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%F=36,\PARA{038<p type="texpara" tag="Body Text" >\QTR{small}{Matematick\U{e1} anal\U{fd}za III online - D\U{f4}kazy\dotfill \thepage }}
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\input{tcilatex}

\begin{document}

\author{A. U. Thor}
\title{Lab Report}
\date{The Date }
\maketitle

\begin{abstract}
A Laboratory report created with Scientific Notebook
\end{abstract}

\section{Komplexn\'{e} \v{c}\'{\i}sla a funkcie komplexnej premennej}

\subsection{D\^{o}kaz vety}

\textbf{D\^{o}kaz: }a) Pre 
\[
\left| z-a\right| <\left| z_{0}-a\right| \Longrightarrow \left| \frac{z-a}{%
z_{0}-a}\right| <1. 
\]%
Potom 
\begin{equation}
\left| c_{n}(z-a)^{n}\right| =\left| c_{n}(z_{0}-a)^{n}\right| \left| \frac{%
z-a}{z_{0}-a}\right| ^{n}<\left| c_{n}(z_{0}-a)^{n}\right|  \tag{(a)}
\end{equation}%
Pod\v{l}a predpokladu rad $\sum_{n=0}^{\infty }c_{n}\left( z_{0}-a\right)
^{n}$ konverguje, potom $\lim_{n\longrightarrow \infty }c_{n}\left(
z_{0}-a\right) ^{n}=0,$ \v{c}o implikuje, \v{z}e existuje $C>0$ tak\'{e}, 
\v{z}e 
\[
\left| c_{n}(z_{0}-a)^{n}\right| <C,\,n=0,1,2,\dots 
\]%
Potom z (a) m\'{a}me 
\[
\left| c_{n}(z-a)^{n}\right| <C\left| \frac{z-a}{z_{0}-a}\right| ^{n} 
\]%
a preto\v{z}e geometrick\'{y} rad 
\[
\sum_{n=0}^{\infty }\left| \frac{z-a}{z_{0}-a}\right| ^{n} 
\]%
konverguje tak rad $\sum_{n=0}^{\infty }c_{n}\left( z-a\right) ^{n}$ absol%
\'{u}tne konverguje.

b) Nech $\left| z-a\right| =\rho \leq r<\left| z_{0}-a\right| .$ Preto\v{z}e
rad $\sum_{n=0}^{\infty }c_{n}\left( z_{0}-a\right) ^{n}$ konverguje, tak aj
rad $\sum_{n=0}^{\infty }\left| c_{n}\right| \rho ^{n}$ konverguje. Z odhadu 
\[
\left| c_{n}(z-a)^{n}\right| =\left| c_{n}\right| \left| z-a\right| ^{n}\leq
\left| c_{n}\right| \rho ^{n}, 
\]%
pou\v{z}it\'{\i}m Weierstrassovho krit\'{e}ria dost\'{a}vame, \v{z}e rad $%
\sum_{n=0}^{\infty }c_{n}\left( z-a\right) ^{n}$ je rovnomerne konvergentn%
\'{y}. $\blacksquare $

\begin{center}
\begin{tabular}{|c|}
\hline
{\small \hyperref{Sp\"{a}\v{t}}{}{}{K13.tex#3}} \\ \hline
\end{tabular}
\end{center}

\rule{6.5in}{0.04in}

\textsl{Matematick\'{a} anal\'{y}za III}

\end{document}
