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\QTR{small}{Matematick\U{e1} anal\U{fd}za III online - Laplaceova transform\U{e1}cia - Vlastnosti Laplaceovej transform\U{e1}cie\dotfill \thepage }}
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\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|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{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{K43.tex}}%
%BeginExpansion
\msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{K43.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{K45.tex}}%
%BeginExpansion
\msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{K45.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{Vlastnosti Laplaceovej transform\'{a}cie}
\begin{theorem}
\label{13}(Line\'{a}rnos\v{t} Laplaceovej transform\'{a}cie) Nech $f\left(
t\right) ,g\left( t\right) $ s\'{u} origin\'{a}ly, $\mathbf{L}\left[ f\right]
=F$ a $\mathbf{L}\left[ g\right] =G,$ potom pre ka\v{z}d\'{e} $\lambda
,\vartheta \in \mathbf{C}$ plat\'{\i}
\begin{equation*}
\mathbf{L}\left[ \lambda f\left( t\right) +\vartheta g\left( t\right) \right]
=\lambda F\left( p\right) +\vartheta G\left( p\right) .
\end{equation*}
\end{theorem}
\begin{tabular}{|c|}
\hline
\textbf{%
%TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO441.tex}}%
%BeginExpansion
\msihyperref{D\^{o}kaz}{}{}{DO441.tex}%
%EndExpansion
} \\ \hline
\end{tabular}%
\label{1}
\begin{theorem}
\label{14}(O deriv\'{a}cii obrazu) Nech $f$ je origin\'{a}l s indexom rastu $%
\alpha _{0},$ $F=\mathbf{L}\left( f\right) $ a $n\in \mathbf{N}.$ Potom aj
funkcia $g:\mathbf{R}\longrightarrow \mathbf{C},$ $g\left( t\right)
=t^{n}f\left( t\right) $ je origin\'{a}l s indexom rastu $\alpha _{0}$ a
\begin{equation*}
\mathbf{L}\left[ t^{n}f\left( t\right) \right] =(-1)^{n}F^{(n)}\left(
p\right) ,\,\func{Re}p>\alpha _{0}.
\end{equation*}
\end{theorem}
\begin{tabular}{|c|}
\hline
\textbf{%
%TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO442.tex}}%
%BeginExpansion
\msihyperref{D\^{o}kaz}{}{}{DO442.tex}%
%EndExpansion
} \\ \hline
\end{tabular}%
\label{2}
\begin{example}
N\'{a}jdite obraz funkcie $f:\mathbf{R\longrightarrow C},$ $f\left( t\right)
=\eta \left( t\right) t^{n},\,n=1,2,3,\dots $
\end{example}
\begin{solution}
Pod\v{l}a predch\'{a}dzaj\'{u}cej vety o deriv\'{a}cii obrazu s\'{u} funkcie
$f:\mathbf{R\longrightarrow C},$ $f\left( t\right) =\eta \left( t\right)
t^{n},\,n=1,2,3,\dots $ origin\'{a}ly s indexom rastu $\alpha _{0}=0,$ pri%
\v{c}om
\begin{equation*}
\mathbf{L}\left[ t^{n}\right] =\frac{n!}{p^{n+1}},\,\func{Re}p>0.\,\square
\end{equation*}
\end{solution}
\begin{theorem}
\label{15}(O posunut\'{\i} v obraze) Nech $f$ je origin\'{a}l s indexom
rastu $\alpha _{0},$ $a\in \mathbf{C}$ a $\mathbf{L}\left( f\right) =F.$
Potom funkcia
\begin{equation*}
g:\mathbf{R\longrightarrow C},\,g\left( t\right) =e^{at}f\left( t\right)
\end{equation*}%
je origin\'{a}l s indexom rastu $\alpha _{0}+\func{Re}a,$ pri\v{c}om
\begin{equation*}
\mathbf{L}\left[ e^{at}f\left( t\right) \right] =F(p-a),\,\func{Re}p>\alpha
_{0}+\func{Re}a.
