%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} \usepackage{amsmath} \setcounter{MaxMatrixCols}{10} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Wednesday, February 10, 1999 13:29:48} %TCIDATA{LastRevised=Wednesday, November 28, 2007 09:43:48} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za III online - Laplaceova transform\U{e1}cia - Laplaceova transform\U{e1}cia\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{Laplaceova transform\'{a}cia} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{mcindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{mcindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{K4.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{K4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{K44.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{K44.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{Ot4.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{Ot4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{Cv4.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{Cv4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Ind.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Ind.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \section{Laplaceova transform\'{a}cia} Opera\v{c}n\'{y} po\v{c}et sa stal jednou z d\^{o}le\v{z}it\'{y}ch \v{c}ast% \'{\i} matematickej anal\'{y}zy, pova\v{z}uje sa za abecedu automatiz\'{a}% cie. Met\'{o}dy opera\v{c}n\'{e}ho po\v{c}tu sa pou\v{z}\'{\i}vaj\'{u} pri rie\v{s}en\'{\i} oby\v{c}ajn\'{y}ch diferenci\'{a}lnych rovn\'{\i}c a syst% \'{e}mov oby\v{c}ajn\'{y}ch diferenci\'{a}lnych rovn\'{\i}c. Mo\v{z}no ich v% \v{s}ak pou\v{z}i\v{t} aj pri rie\v{s}en\'{\i} parci\'{a}lnych diferenci\'{a}% lnych rovn\'{\i}c. Z\'{a}kladom tzv. \emph{opera\v{c}n\'{e}ho po\v{c}tu} je bu\v{d} \emph{% Carsonova transform\'{a}cia}, ktor\'{a} zobraz\'{\i} funkciu $f:\mathbf{% R\longrightarrow C},\,\,f\left( t\right) $ na funkciu komplexnej premennej $% K\left( p\right) $ pomocou formuly \begin{equation} K\left( p\right) =p\int_{0}^{\infty }f\left( t\right) e^{-pt}dt \tag{(1)} \end{equation}% alebo integr\'{a}lna \textsl{Laplaceova transform\'{a}cia,\label{4}} ktor% \'{a} zobraz\'{\i} funkciu $f:\mathbf{R\longrightarrow C},\,\,f\left( t\right) $ na funkciu komplexnej premennej $p$ - $F\left( p\right) $ pomocou formuly \begin{equation} \mathbf{L}\left[ f\left( t\right) \right] =F\left( p\right) =\int_{0}^{\infty }f\left( t\right) e^{-pt}dt \tag{(2)} \end{equation}% kde $t$ je re\'{a}lna premenn\'{a} a $p=\alpha +i\beta $ je komplexn\'{a} premenn\'{a}. Komplexn\'{a} alebo re\'{a}lna funkcia re\'{a}lneho argumentu \ $f\left( t\right) $\ sa naz\'{y}va \emph{origin\'{a}l}\ a jej obraz, funkcia komplexnej premennej $F\left( p\right) ,$\ sa na naz\'{y}va Laplaceov\'{y}m zobrazen\'{\i}m (transform\'{a}ciou) funkcie $f\left( t\right) .