\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Friday, January 24, 2003 22:57:13} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za III online - D\U{f4}kazy\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \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{Cauchyho integr\'{a}lna formula} \subsection{D\^{o}kaz vety} \textbf{D\^{o}kaz: }Dok\'{a}\v{z}eme t\'{u}to vetu ak budeme okrem predpokladov vety predpoklada\v{t}, \v{z}e deriv\'{a}cia $f\,^{\prime }\left( z\right) $\ je spojit\'{a} v oblasti $D.% \CustomNote{Note}{% Tento dopl\v{n}uj\'{u}ci predpoklad nie je umel\'{y}, preto\v{z}e \hyperref{% nesk\^{o}r uk\'{a}\v{z}eme}{}{}{K33.tex#7}, \v{z}e analytick\'{a} funkcia m% \'{a} v oblasti deriv\'{a}cie \v{l}ubovo\v{l}n\'{e}ho r\'{a}du a teda jej deriv\'{a}cia prv\'{e}ho r\'{a}du je spojit\'{a}.}$\ Pou\v{z}ijeme Greenovu vetu: \[ \int_{C}P\left( x,y\right) dx+Q\left( x,y\right) dy=\iint_{A}\left( \frac{% \partial Q}{\partial x}-\frac{\partial P}{\partial y}\right) dxdy, \]% kde $A$\ je oblas\v{t} ohrani\v{c}en\'{a} krivkou $C.$\ Sk\'{u}majme rovnos% \v{t}: \begin{equation} \int_{C}f\left( z\right) dz=\int_{C}u\left( x,y\right) dx-v\left( x,y\right) dy+i\int_{C}v\left( x,y\right) dx+u\left( x,y\right) dy. \tag{(a)} \end{equation}% Z analytickosti funkcie $f\left( z\right) $\ v oblasti $D$\ plynie, \v{z}e funkcie $u\left( x,y\right) ,\,v\left( x,y\right) $ sp\'{l}\v{n}aj\'{u} Cauchyho - Riemannove rovnosti: \[ \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\,\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}, \]% $\,$a z predpokladu o spojitosti $f\,^{\prime }\left( z\right) $\ dostaneme \[ \int_{C}f\left( z\right) dz=\iint_{A}\left( \frac{\partial v}{\partial x}+% \frac{\partial u}{\partial y}\right) dxdy+i\iint_{A}\left( \frac{\partial u}{% \partial x}-\frac{\partial v}{\partial y}\right) dxdy=0. \]% Tak sme vetu dok\'{a}zali. $\,\blacksquare $ \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{K32.tex#7}} \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za III} \end{document}