\end{equation*}
\end{theorem}
\begin{tabular}{|c|}
\hline
%TCIMACRO{\hyperref{\textbf{D\^{o}kaz}}{}{}{DO443.tex} }%
%BeginExpansion
\msihyperref{\textbf{D\^{o}kaz}}{}{}{DO443.tex}
%EndExpansion
\\ \hline
\end{tabular}%
\label{3}
\begin{example}
N\'{a}jdite obraz funkci\'{\i} $f,h:\mathbf{R\longrightarrow C},$ $f\left(
t\right) =e^{at},\,h\left( t\right) =t^{n}e^{at},\,n=1,2,3,\dots $
\end{example}
\begin{solution}
Pod\v{l}a vety o posunut\'{\i} v obraze m\'{a}me
\begin{equation*}
\mathbf{L}\left[ e^{at}\right] =\frac{1}{p-a},\,\func{Re}p>\func{Re}a.
\end{equation*}%
\begin{equation*}
\mathbf{L}\left[ t^{n}e^{at}\right] =\frac{n!}{\left( p-a\right) ^{n+1}},\,%
\func{Re}p>\func{Re}a.\,\square
\end{equation*}
\end{solution}
\begin{example}
N\'{a}jdite obraz funkci\'{\i} $\cos t,\sin t.$
\end{example}
\begin{solution}
Plat\'{\i}
\begin{equation*}
\mathbf{L}\left[ \cos t\right] =\mathbf{L}\left[ \frac{e^{it}+e^{-it}}{2}%
\right] =\frac{1}{2}\left( \frac{1}{p-i}+\frac{1}{p+i}\right) =\frac{p}{%
p^{2}+1},\,\func{Re}p>0.
\end{equation*}%
\begin{equation*}
\mathbf{L}\left[ \sin t\right] =\mathbf{L}\left[ \frac{e^{it}-e^{-it}}{2i}%
\right] =\frac{1}{2i}\left( \frac{1}{p-i}-\frac{1}{p+i}\right) =\frac{1}{%
p^{2}+1},\,\func{Re}p>0.\,\square
\end{equation*}
\end{solution}
\begin{theorem}
\label{16}(Veta o zmene mierky) Nech $f$ je origin\'{a}l s indexom rastu $%
\alpha _{0},$ $b\in \mathbf{R}^{+}$ a $\mathbf{L}\left[ f\right] =F.$ Potom
\begin{equation*}
\mathbf{L}\left[ f\left( bt\right) \right] =\frac{1}{b}F\left( \frac{p}{b}%
\right) ,\,\func{Re}\left( \frac{p}{b}\right) >\alpha _{0}.
\end{equation*}
\end{theorem}
\begin{tabular}{|c|}
\hline
\textbf{%
%TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO444.tex}}%
%BeginExpansion
\msihyperref{D\^{o}kaz}{}{}{DO444.tex}%
%EndExpansion
} \\ \hline
\end{tabular}%
\label{4}
\begin{example}
N\'{a}jdite obraz funkci\'{\i} $\cos \left( \omega t\right) ,\sin \left(
\omega t\right) ,\,\omega >0.$
\end{example}
\begin{solution}
Plat\'{\i}
\begin{equation*}
\mathbf{L}\left[ \cos \left( \omega t\right) \right] =\frac{1}{\omega }\frac{%
\frac{p}{\omega }}{\left( \frac{p}{\omega }\right) ^{2}+1}=\frac{p}{%
p^{2}+\omega ^{2}},\,\func{Re}p>0.
\end{equation*}%
podobne
\begin{equation*}
\mathbf{L}\left[ \sin \left( \omega t\right) \right] =\frac{\omega }{%
p^{2}+\omega ^{2}},\,\func{Re}p>0.\,\square
\end{equation*}
\end{solution}
\subsection{Posun v origin\'{a}le.}
Ak $f:\mathbf{R\longrightarrow C}$ je origin\'{a}l s indexom rastu $\alpha
_{0}$ a $\tau \in \mathbf{R}$ definujeme funkciu $f_{\tau }:\mathbf{%
R\longrightarrow C}$ vz\v{t}ahom $f_{\tau }\left( t\right) =\eta (t-\tau
)f(t-\tau ).$
\begin{theorem}
\label{10}(Veta o posune v origin\'{a}le) Nech $f$ je origin\'{a}l s indexom
rastu $\alpha _{0},$ $\tau \in \mathbf{R}$ a $\mathbf{L}\left[ f\right] =F.$
Potom aj posunut\'{a} funkcia $f_{\tau },$ $\tau >0$ je origin\'{a}l s
indexom rastu $\alpha _{0}$ a plat\'{\i}
\begin{equation*}
\mathbf{L}\left[ \eta (t-\tau )f(t-\tau )\right] =e^{-\tau p}F\left(
p\right) ,\,\func{Re}p>\alpha _{0}.
