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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Wed Mar 26 12:35:25 1997}
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\input{tcilatex}
\begin{document}


\section{Table of Laplace Transforms\label{table of integrals}\qquad}

\paragraph{
\[
F\left( s\right) =L\left\{ f\right\} \left( s\right) =\int_{0}^{\infty
}e^{-st}f(t)dt 
\]
\protect\medskip}

\subsubsection{\ Powers}

\begin{center}
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
f\right\} \left( s\right) $ \\ \hline\hline
$1$ & $\dfrac{1}{s}$\medskip \\ 
$t$ & $\dfrac{1}{s^{2}}$\medskip \\ 
$t^{n}$ & $\dfrac{n!}{s^{n+1}}$\medskip , $n$ a positive integer \\ 
$t^{-1/2}\quad $ & $\sqrt{\dfrac{\pi }{s}}$\medskip \\ 
$t^{1/2}$ & $\dfrac{\sqrt{\pi }}{2s^{3/2}}$\medskip \\ 
$t^{\alpha }$ & $\dfrac{\Gamma \left( \alpha +1\right) }{s^{\alpha +1}}$, $%
\alpha >-1$%
\end{tabular}
\medskip
\end{center}

\subsubsection{\ Trigonometric Functions}

\begin{center}
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
f\right\} \left( s\right) $ \\ \hline\hline
$\sin kt$ & $\dfrac{k}{s^{2}+k^{2}}$\medskip \\ 
$\cos kt$ & $\dfrac{s}{s^{2}+k^{2}}$\medskip \\ 
$\sin ^{2}kt$ & $\dfrac{2k^{2}}{s\left( s^{2}+4k^{2}\right) }$\medskip \\ 
$\cos ^{2}kt$ & $\dfrac{s^{2}+2k^{2}}{s\left( s^{2}+4k^{2}\right) }$\medskip
\\ 
$t\sin kt$ & $\dfrac{2ks}{\left( s^{2}+k^{2}\right) ^{2}}$\medskip \\ 
$t\cos kt$ & $\dfrac{s^{2}-k^{2}}{\left( s^{2}+k^{2}\right) ^{2}}$\medskip
\\ 
$\dfrac{2\left( 1-\cos kt\right) }{t}\quad $ & $\ln \dfrac{s^{2}+k^{2}}{s^{2}%
}$\medskip \\ 
$\dfrac{\sin at}{t}$ & $\arctan \left( \dfrac{a}{s}\right) $%
\end{tabular}
\qquad \qquad 
\begin{tabular}{ll}
$f\left( t\right) $ & $F\left( s\right) =L\left\{ f\right\} \left( s\right) 
\rule[-0.07in]{0in}{0.25in}$ \\ \hline\hline
$\sin kt+kt\cos kt$ & $\dfrac{2ks^{2}}{\left( s^{2}+k^{2}\right) ^{2}}$%
\medskip \\ 
$\sin kt-kt\cos kt$ & $\dfrac{2k^{3}}{\left( s^{2}-k^{2}\right) ^{2}}$%
\medskip \\ 
$1-\cos kt$ & $\dfrac{k^{2}}{s\left( s^{2}+k^{2}\right) }$\medskip \\ 
$kt-\sin kt$ & $\dfrac{k^{3}}{s^{2}\left( s^{2}+k^{2}\right) }$\medskip \\ 
$\dfrac{a\sin bt-b\sin at}{ab\left( a^{2}-b^{2}\right) }\quad $ & $\dfrac{1}{%
\left( s^{2}+a^{2}\right) \left( s^{2}+b^{2}\right) }$\medskip \\ 
$\dfrac{\cos bt-\cos at}{a^{2}-b^{2}}$ & $\dfrac{s}{\left(
s^{2}+a^{2}\right) \left( s^{2}+b^{2}\right) }$\medskip \\ 
$\dfrac{\sin at\cos bt}{t}$ & $\dfrac{1}{2}\arctan \dfrac{a+b}{s}+\dfrac{1}{2%
}\arctan \dfrac{a-b}{s}$ \\ 
& 
\end{tabular}
\medskip
\end{center}

