%
% 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}
\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}
\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}
\begin{block}{Stammfunktion}
\begin{align*}
\int H_n(x) e^{-x^2}\,dx
&=
\int \bigl({\color{red}2x}H_{n-1}(x)
\\
&\qquad -H_{n-1}'(x)\bigr) e^{-x^2}\,dx
\\
{\color{gray}((e^{-x^2})'=-2x)}
&=
{\color{red}-}\int {\color{red}(e^{-x^2})'} H_{n-1}(x)\,dx
\\
&\qquad
-
\int H_{n-1}'(x) e^{-x^2}\,dx
\\
\text{\color{gray}(Produktregel)}
&=
\int (e^{-x^2}H_{n-1}(x))'\,dx
\\
\text{\color{gray}(Ableitung)}
&=
e^{-x^2}H_{n-1}(x)
\end{align*}
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
