PlanetMath: table of Laplace transforms

Below are tables of Laplace transforms; one lists some of the common properties, and the other lists some common examples.

Original	Transformed	comment	derivation
$af(t)+bg(t)$	$a\mathcal{L}\{f(t)\}+b\mathcal{L}\{g(t)\}$	linearity
$f(t)*g(t)$	$\mathcal{L}\{f(t)\}\mathcal{L}\{g(t)\}$	convolution property	here
$\displaystyle{\int_a^bf(t,\,x)\,dx}$	$\displaystyle{\int_a^b\mathcal{L}\{f(t,\,x)\}\,dx}$	integration with respect to a parametre	here
$\displaystyle{\frac{\partial}{\partial x}f(t,\,x)}$	$\displaystyle{\frac{\partial}{\partial x}\mathcal{L}\{f(t,\,x)\}}$	diffentiation with respect to a parameter
$f(\displaystyle{\frac{t}{a}})$	$aF(as)$	$\mathcal{L}\{f(t)\} = F(s)$	here
$e^{at}f(t)$	$F(s-a)$	$\mathcal{L}\{f(t)\} = F(s)$	here
$f(t-a)$	$e^{-as}F(s)$	$\mathcal{L}\{f(t)\} = F(s)$	here
$t^nf(t)$	$(-1)^nF^{(n)}(s)$	$\mathcal{L}\{f(t)\} = F(s)$	here
$\displaystyle\frac{f(t)}{t}$	$\displaystyle\int_s^\infty F(u)\,du$	$\mathcal{L}\{f(t)\} = F(s)$	here
$\displaystyle{\int_0^tf(u)\,du}$	$\displaystyle{\frac{F(s)}{s}}$	$\mathcal{L}\{f(t)\} = F(s)$	here
$f'(t)$	$sF(s)-\lim_{x\to0+}f(x)$	$\mathcal{L}\{f(t)\} = F(s)$	here
$f''(t)$	$s^2F(s)-s\lim_{x\to0+}f'(x)-\lim_{x\to0+}f(x)$	$\mathcal{L}\{f(t)\} = F(s)$

$f(t)$	$\mathcal{L}\{f(t)\}$	conditions	explanation	derivation
$e^{at}$	$\displaystyle{\frac{1}{s-a}}$	$s>a$		trivial
$\cos{at}$	$\displaystyle{\frac{s}{s^{2}+a^{2}}}$	$s>0$		here
$\sin{at}$	$\displaystyle{\frac{a}{s^{2}+a^{2}}}$	$s>0$		here
$\cosh{at}$	$\displaystyle{\frac{s}{s^{2}-a^{2}}}$	$s>\|a\|$		here
$\sinh{at}$	$\displaystyle{\frac{a}{s^{2}-a^{2}}}$	$s>\|a\|$		here
$\displaystyle\frac{\sin{t}}{t}$	$\displaystyle\arctan\frac{1}{s}$	$s>0$	See sinc function	here
$t^r$	$\displaystyle{\frac{\Gamma(r+1)}{s^{r+1}}}$	$r>-1,\;\;s>0$	gamma function $\Gamma$	here
$\displaystyle e^{a^2t}\,{\rm erf}\,a\sqrt{t}$	$\displaystyle\frac{a}{(s\!-\!a^2)\sqrt{s}}$	$s>a^2$	See error function	here
$\displaystyle e^{a^2t}\,{\rm erfc}\,a\sqrt{t}$	$\displaystyle\frac{1}{(a\!+\!\sqrt{s})\sqrt{s}}$	$s>0$	See error function	here
$\displaystyle\frac{1}{\sqrt{t}}$	$\displaystyle\sqrt{\frac{\pi}{s}}$	$s>0$		here
$J_0(at)$	$\displaystyle\frac{1}{\sqrt{s^2+a^2}}$	$s>0$	Bessel function $J_0$	here
$e^{-t^2}$	$\displaystyle\frac{\sqrt{\pi}}{2}e^\frac{s^2}{4}\mathrm{erfc}\Big(\frac{s}{2}\Big)$	$s>0$	See error function	here
$\ln{t}$	$\displaystyle-\frac{\gamma+\ln{s}}{s}$	$s>0$	Euler'sconstant $\gamma$	here
$\delta(t)$	$1$		Dirac delta function

$f(t)$	$\mathcal{L}\{f(t)\}$	conditions	explanation	derivation
1	$\displaystyle{1 \over s}$
$t$	$\displaystyle{1 \over s^2}$			here
$\displaystyle{t^{n-1} \over (n-1)!}$	$\displaystyle{1 \over s^n}$			here
$\displaystyle{1 \over t+a}$	$e^{as} {\rm E}_1(as)$	$a > 0$	exponential integral ${\rm E}_1$	here
$\displaystyle{1 \over (t+a)^2}$	$\displaystyle{1 \over a}-se^{as}{\rm E}_1(as)$	$a > 0$		here
$\displaystyle{1 \over (t+a)^n}$	$a^{1-n} e^{as} E_n (as)$	$a > 0,\;\; n \in \mathbb{N}$	?
$L_n(t)$	$\displaystyle\frac{1}{s}\!\left(\!\frac{s-1}{s}\!\right)^n$	$s > 0$	Laguerre polynomial $L_n$