\subsubsection{\ Hyperbolic Functions}

\begin{center}
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
f\right\} \left( s\right) $ \\ \hline\hline
$\sinh kt$ & $\dfrac{k}{s^{2}-k^{2}}\medskip $ \\ 
$\cosh kt$ & $\dfrac{s}{s^{2}-k^{2}}\medskip $ \\ 
$\sinh ^{2}kt$ & $\dfrac{2k^{2}}{s\left( s^{2}-4k^{2}\right) }\medskip $ \\ 
$\cosh ^{2}kt\quad $ & $\dfrac{s^{2}-2k^{2}}{s\left( s^{2}-4k^{2}\right) }$%
\end{tabular}
\qquad \qquad 
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
f\right\} \left( s\right) $ \\ \hline\hline
$t\sinh kt$ & $\dfrac{2ks}{\left( s^{2}-k^{2}\right) ^{2}}\medskip $ \\ 
$t\cosh kt$ & $\dfrac{s^{2}+k^{2}}{\left( s^{2}-k^{2}\right) ^{2}}\medskip $
\\ 
$\dfrac{2\left( 1-\cosh kt\right) }{t}$\quad & $\ln \dfrac{s^{2}-k^{2}}{s^{2}%
}$ \\ 
& 
\end{tabular}
\medskip
\end{center}

\subsubsection{\ Exponential Functions}

\begin{center}
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
f\right\} \left( s\right) $ \\ \hline\hline
$e^{at}$ & $\dfrac{1}{s-a}\medskip $ \\ 
$te^{at}$ & $\dfrac{1}{\left( s-a\right) ^{2}}\medskip $ \\ 
$t^{n}e^{at}$ & $\dfrac{n!}{\left( s-a\right) ^{n+1}}\medskip $, $n$ a
positive integer \\ 
$\dfrac{e^{bt}-e^{at}}{t}\quad $ & $\ln \frac{s-a}{s-b}$%
\end{tabular}
\qquad \qquad 
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
f\right\} \left( s\right) $ \\ \hline\hline
$\dfrac{1}{\sqrt{\pi t}}e^{-a^{2}/4t}$ & $\dfrac{e^{-a\sqrt{s}}}{\sqrt{s}}%
\medskip $ \\ 
$\dfrac{a}{2\sqrt{\pi t^{3}}}e^{-a^{2}/4t}\quad $ & $e^{-a\sqrt{s}}\medskip $
\\ 
$\dfrac{e^{at}-e^{bt}}{a-b}$ & $\dfrac{1}{\left( s-a\right) \left(
s-b\right) }\medskip $ \\ 
$\dfrac{ae^{at}-be^{bt}}{a-b}$ & $\dfrac{s}{\left( s-a\right) \left(
s-b\right) }$%
\end{tabular}
\medskip
\end{center}

\subsubsection{\ Exponential and Trigonometric Functions}

\begin{center}
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
f\right\} \left( s\right) $ \\ \hline\hline
$e^{at}\sin kt$ & $\dfrac{k}{\left( s-a\right) ^{2}+k^{2}}$\medskip \\ 
$e^{at}\cos kt\quad $ & $\dfrac{s-a}{\left( s-a\right) ^{2}+k^{2}}$%
\end{tabular}
\medskip
\end{center}

\subsubsection{\ Exponential and Hyperbolic Functions}

\begin{center}
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
ft\right\} \left( s\right) $ \\ \hline\hline
$e^{at}\sinh kt$ & $\dfrac{k}{\left( s-a\right) ^{2}-k^{2}}$\medskip \\ 
$e^{at}\cosh kt\quad $ & $\dfrac{s-a}{\left( s-a\right) ^{2}-k^{2}}$%
\end{tabular}
\medskip
\end{center}

\subsubsection{\ Trigonometric and Hyperbolic Functions}

\begin{center}
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
f\right\} \left( s\right) $ \\ \hline\hline
$\sin kt\sinh kt$ & $\dfrac{2k^{2}s}{s^{2}+4k^{4}}$\medskip \\ 
$\sin kt\cosh kt$ & $\dfrac{k\left( s^{2}+2k^{2}\right) }{s^{4}+4k^{4}}$%
\medskip \\ 
$\cos kt\sinh kt$ & $\dfrac{k\left( s^{2}-2k^{2}\right) }{s^{4}+4k^{4}}$%
\medskip \\ 
$\cos kt\cosh kt\quad $ & $\dfrac{s^{3}}{s^{4}+4k^{4}}$%
\end{tabular}
\medskip
\end{center}

