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author | Nicolas Tobler <nicolas.tobler@ost.ch> | 2022-05-30 00:06:46 +0200 |
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committer | Nicolas Tobler <nicolas.tobler@ost.ch> | 2022-05-30 00:06:46 +0200 |
commit | 65a3fc106c36dfd1750f8caf8b3d1b5fb0fe71f9 (patch) | |
tree | 30791dc17973690a6d761589de357c452ba9fa29 /vorlesungen/slides/hermite/normalhermite.tex | |
parent | Added content, presentation (diff) | |
parent | beispiel korrektur (diff) | |
download | SeminarSpezielleFunktionen-65a3fc106c36dfd1750f8caf8b3d1b5fb0fe71f9.tar.gz SeminarSpezielleFunktionen-65a3fc106c36dfd1750f8caf8b3d1b5fb0fe71f9.zip |
Merge branch 'master' of https://github.com/AndreasFMueller/SeminarSpezielleFunktionen
Diffstat (limited to 'vorlesungen/slides/hermite/normalhermite.tex')
-rw-r--r-- | vorlesungen/slides/hermite/normalhermite.tex | 88 |
1 files changed, 88 insertions, 0 deletions
diff --git a/vorlesungen/slides/hermite/normalhermite.tex b/vorlesungen/slides/hermite/normalhermite.tex new file mode 100644 index 0000000..16a314c --- /dev/null +++ b/vorlesungen/slides/hermite/normalhermite.tex @@ -0,0 +1,88 @@ +% +% 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 |