->{ +L_n(x) + & = +\sum_{k=0}^{n} \frac{(-1)^k}{k!} \binom{n}{k} x^k +} +\end{align*} +\uncover<4->{ +\begin{itemize} + \item Die Lösungen der DGL sind die Laguerre-Polynome +\end{itemize} +} +\end{frame} + +\begin{frame} +\begin{figure}[h] +\centering +\scalebox{0.66}{\input{../images/laguerre_polynomes.pgf}} +\caption{Laguerre-Polynome vom Grad $0$ bis $7$} +\end{figure} +\end{frame} + +\begin{frame}{Orthogonalität} +\begin{itemize}[<+->] +\item Beweis: Umformen in Sturm-Liouville-Problem (siehe Paper) +\begin{alignat*}{5} +((p(x) &y'(x)))' + q(x) &y(x) +&= +\lambda &w(x) &y(x) +\\ +((x e^{-x} &y'(x)))' + 0 &y(x) +&= +n &e^{-x} &y(x) +\end{alignat*} +\item Definitionsbereich $(0, \infty)$ +\item Gewichtsfunktion $w(x) = e^{-x}$ +\end{itemize} + +\only<4>{ +\begin{align*} +\int_0^\infty e^{-x} L_n(x) L_m(x) \, dx += +0 +,\quad +n, m \in \mathbb{N} +\end{align*} +} +\end{frame} \ No newline at end of file -- cgit v1.2.1 From fac45f54d4cee5018c063b4a720695cbf3040fa9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Patrik=20M=C3=BCller?= Date: Thu, 2 Jun 2022 15:53:49 +0200 Subject: Correct typos in presentation --- buch/papers/laguerre/presentation/sections/laguerre.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'buch/papers/laguerre/presentation/sections/laguerre.tex') diff --git a/buch/papers/laguerre/presentation/sections/laguerre.tex b/buch/papers/laguerre/presentation/sections/laguerre.tex index cba9ffb..faa50e5 100644 --- a/buch/papers/laguerre/presentation/sections/laguerre.tex +++ b/buch/papers/laguerre/presentation/sections/laguerre.tex @@ -34,7 +34,7 @@ x y''(x) + (1 - x) y'(x) + n y(x) \begin{tikzpicture}[remember picture,overlay] %% use here too \path[draw=mainColor, very thick,->](0, 1.1) to -node[anchor=west]{Potenreihenansatz} (0, -0.8); +node[anchor=west]{Potenzreihenansatz} (0, -0.8); \end{tikzpicture} } @@ -76,7 +76,7 @@ n &e^{-x} &y(x) \item Gewichtsfunktion $w(x) = e^{-x}$ \end{itemize} -\only<4>{ +\uncover<4->{ \begin{align*} \int_0^\infty e^{-x} L_n(x) L_m(x) \, dx = -- cgit v1.2.1 From fb20b12bd912595deb6ad98a6428842f893edcda Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Patrik=20M=C3=BCller?= Date: Thu, 2 Jun 2022 16:13:14 +0200 Subject: Add n != m to presentation at orthogonality section --- buch/papers/laguerre/presentation/sections/laguerre.tex | 2 ++ 1 file changed, 2 insertions(+) (limited to 'buch/papers/laguerre/presentation/sections/laguerre.tex') diff --git a/buch/papers/laguerre/presentation/sections/laguerre.tex b/buch/papers/laguerre/presentation/sections/laguerre.tex index faa50e5..1add511 100644 --- a/buch/papers/laguerre/presentation/sections/laguerre.tex +++ b/buch/papers/laguerre/presentation/sections/laguerre.tex @@ -81,6 +81,8 @@ n &e^{-x} &y(x) \int_0^\infty e^{-x} L_n(x) L_m(x) \, dx = 0 +,\quad +n \neq m ,\quad n, m \in \mathbb{N} \end{align*} -- cgit v1.2.1 From 8fb46098cb8e42a94b8e01ecc809f536d5c7efaf Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Patrik=20M=C3=BCller?= Date: Fri, 3 Jun 2022 07:23:21 +0200 Subject: Minor tweaks of presentation --- buch/papers/laguerre/presentation/sections/laguerre.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'buch/papers/laguerre/presentation/sections/laguerre.tex') diff --git a/buch/papers/laguerre/presentation/sections/laguerre.tex b/buch/papers/laguerre/presentation/sections/laguerre.tex index 1add511..07cafb8 100644 --- a/buch/papers/laguerre/presentation/sections/laguerre.tex +++ b/buch/papers/laguerre/presentation/sections/laguerre.tex @@ -55,7 +55,7 @@ L_n(x) \begin{frame} \begin{figure}[h] \centering -\scalebox{0.66}{\input{../images/laguerre_polynomes.pgf}} +\resizebox{0.74\textwidth}{!}{\input{../images/laguerre_polynomes.pgf}} \caption{Laguerre-Polynome vom Grad $0$ bis $7$} \end{figure} \end{frame} -- cgit v1.2.1 From fde57297b3efbef28d09a532e1b3895d2b2ad917 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Patrik=20M=C3=BCller?= Date: Thu, 14 Jul 2022 15:03:28 +0200 Subject: Correct Makefile, add text to gamma.tex, separate python-scripts for each image --- buch/papers/laguerre/presentation/sections/laguerre.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'buch/papers/laguerre/presentation/sections/laguerre.tex') diff --git a/buch/papers/laguerre/presentation/sections/laguerre.tex b/buch/papers/laguerre/presentation/sections/laguerre.tex index 07cafb8..ed29387 100644 --- a/buch/papers/laguerre/presentation/sections/laguerre.tex +++ b/buch/papers/laguerre/presentation/sections/laguerre.tex @@ -55,7 +55,7 @@ L_n(x) \begin{frame} \begin{figure}[h] \centering -\resizebox{0.74\textwidth}{!}{\input{../images/laguerre_polynomes.pgf}} +\resizebox{0.74\textwidth}{!}{\input{../images/laguerre_poly.pgf}} \caption{Laguerre-Polynome vom Grad $0$ bis $7$} \end{figure} \end{frame} -- cgit v1.2.1 From e1f5d6267540ea8dc758696fb08cb7540362cf8f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Patrik=20M=C3=BCller?= Date: Mon, 18 Jul 2022 17:34:37 +0200 Subject: First complete draft of Laguerre chapter --- buch/papers/laguerre/presentation/sections/laguerre.tex | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) (limited to 'buch/papers/laguerre/presentation/sections/laguerre.tex') diff --git a/buch/papers/laguerre/presentation/sections/laguerre.tex b/buch/papers/laguerre/presentation/sections/laguerre.tex index ed29387..f99214e 100644 --- a/buch/papers/laguerre/presentation/sections/laguerre.tex +++ b/buch/papers/laguerre/presentation/sections/laguerre.tex @@ -55,7 +55,8 @@ L_n(x) \begin{frame} \begin{figure}[h] \centering -\resizebox{0.74\textwidth}{!}{\input{../images/laguerre_poly.pgf}} +% \resizebox{0.74\textwidth}{!}{\input{../images/laguerre_poly.pgf}} +\includegraphics[width=0.7\textwidth]{../images/laguerre_poly.pdf} \caption{Laguerre-Polynome vom Grad $0$ bis $7$} \end{figure} \end{frame} -- cgit v1.2.1