\end{equation*}
\end{theorem}
\begin{tabular}{|c|}
\hline
\textbf{%
%TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO445.tex}}%
%BeginExpansion
\msihyperref{D\^{o}kaz}{}{}{DO445.tex}%
%EndExpansion
} \\ \hline
\end{tabular}%
\label{5}
Pomocou vety o obraze posunutej funkcie m\^{o}\v{z}me dosta\v{t} obrazy
funkci\'{\i}, ktor\'{e} sa \v{c}asto pou\v{z}\'{\i}vaj\'{u} v
elektrotechnike ako s\'{u} napr\'{\i}klad \emph{kone\v{c}n\'{e} impulzy.}
\begin{example}
N\'{a}jdite obraz obd\'{l}\v{z}nikov\'{e}ho impulzu - funkcie $f,$ ktor\'{a}
m\'{a} tvar
\begin{equation*}
f\left( t\right) =\left\{
\begin{tabular}{ccc}
$0$ & ak & $t<0$ \\
$M$ & ak & $0\leq ta$%
\end{tabular}%
\right. .
\end{equation*}
\end{example}
\begin{solution}
Funkciu $f$ mo\v{z}no vyjadri\v{t} v tvare
\begin{equation*}
f\left( t\right) =M\left[ \eta \left( t-a\right) -\eta \left( t-b\right) %
\right] ,\,t\in \mathbf{R}.
\end{equation*}%
Potom
\begin{equation*}
\mathbf{L}\left[ f\left( t\right) \right] =\frac{M}{p}\left(
e^{-ap}-e^{-bp}\right) ,\,\func{Re}p>0.\,\square
\end{equation*}
\end{solution}
\begin{example}
N\'{a}jdite obraz lichobe\v{z}n\'{\i}kov\'{e}ho impulzu - funkcie $f,$ ktor%
\'{a} m\'{a} tvar
\begin{equation*}
f\left( t\right) =\left\{
\begin{tabular}{ccc}
$0$ & ak & $t0.\,\square
\end{equation*}
\end{solution}
\begin{example}
\label{12}N\'{a}jdite obraz s\'{\i}nusov\'{e}ho impulzu - funkcie $f,$ ktor%
\'{a} m\'{a} tvar
\begin{equation*}
f\left( t\right) =\left\{
\begin{tabular}{ccc}
$0$ & ak & $t<0$ \\
$\sin t$ & ak & $0\leq t<\pi $ \\
$0$ & ak & $\pi \leq t$%
\end{tabular}%
.\right.
\end{equation*}
\end{example}
\begin{solution}
Funkciu $f$ mo\v{z}no vyjadri\v{t} v tvare
\begin{equation*}
f\left( t\right) =\eta \left( t\right) \sin t+\eta \left( t-\pi \right) \sin
\left( t-\pi \right) .
\end{equation*}%
Potom
\begin{equation*}
\mathbf{L}\left[ f\left( t\right) \right] =F\left( p\right) =\left(
1+e^{-\pi p}\right) \frac{1}{p^{2}+1},\,\func{Re}p>0.\,\square
\end{equation*}
\end{solution}
\subsection{Obraz periodickej funkcie.}
\begin{theorem}
\label{11}(Veta o obraze periodickej funkcie) Ak $f_{1}:\mathbf{%
R\longrightarrow C}$ je nenulov\'{a} po \v{c}astiach periodick\'{a} funkcia
s periodou $T$, tak funkcia $f:\mathbf{R\longrightarrow C,}$ $f\left(
t\right) =\eta \left( t\right) f_{1}\left( t\right) $ je origin\'{a}l s
indexom rastu $\alpha _{0}=0$ a plat\'{\i} vz\v{t}ah
\begin{equation*}
\mathbf{L}\left[ f\left( t\right) \right] =\frac{\int_{0}^{T}f\left(
t\right) e^{-pt}dt}{1-e^{-pT}}=\frac{F_{T}\left( p\right) }{1-e^{-pT}},\,%
\func{Re}p>0,
\end{equation*}%
kde $F_{T}$ je obraz kone\v{c}n\'{e}ho impulzu $f_{\left\langle
0,T\right\rangle }:$
\begin{equation*}
f_{\left\langle 0,T\right\rangle }\left( t\right) =\left\{
\begin{tabular}{ccc}
$0$ & ak & $t<0$ \\
$f\left( t\right) $ & ak & $0\leq t