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author | Runterer <37069007+Runterer@users.noreply.github.com> | 2022-05-26 15:44:34 +0200 |
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committer | GitHub <noreply@github.com> | 2022-05-26 15:44:34 +0200 |
commit | 97f9df441a8cf9e86545a7db2e0c1e935937638f (patch) | |
tree | 43adb2438e1e316c34997b13bb2d0fa3044f8f1b /vorlesungen/slides/hermite/loesung.tex | |
parent | added tikz -> kudos nic (diff) | |
parent | Merge branch 'master' of github.com:AndreasFMueller/SeminarSpezielleFunktionen (diff) | |
download | SeminarSpezielleFunktionen-97f9df441a8cf9e86545a7db2e0c1e935937638f.tar.gz SeminarSpezielleFunktionen-97f9df441a8cf9e86545a7db2e0c1e935937638f.zip |
Merge branch 'AndreasFMueller:master' into master
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-rw-r--r-- | vorlesungen/slides/hermite/loesung.tex | 56 |
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diff --git a/vorlesungen/slides/hermite/loesung.tex b/vorlesungen/slides/hermite/loesung.tex new file mode 100644 index 0000000..7d4741f --- /dev/null +++ b/vorlesungen/slides/hermite/loesung.tex @@ -0,0 +1,56 @@ +% +% 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} +\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} +\begin{block}{Allgemein} +\begin{align*} +\int P(x)e^{-x^2}\,dx +&= +\int \sum_{k=0}^n a_kH_k(x)e^{-x^2}\,dx +\\ +&= +\sum_{k=0}^n +a_k +\int +H_k(x)e^{-x^2}\,dx +\\ +&= +a_0\operatorname{erf}(x) + C +\\ +&\hspace*{2mm} + \sum_{k=1}^n a_k\int H_k(x)e^{-x^2}\,dx +\end{align*} +\end{block} +\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 |