%\input file to produce a list of common Laplace transforms
%11/18/91; ed: 5/1/95; ed: 2008/11/16
%insert "\input{laplace.tex}" into your latex-file
\newcommand{\la}[1]{{\cal L}(#1)(s)&=&}
%
%\newpage
\begin{center}\bf Elementary Laplace Transforms\end{center}
\begin{eqnarray}
\la{1} \frac{1}{s} \ \  (s>0)\\[.50\baselineskip]
\la{e^{at}}   \frac{1}{s-a}\ \  (s>a)\\[.50\baselineskip]
\la{t^n} \frac{n!}{s^{n+1}}\ \  (s>0,n \mbox{ a positive integer})\\[.50\baselineskip]
\la{t^p} \frac{\Gamma(p+1)}{s^{p+1}}\ \  (s>0,p>-1)\\[.50\baselineskip]
\la{\sin(bt)}  \frac{b}{s^2+b^2}\ \   (s>0)\\[.50\baselineskip]
\la{\cos(bt)}  \frac{s}{s^2+b^2} \ \  (s>0)\\[.50\baselineskip]
\la{e^{at}\cdot \sin(bt)} \frac{b}{(s-a)^2+b^2}\ \ (s>a)\\[.50\baselineskip]
\la{e^{at}\cdot \cos(bt)} \frac{s-a}{(s-a)^2+b^2}\ \ (s>a)\\[.50\baselineskip]
\la{t^n\cdot e^{at}} \frac{n!}{(s-a)^{n+1}}\ \ (s>a)\\[.50\baselineskip]
\la{t^n \cdot f(t)} (-1)^n \frac{d^n}{ds^n}\left({\cal L}(f(t))\right)(s)\\[.50\baselineskip]
\la{f'(t)} s\cdot {\cal L}(f(t))(s) - f(0)\\[.50\baselineskip]
\la{u_c(t)}\frac{e^{-cs}}{s}\ \ (s>0)\\[.50\baselineskip]
\la{u_c(t)\cdot f(t-c)} e^{-cs}{\cal L}(f(t))(s)\\[.50\baselineskip]
\la{u_c(t)\cdot f(t)}{e^{-cs}{\cal L}(f(t+c))(s)}\\[.50\baselineskip]
\la{\delta_c(t)} e^{-cs}\\[.50\baselineskip]
\la{e^{ct}\cdot f(t)} {\cal L}(f(t))(s-c)
\end{eqnarray}
