%
% 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}