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%TCIDATA{Created=Monday, June 25, 2001 17:45:57}
%TCIDATA{LastRevised=Friday, January 24, 2003 22:37:33}
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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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\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{Analytickos\v{t} s\'{u}\v{c}tu mocninov\'{e}ho radu}

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

\textbf{D\^{o}kaz: }Z \hyperref{vety}{}{}{K13.tex#8} plynie, \v{z}e $f\left(
z\right) $ je spojit\'{a} funkcia na $C,$ \v{c}o znamen\'{a}, \v{z}e integr%
\'{a}l je definovan\'{y} (existuje). Ozna\v{c}me 
\begin{equation}
r_{n}=\int_{C}f\left( z\right) dz-\sum_{k=1}^{n}\int_{C}f_{k}\left( z\right)
dz=\int_{C}\left[ f\left( z\right) -\sum_{k=1}^{n}f_{k}\left( z\right) %
\right] dz\,  \tag{(i)}
\end{equation}%
Preto\v{z}e rad $\sum_{n=1}^{\infty }f_{n}\left( z\right) $ je rovnomerne
konvergentn\'{y} na $C,$ potom pre ka\v{z}d\'{e} $\varepsilon >0$ existuje $%
n_{0}\in \mathbf{N}$ tak\'{e}, \v{z}e pre ka\v{z}d\'{e} $n\geq n_{0}$ a ka%
\v{z}d\'{e} 
\[
z\in C,\,\left| f\left( z\right) -\sum_{k=1}^{n}f_{k}\left( z\right) \right|
<\varepsilon 
\]%
potom 
\[
\left| r_{n}\right| =\left| \int_{C}\left[ f\left( z\right)
-\sum_{k=1}^{n}f_{k}\left( z\right) \right] dz\right| \leq \int_{C}\left| %
\left[ f\left( z\right) -\sum_{k=1}^{n}f_{k}\left( z\right) \right] \right|
dz\leq \varepsilon d(C), 
\]%
t.j. $\lim_{n\longrightarrow \infty }r_{n}=0$ a (i) plat\'{\i}. $%
\blacksquare $

\begin{center}
\begin{tabular}{|c|}
\hline
{\small \hyperref{Sp\"{a}\v{t}}{}{}{K33.tex#4}} \\ \hline
\end{tabular}
\end{center}

\rule{6.5in}{0.04in}

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

\end{document}
