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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-03-09 21:34:40 +0100 |
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committer | Andreas Müller <andreas.mueller@ost.ch> | 2021-03-09 21:34:40 +0100 |
commit | 5f146c5b38ad91022dd0854bc52d1b5086bf9e13 (patch) | |
tree | 9c3a58ce41edd11a27ef45f6305d605510c7e0e7 /vorlesungen/slides/5/logarithmusreihe.tex | |
parent | potenzreihenmethode (diff) | |
download | SeminarMatrizen-5f146c5b38ad91022dd0854bc52d1b5086bf9e13.tar.gz SeminarMatrizen-5f146c5b38ad91022dd0854bc52d1b5086bf9e13.zip |
add two slides
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-rw-r--r-- | vorlesungen/slides/5/logarithmusreihe.tex | 53 |
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diff --git a/vorlesungen/slides/5/logarithmusreihe.tex b/vorlesungen/slides/5/logarithmusreihe.tex new file mode 100644 index 0000000..85ba0ef --- /dev/null +++ b/vorlesungen/slides/5/logarithmusreihe.tex @@ -0,0 +1,53 @@ +% +% logarithmus.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Logarithmusreihe} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Integralgleichung} +\vspace{-5pt} +\begin{align*} +\log(1+x)&=\int_0^x \frac{1}{1+t}\,dt +\\ +&\uncover<5->{= +\int_0^x +1-t+t^2-t^3+\dots\,dt +} +\\ +\uncover<6->{ +&= +x-\frac{x^2}2+\frac{x^3}{3}-\frac{x^4}{4}+\dots +} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Geometrische Reihe} +\vspace{-5pt} +\begin{align*} +\frac{1}{1-q}&=1+q+q^2+q^3+\dots +\\ +\uncover<3->{ +\frac{1}{1+q}&=1-q+q^2-q^3+\dots +} +\end{align*} +\uncover<4->{Konvergenzradius $1$} +\end{block}} +\end{column} +\end{columns} +\uncover<7->{% +\begin{block}{Matrix-Logarithmus} +Für $\operatorname{Sp}(A)\subset \{z\in\mathbb{C}\;|\;|z-1|<1\}$ konvergiert +\[ +\log A += +(A-I) - \frac12(A-I)^2 + \frac13(A-I)^3 - \frac14(A-I)^4 + \dots +\] +\end{block}} +\end{frame} |