\subsubsection{\ Bessel Functions}

\begin{center}
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
f\right\} \left( s\right) $ \\ \hline\hline
$J_{0}\left( kt\right) \quad $ & $\dfrac{1}{\sqrt{s^{2}+k^{2}}}$%
\end{tabular}
\medskip
\end{center}

\subsubsection{\ Complementary Error Function}

\begin{center}
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
f\right\} \left( s\right) $ \\ \hline\hline
$\limfunc{erfc}\left( \frac{a}{2\sqrt{t}}\right) =1-\func{erf}\left( \frac{a%
}{2\sqrt{t}}\right) $ & $\dfrac{e^{-a\sqrt{s}}}{s}$\medskip \\ 
$2\sqrt{\frac{t}{\pi }}e^{-a^{2}/4t}-a\limfunc{erfc}\left( \frac{a}{2\sqrt{t}%
}\right) $ & $\dfrac{e^{-a\sqrt{s}}}{s\sqrt{s}}$\medskip \\ 
$e^{ab}e^{b^{2}t}\limfunc{erfc}\left( b\sqrt{t}+\frac{a}{2\sqrt{t}}\right) $
& $\dfrac{e^{-a\sqrt{s}}}{\sqrt{s}\left( \sqrt{s}+b\right) }$\medskip \\ 
$-e^{ab}e^{b^{2}t}\limfunc{erfc}\left( b\sqrt{t}+\frac{a}{2\sqrt{t}}\right) +%
\limfunc{erfc}\left( \frac{a}{2\sqrt{t}}\right) \quad $ & $\frac{be^{-a\sqrt{%
s}}}{s\left( \sqrt{s}+b\right) }$%
\end{tabular}
\medskip
\end{center}

\subsubsection{\ Dirac Function}

\begin{center}
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
f\right\} \left( s\right) $ \\ \hline\hline
$\delta \left( t\right) $ & $1$\medskip \\ 
$\delta \left( t-t_{0}\right) $ & $e^{-st_{0}}$%
\end{tabular}
\medskip
\end{center}

\subsubsection{\ Unit Step Function}

\begin{center}
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
f\right\} \left( s\right) $ \\ \hline\hline
$e^{at}f\left( t\right) $ & $F\left( s-a\right) $\medskip \\ 
$f\left( t-a\right) u\left( t-a\right) \quad $ & $e^{-as}F\left( s\right) $%
\medskip \\ 
$u\left( t-a\right) $ & $\dfrac{e^{-as}}{s}$%
\end{tabular}
\medskip
\end{center}

\subsubsection{\ General Functions}

\begin{center}
\begin{tabular}{ll}
$f\left( t\right) \rule[-0.07in]{0in}{0.25in}$ & $F\left( s\right) =L\left\{
f\right\} \left( s\right) $ \\ \hline\hline
$e^{at}f\left( t\right) $ & $F\left( s-a\right) $\medskip \\ 
$f\left( t-a\right) u\left( t-a\right) $ & $e^{-as}F\left( s\right) $\medskip
\\ 
$f^{\left( n\right) }\left( t\right) $ & $s^{n}F\left( s\right) -s^{\left(
n-1\right) }f\left( 0\right) -...-f^{\left( n-1\right) }\left( 0\right) $%
\medskip \\ 
$t^{n}f\left( t\right) $ & $\left( -1\right) ^{n}\frac{d^{n}}{ds^{n}}F\left(
s\right) $\medskip \\ 
$\int_{0}^{t}f\left( \tau \right) g\left( t-\tau \right) d\tau \quad $ & $%
F\left( s\right) G\left( s\right) $%
\end{tabular}
\medskip
\end{center}

\ \bigskip 

\noindent \hrulefill 

\begin{center}
{\small See the Reference Library (under Help) for more information about
mathematics and science.}
\end{center}

\noindent \hrulefill 

\end{document}
