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Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/hermite/Makefile.inc | 5 | ||||
| -rw-r--r-- | vorlesungen/slides/hermite/hermiteentwicklung.tex | 72 | ||||
| -rw-r--r-- | vorlesungen/slides/hermite/loesung.tex | 65 | ||||
| -rw-r--r-- | vorlesungen/slides/hermite/normalhermite.tex | 103 | ||||
| -rw-r--r-- | vorlesungen/slides/hermite/normalintegrale.tex | 57 | ||||
| -rw-r--r-- | vorlesungen/slides/hermite/skalarprodukt.tex | 82 |
6 files changed, 384 insertions, 0 deletions
diff --git a/vorlesungen/slides/hermite/Makefile.inc b/vorlesungen/slides/hermite/Makefile.inc index 5c55467..58c21f2 100644 --- a/vorlesungen/slides/hermite/Makefile.inc +++ b/vorlesungen/slides/hermite/Makefile.inc @@ -4,4 +4,9 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # chapterhermite = \ + ../slides/hermite/normalintegrale.tex \ + ../slides/hermite/normalhermite.tex \ + ../slides/hermite/hermiteentwicklung.tex \ + ../slides/hermite/loesung.tex \ + ../slides/hermite/skalarprodukt.tex \ ../slides/hermite/test.tex diff --git a/vorlesungen/slides/hermite/hermiteentwicklung.tex b/vorlesungen/slides/hermite/hermiteentwicklung.tex new file mode 100644 index 0000000..5f6e1c9 --- /dev/null +++ b/vorlesungen/slides/hermite/hermiteentwicklung.tex @@ -0,0 +1,72 @@ +% +% hermiteentwicklung.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Beliebige Polynome} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Polynom} +\[ +P(x) += +p_0 + p_1x + p_2x^2 + \dots + p_nx^n +\] +\uncover<2->{% +als Linearkombination von Hermite-Polynome schreiben: +\begin{align*} +P(x) +&= +a_0H_0(x)% + a_1H_1(x) ++ \dots + a_nH_n(x) +\\ +&= +a_0\cdot 1 +\\ +&\quad + a_1\cdot 2x +\\ +&\quad + a_2\cdot(4x^2-2) +\\ +&\quad + a_3\cdot(8x^3-12x) +\\ +&\quad + a_4\cdot(16x^4-48x^2+12) +\\ +&\quad\;\;\vdots +\\ +&\quad + a_n(2^nx^n + \dots) +\end{align*}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Koeffizientenvergleich} +führt auf ein Gleichungssystem +\begin{center} +\begin{tabular}{|>{$}r<{$}>{$}r<{$}>{$}r<{$}>{$}r<{$}>{$}r<{$}>{$}c<{$}|>{$}c<{$}|} +\hline +a_0&a_1&a_2&a_3&a_4&\dots&\\ +\hline + 1& 0& 0& 0& 0&\dots&p_0\\ + 0& 2& 0& 0& 0&\dots&p_1\\ +-2& 0& 4& 0& 0&\dots&p_2\\ + 0&-12& 0& 8& 0&\dots&p_3\\ +12& 0&-48& 0& 16&\dots&p_4\\ +\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ +\hline +\end{tabular} +\end{center} +\uncover<4->{% +Dreiecksmatrix}\uncover<5->{, Diagonalelement +$\ne 0$} +\uncover<6->{$\Rightarrow$ +$\exists$ eindeutige Lösung} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/hermite/loesung.tex b/vorlesungen/slides/hermite/loesung.tex new file mode 100644 index 0000000..68ee32e --- /dev/null +++ b/vorlesungen/slides/hermite/loesung.tex @@ -0,0 +1,65 @@ +% +% loesung.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lösung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Frage} +Für welche Polynome $P(t)$ kann man eine Stammfunktion +\[ +\int +P(t)e^{-\frac{t^2}2} +\,dt +\] +in geschlossener Form angeben? +\end{block} +\uncover<2->{% +\begin{block}{``Hermite-Antwort''} +\[ +\int H_n(x)e^{-x^2}\,dx +\] +kann genau für $n>0$ in geschlossener Form angegeben werden. +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Allgemein} +\begin{align*} +\int P(x)e^{-x^2}\,dx +&\uncover<4->{= +\int \sum_{k=0}^n a_kH_k(x)e^{-x^2}\,dx} +\\ +\uncover<5->{ +&= +\sum_{k=0}^n +a_k +\int +H_k(x)e^{-x^2}\,dx +} +\\ +\uncover<6->{ +&= +a_0\operatorname{erf}(x) + C +} +\\ +\uncover<6->{ +&\hspace*{2mm} + \sum_{k=1}^n a_k\int H_k(x)e^{-x^2}\,dx +} +\end{align*} +\end{block}} +\uncover<7->{% +\begin{theorem} +Das Integral von $P(x)e^{-x^2}$ ist genau dann elementar darstellbar, wenn +$a_0=0$ +\end{theorem}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/hermite/normalhermite.tex b/vorlesungen/slides/hermite/normalhermite.tex new file mode 100644 index 0000000..98721dc --- /dev/null +++ b/vorlesungen/slides/hermite/normalhermite.tex @@ -0,0 +1,103 @@ +% +% normalhermite.tex -- integrability of hermite polynomials +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Hermite-Polynome} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition (Rodrigues-Formel)} +\[ +H_n(x) += +(-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} +\] +\end{block} +\vspace{-10pt} +\uncover<2->{% +\begin{block}{Orthogonalität} +$H_n(x)$ sind orthogonale Polynome bezüglich $w(x)=e^{-x^2}$, d.