ap-laplace-list.tex

\chapter{Table of Laplace Transforms} \label{laplacelist:appendix}

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The function $u$ is the
Heaviside function, $\delta$ is the Dirac delta function, and 
\begin{equation*}
\Gamma(t) =
\int_0^\infty e^{-\tau} \tau^{t-1} \, d\tau ,
\qquad
\operatorname{erf}(t) =
\frac{2}{\sqrt{\pi}} \int_0^t e^{-\tau^2} \, d\tau ,
\qquad
\operatorname{erfc}(t) =
%\frac{2}{\sqrt{\pi}} \int_t^\infty e^{-\tau^2} \, d\tau =
1 - \operatorname{erf}(t) .
\end{equation*}

\begin{center}
\begin{tabular}{@{}lllll@{}}
\toprule
$f(t)$ &
$F(s) = \mathcal{L} \bigl\{ f(t) \bigr\}= \int_0^\infty e^{-st} f(t) \, dt$ \\
\midrule
$C$ & $\frac{C}{s}$
\\[6pt]
$t$ & $\frac{1}{s^2}$
\\[6pt]
$t^2$ & $\frac{2}{s^3}$
\\[6pt]
$t^n$ & $\frac{n!}{s^{n+1}}$
\\[6pt]
$t^p \quad (p > 0)$ & $\frac{\Gamma(p+1)}{s^{p+1}}$
\\[6pt]
$e^{-at}$ & $\frac{1}{s+a}$
\\[6pt]
$\sin (\omega t)$ & $\frac{\omega}{s^2+\omega^2}$
\\[6pt]
$\cos (\omega t)$ & $\frac{s}{s^2+\omega^2}$
\\[6pt]
$\sinh (\omega t)$ & $\frac{\omega}{s^2-\omega^2}$
\\[6pt]
$\cosh (\omega t)$ & $\frac{s}{s^2-\omega^2}$
\\[6pt]
$u(t-a)$ & $\frac{e^{-as}}{s}$
\\[6pt]
$\delta(t)$ & $1$
\\[6pt]
$\delta(t-a)$ & $e^{-as}$
\\[6pt]
$\operatorname{erf}\left( \frac{t}{2a} \right)$ & $\frac{1}{s} e^{(as)^2} \operatorname{erfc}(as)$
\\[6pt]
$\frac{1}{\sqrt{\pi t}} \exp\left(\frac{-a^2}{4t}\right) \quad (a \geq 0)$ &
$\frac{e^{-as}}{\sqrt{s}}$
\\[6pt]
$\frac{1}{\sqrt{\pi t}} - a e^{a^2 t} \operatorname{erfc}(a \sqrt{t}) \quad (a>0)$ &
$\frac{1}{\sqrt{s}+a}$
\\[6pt]
%mbxSTARTIGNORE
\bottomrule
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{@{}lllll@{}}
\toprule
$f(t)$ &
$F(s) = \mathcal{L} \bigl\{ f(t) \bigr\}= \int_0^\infty e^{-st} f(t) \, dt$ \\
\midrule
%mbxENDIGNORE
$a f(t) + b g(t)$ & $a F(s) + bG(s)$
\\[6pt]
$f(at) \quad (a > 0)$ & $\frac{1}{a}F\left( \frac{s}{a} \right)$
\\[6pt]
$f(t-a)u(t-a)$ & $e^{-as} F(s)$
\\[6pt]
$e^{-at} f(t)$ & $F(s+a)$
\\[6pt]
$g'(t)$ & $sG(s)-g(0)$
\\[6pt]
$g''(t)$ & $s^2G(s)-sg(0)-g'(0)$
\\[6pt]
$g'''(t)$ & $s^3G(s)-s^2g(0)-sg'(0)-g''(0)$
\\[6pt]
$g^{(n)}(t)$ & $s^nG(s)-s^{n-1}g(0)-\cdots-g^{(n-1)}(0)$
\\[6pt]
$(f * g)(t) = \int_0^t f(\tau) g(t-\tau) \, d\tau$ & $F(s)G(s)$
\\[6pt]
$tf(t)$ & $-F'(s)$
\\[6pt]
$t^nf(t)$ & ${(-1)}^nF^{(n)}(s)$
\\[6pt]
$\int_0^t f(\tau) d\tau$ & $\frac{1}{s} F(s)$
\\[6pt]
$\frac{f(t)}{t}$ & $\int_s^\infty F(\sigma) d\sigma$
\\[6pt]
\bottomrule
\end{tabular}
\end{center}