$ Zo vz\v{t}ahov (1) a (2) je vidie\v{t}, \v{z}e medzi Carsonovou a Laplaceovou transform\'{a}ciou plat\'{\i} vz\v{t}ah \begin{equation*} pF\left( p\right) =K\left( p\right) . \end{equation*}% \ Preto sa budeme zaobera\v{t} Laplaceovou transform\'{a}ciou. Na ozna\v{c}% enie toho, \v{z}e $F\left( p\right) $ je Laplaceovou transform\'{a}ciou $% f\left( t\right) ,$\ budeme pou\v{z}\'{\i}va\v{t} z\'{a}pis $F\left( p\right) =\mathbf{L}\left[ f\left( t\right) \right] .$ Pre ka\v{z}d\'{y} z t% \'{y}chto symbolick\'{y}ch z\'{a}pisov budeme mal\'{y}mi p\'{\i}smenami ozna% \v{c}ova\v{t} origin\'{a}ly, ve\v{l}k\'{y}mi p\'{\i}smenami ich Laplaceove obrazy. Laplaceova transform\'{a}cia je charakteristick\'{a} t\'{y}m, \v{z}e mnoh\'{y}m oper\'{a}ci\'{a}m s origin\'{a}lmi $f\left( t\right) $\ zodpovedaj% \'{u} omnoho jednoduch\v{s}ie oper\'{a}cie s ich obrazmi. Najsk\^{o}r definujeme, \v{c}o rozumieme pod pojmom (Laplaceov ) origin\'{a}l. \begin{definition} Funkciu $f:\mathbf{R\longrightarrow C}$ naz\'{y}vame \emph{(Laplaceov\'{y}m) origin\'{a}lom\label{5}} ak 1. funkcia $f\left( t\right) $ je po \v{c}astiach spojit\'{a}, 2. $f\left( t\right) =0$ pre $t<0,$ 3. existuje $M>0$ re\'{a}lne a $\alpha \in \mathbf{R}$ tak\'{e}, \v{z}e plat% \'{\i} odhad \begin{equation} \left| f\left( t\right) \right| \leq Me^{\alpha t},\,\forall t\in \mathbf{R} \tag{(3)} \end{equation}% Mno\v{z}inu v\v{s}etk\'{y}ch origin\'{a}lov ozna\v{c}\'{\i}me symbolom $% \widetilde{A}$. \end{definition} Ak je funkcia $f$ origin\'{a}l, tak hodnotu \begin{equation*} \alpha _{0}=\inf \left\{ \alpha \in \mathbf{R};\left| f\left( t\right) \right| \leq M_{\alpha }e^{\alpha t},\,\forall \,t\in \mathbf{R}\right\} \end{equation*}% naz\'{y}vame \emph{indexom rastu\label{6}} funkcie $f$. Origin\'{a}l $f$ m% \'{a} index rastu $\alpha =-\infty ,$ ak pre ka\v{z}d\'{e} $\alpha \in \mathbf{R}$ existuje kon\v{s}tanta $M_{\alpha }$ s vlastnos\v{t}ou \begin{equation*} \left| f\left( t\right) \right| \leq M_{\alpha }e^{\alpha t},\,\forall t\in \mathbf{R}. \end{equation*} \begin{example} \label{3}Funkcia $\eta :\mathbf{R\longrightarrow C}$ definovan\'{a} predpisom \begin{equation*} \eta \left( t\right) =\left\{ \begin{tabular}{ccc} $0$ & ak & $t<0$ \\ $1$ & ak & $t\geq 0$% \end{tabular}% \right. , \end{equation*}% naz\'{y}van\'{a} \emph{Heavisideova funkcia\label{7}} (funkcia jednotkov\'{e}% ho skoku) je origin\'{a}l s indexom rastu $\alpha _{0}=0,$ preto\v{z}e \begin{equation*} \left| \eta \left( t\right) \right| \leq 1e^{\alpha t},\,\forall \alpha >0,\,\forall t\in \mathbf{R}. \end{equation*} \end{example} Ak \v{l}ubovo\v{l}n\'{u} funkciu $f:\mathbf{R\longrightarrow C}$ vyn\'{a}sob% \'{\i}me Heavisideovou funkciou $\eta ,$ tak dostaneme funkciu rovn\'{u} nule na z\'{a}pornej \v{c}asti re\'{a}lnej osi. \begin{example} Ak $n\in \mathbf{N},$ tak funkcia $f\left( t\right) =\eta \left( t\right) t^{n}$ je origin\'{a}l s indexom rastu $\alpha _{0}=0.