~h. +\begin{align*} +\langle H_n,H_m\rangle_w +&= +\int H_n(x)H_m(x)e^{-x^2}\,dx +\\ +&= +\biggl\{ +\renewcommand{\arraycolsep}{1pt} +\begin{array}{l@{\quad}l} +1&\text{falls $n=m$}\\ +0&\text{sonst} +\end{array} +\biggr\} += +\delta_{mn} +\end{align*} +\end{block}} +\vspace{-10pt} +\uncover<3->{% +\begin{block}{Rekursion: Auf-/Absteigeoperatoren} +Rekursionsformel: +\[ +H_n(x) += +2x\cdot H_{n-1}(x) - H_{n-1}'(x) +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Stammfunktion} +\begin{align*} +\uncover<4->{ +\int H_n(x) e^{-x^2}\,dx} +&\uncover<5->{= +\int \bigl({\color{red}2x}H_{n-1}(x)} +\\ +\uncover<5->{ +&\qquad -H_{n-1}'(x)\bigr) e^{-x^2}\,dx +} +\\ +\uncover<6->{ +{\color{gray}((e^{-x^2})'=-2x)} +&= +{\color{red}-}\int {\color{red}(e^{-x^2})'} H_{n-1}(x)\,dx +} +\\ +\uncover<6->{ +&\qquad +- +\int H_{n-1}'(x) e^{-x^2}\,dx +} +\\ +\uncover<7->{ +\text{\color{gray}(Produktregel)} +&= +\int (e^{-x^2}H_{n-1}(x))'\,dx +} +\\ +\uncover<8->{ +\text{\color{gray}(Ableitung)} +&= +e^{-x^2}H_{n-1}(x) +} +\end{align*} +\uncover<9->{% +ausser für $n=0$: +\[ +\int +H_0(x)e^{-x^2}\,dx += +\int +e^{-x^2}\,dx +\]} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/hermite/normalintegrale.tex b/vorlesungen/slides/hermite/normalintegrale.tex new file mode 100644 index 0000000..32333cd --- /dev/null +++ b/vorlesungen/slides/hermite/normalintegrale.tex @@ -0,0 +1,57 @@ +% +% normalintegrale.tex -- +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Integranden $P(t)e^{-t^2}$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Frage} +Für welche Polynome $P(t)$ kann man eine Stammfunktion +\[ +\int +P(t)e^{-t^2} +\,dt +\] +in geschlossener Form angeben? +\end{block} +\uncover<4->{% +\begin{block}{Allgemeine Antwort} +Satz von Liouville und +Risch- Algorithmus können entscheiden, ob es eine elementare Stammfunktion gibt +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Negativbeispiel} +$P(t) = 1$, das Normalverteilungsintegral +\[ +F(x) += +\frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2}\,dt +\] +ist nicht elementar darstellbar. +\end{block}} +\uncover<3->{% +\begin{block}{Positivbeispiel} +$P(t)=t$. Wegen +\begin{align*} +\frac{d}{dx}e^{-x^2} +&= +-xe^{-x^2} +\intertext{ist} +\int te^{-t^2}\,dt +&= +-e^{-x^2}+C +\end{align*} +elementar darstellbar. +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/hermite/skalarprodukt.tex b/vorlesungen/slides/hermite/skalarprodukt.tex new file mode 100644 index 0000000..a51e9f6 --- /dev/null +++ b/vorlesungen/slides/hermite/skalarprodukt.tex @@ -0,0 +1,82 @@ +% +% skalarprodukt.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Skalarprodukt} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Orthogonale Zerlegung} +Orthogonale $H_k$ normalisieren: +\[ +\tilde{H}_k(x) = \frac{1}{\|H_k\|_w} H_k(x) +\] +mit Gewichtsfunktion $w(x)=e^{-x^2}$ +\end{block} +\uncover<2->{% +\begin{block}{``Hermite''-Analyse} +\begin{align*} +P(x) +&= +\sum_{k=1}^\infty a_k H_k(x) += +\sum_{k=1}^\infty \tilde{a}_k \tilde{H}_k(x) +\\ +\uncover<3->{ +\tilde{a}_k +&= +\| H_k\|_w\, a_k +} +\\ +\uncover<4->{ +a_k +&= +\frac{1}{\|H_k\|} +\langle \tilde{H}_k, P\rangle_w +}\uncover<5->{= +\frac{1}{\|H_k\|^2} +\langle H_k, P\rangle_w +} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Integrationsproblem} +Bedingung: +\begin{align*} +a_0=0 +\uncover<7->{% +\qquad\Leftrightarrow\qquad +\langle H_0,P\rangle_w +&= +0} +\\ +\uncover<8->{% +\int_{-\infty}^\infty +P(t) w(t) \,dt +}\uncover<9->{% += +\int_{-\infty}^\infty +P(t) e^{-t^2} \,dt +&= +0} +\end{align*} +\end{block}} +\uncover<10->{% +\begin{theorem} +Das Integral von $P(t)e^{-t^2}$ ist in geschlossener Form darstellbar +genau dann, wenn +\[ +\int_{-\infty}^\infty P(t)e^{-t^2}\,dt = 0 +\] +\end{theorem}} +\end{column} +\end{columns} +\end{frame} +\egroup |