\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amsmath} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Friday, January 24, 2003 21:48:05} %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{Vlastnosti Laplaceovej transform\'{a}cie} \subsection{D\^{o}kaz vety o inverznej Laplaceovej transform\'{a}cii.} \textbf{D\^{o}kaz: }Nech $\gamma _{r}$ je jednoduch\'{a} uzavret\'{a} kladne orientovan\'{a} krivka, ktor\'{a} sa sklad\'{a} s polkru\v{z}nice $\varphi _{r}$ so stredom v bode $x\in \mathbf{R},$ $x>\alpha _{0}$ a \'{u}se\v{c}ky $% \overline{AB}$ sp\'{a}jaj\'{u}cej body $A=x-ir,\,B=x+ir.$ Vyberieme $r$ tak, aby v\v{s}etky izolovan\'{e} singul\'{a}rne body $p_{1},\dots ,p_{n}$ le\v{z}% ali v $Int\gamma _{r}.$ Potom \begin{equation*} \frac{1}{2\pi i}\int_{\gamma _{r}}F\left( p\right) e^{pt}dp=\sum_{k=1}^{n}% \limfunc{res}_{p_{k}}\left[ F\left( p\right) e^{pt}\right] . \end{equation*}% Ak vezmeme do \'{u}vahy, \v{z}e funkcie $F,\,G_{t}=F\left( p\right) e^{pt}$ maj\'{u} tak\'{e} ist\'{e} izolovan\'{e} singul\'{a}rne body, pre funkciu $f$ m\'{a}me \begin{equation*} f\left( t\right) =\frac{1}{2\pi i}\lim_{r\longrightarrow \infty }\int_{x-ir}^{x+ir}F\left( p\right) e^{pt}dp,\,x>\alpha _{0},\,r\in \mathbf{R% }. \end{equation*}% Aby sme to dok\'{a}zali, sta\v{c}\'{\i} uk\'{a}za\v{t}, \v{z}e \begin{equation*} \lim_{r\longrightarrow \infty }\int_{\varphi _{r}}F\left( p\right) e^{pt}dp=0\,\text{\ \ pre ka\v{z}d\'{e} }\,t>0. \end{equation*}% Nech $M_{r}=\max \left| F\left( p\right) \right| ,$ potom $% \lim_{r\longrightarrow \infty }M_{r}=0$ (predpoklad b)). Potom% \begin{equation*} \left| \int_{\varphi _{r}}F\left( p\right) e^{pt}dp\right| =\left| \int_{% \frac{\pi }{2}}^{\frac{3\pi }{2}}F\left( x+re^{is}\right) e^{x+re^{is}}ds\right| \leq rM_{r}e^{xt}\int_{\frac{\pi }{2}}^{\frac{3\pi }{2% }}e^{rt\cos s}ds=2rM_{r}e^{xt}\int_{0}^{\frac{\pi }{2}}e^{-rt\cos s}ds= \end{equation*}% \begin{equation*} =2rM_{r}e^{xt}\int_{0}^{\frac{\pi }{2}}e^{-rt\sin s}ds\leq 2rM_{r}e^{xt}\int_{0}^{\frac{\pi }{2}}e^{-\frac{2rts}{\pi }}ds=\frac{\pi }{t}% M_{r}e^{xt}\left( 1-e^{-tr}\right) \end{equation*}% pri\v{c}om sme pou\v{z}ili zn\'{a}mu nerovnos\v{t} \begin{equation*} \sin s\geq \frac{2}{\pi }s\text{ \ pre ka\v{z}d\'{e} }\,s\in \left\langle 0,% \frac{\pi }{2}\right\rangle . \end{equation*}% Ak pou\v{z}ijeme \begin{equation*} \lim_{r\longrightarrow \infty }M_{r}=0,\,\lim_{r\longrightarrow \infty }e^{-rt}=0 \end{equation*}% dostaneme \begin{equation*} \lim_{r\longrightarrow \infty }\int_{\varphi _{r}}F\left( p\right) e^{pt}dp=0\,\text{\ \ pre ka\v{z}d\'{e} }\,t>0.\,\blacksquare \end{equation*} \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{K45.tex#1}} \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za III} \end{document}