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author | LordMcFungus <mceagle117@gmail.com> | 2022-07-22 21:28:45 +0200 |
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committer | GitHub <noreply@github.com> | 2022-07-22 21:28:45 +0200 |
commit | 23f17598c1742c70f442b94044a20aa821022c5a (patch) | |
tree | a945540ee6a4e86b37df2f01e3a91584b4797c4f /vorlesungen/slides/hermite/normalintegrale.tex | |
parent | Merge pull request #2 from AndreasFMueller/master (diff) | |
parent | Merge pull request #25 from JODBaer/master (diff) | |
download | SeminarSpezielleFunktionen-23f17598c1742c70f442b94044a20aa821022c5a.tar.gz SeminarSpezielleFunktionen-23f17598c1742c70f442b94044a20aa821022c5a.zip |
Merge pull request #3 from AndreasFMueller/master
update
Diffstat (limited to 'vorlesungen/slides/hermite/normalintegrale.tex')
-rw-r--r-- | vorlesungen/slides/hermite/normalintegrale.tex | 57 |
1 files changed, 57 insertions, 0 deletions
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 |