$ Vypl\'{y}va to zo vz% \v{t}ahu \begin{equation*} \lim_{t\longrightarrow \infty }\frac{t^{n}}{e^{\alpha t}}=0,\,\forall \,\alpha >0\Longrightarrow \left| \eta \left( t\right) t^{n}\right| \leq \varepsilon e^{\alpha t},\,\forall t\geq T\Longrightarrow \left| \eta \left( t\right) t^{n}\right| \leq Me^{\alpha t},\,\forall t\in \mathbf{R}. \end{equation*} \end{example} \begin{example} Funkcia $f:\mathbf{R\longrightarrow C},f\left( t\right) =\eta \left( t\right) \frac{1}{t-2}$ nie je origin\'{a}lom, preto\v{z}e v bode $t=2$ nem% \'{a} kone\v{c}n\'{u} limitu ani sprava, ani z\v{l}ava, teda nie je splnen% \'{a} vlastnos\v{t} (3) origin\'{a}lu. \end{example} \begin{example} Funkcia $f:\mathbf{R\longrightarrow C},\,f\left( t\right) =\eta \left( t\right) e^{t^{3}}$ nie je origin\'{a}l, preto\v{z}e% \begin{equation*} \lim_{t\longrightarrow \infty }\frac{e^{t^{3}}}{e^{\alpha t}}=\infty \,,\,\forall \alpha \in \mathbf{R,} \end{equation*}% \v{c}o implikuje, \v{z}e nie je splnen\'{a} vlastnos\v{t} (3) origin\'{a}lu. \end{example} Teraz uk\'{a}\v{z}eme, \v{z}e podmienka (3) zabezpe\v{c}uje absol\'{u}tnu konvergenciu integr\'{a}lu (2), ke\v{d} $\func{Re}p>\alpha _{0}.$ \begin{theorem} \label{2}Pre ka\v{z}d\'{y} origin\'{a}l $f\left( t\right) $ je zobrazenie $% F\left( p\right) $ definovan\'{e} vz\v{t}ahom (2) v polrovine $\func{Re}% p>\alpha _{0},$ kde $\alpha _{0}$ je index rastu funkcie $f\left( t\right) ,$ analytickou funkciou, pri\v{c}om \begin{equation*} F\,\,^{\prime }\left( p\right) =-\int_{0}^{\infty }tf\left( t\right) e^{-pt}dt,\,\func{Re}p>\alpha _{0}\,\text{\ a }\,\lim_{\func{Re}% p\longrightarrow \infty }F\left( p\right) =0. \end{equation*} \end{theorem} \begin{tabular}{|c|} \hline %TCIMACRO{\hyperref{\textbf{D\^{o}kaz}}{}{}{DO431.tex} }% %BeginExpansion \msihyperref{\textbf{D\^{o}kaz}}{}{}{DO431.tex} %EndExpansion \\ \hline \end{tabular}% \label{1} \begin{example} N\'{a}jdite obraz Heavisideovej funkcie. \end{example} \begin{solution} V %TCIMACRO{\hyperref{pr\'{\i}klade}{}{}{K43.tex#3} }% %BeginExpansion \msihyperref{pr\'{\i}klade}{}{}{K43.tex#3} %EndExpansion sme uk\'{a}zali, \v{z}e funkcia $\eta $ je origin\'{a}l s indexom rastu $% \alpha _{0}=0.$ Teda Laplaceov obraz bude existova\v{t} pre $\func{Re}p>0$ a m\'{a}me \begin{equation*} \mathbf{L}\left[ \eta \left( t\right) \right] =\int_{0}^{\infty }e^{-pt}dt=% \left[ -\frac{1}{p}e^{-pt}\right] _{0}^{\infty }=\frac{1}{p}.\,\square \end{equation*} \end{solution} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{mcindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{mcindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{K4.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{K4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{K44.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{K44.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{Ot4.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{Ot4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{Cv4.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{Cv4.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Ind.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Ind.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za III} \section{Laplaceova transform\'{a}cia} \end{document}