8.8: A Brief Table of Laplace Transforms

Table $\PageIndex{1}$

$\displaystyle f(t)$	$\displaystyle F(s)$
1	$\displaystyle{ 1\over s}$	$\displaystyle (s > 0)$
$\displaystyle t^n$	$\displaystyle{ n!\over s^{n+1} }$	$\displaystyle (s > 0)$
($\displaystyle n = \mbox{ integer } > 0$)
$\displaystyle t^p,\; p > -1$	$\displaystyle{ \Gamma (p+1) \over s^{(p+1)} }$	$\displaystyle (s>0)$
$\displaystyle e^{at}$	$\displaystyle{ 1 \over s-a }$	$\displaystyle (s > a)$
$\displaystyle t^ne^{at}$	$\displaystyle{ n! \over (s-a)^{n+1} }$	$\displaystyle (s > 0)$
($\displaystyle n= \text{ integer } > 0$)
$\displaystyle \cos \omega t$	$\displaystyle{ \frac{s}{s^{2}+\omega ^{2}} }$	$\displaystyle (s > 0)$
$\displaystyle \sin \omega t$	${ \displaystyle \omega \over s^2+\omega^2 }$	$\displaystyle (s > 0)$
$\displaystyle e^{\lambda t} \cos \omega t$	$\displaystyle{ s - \lambda \over (s-\lambda)^2+\omega^2 }$	$\displaystyle (s > \lambda)$
$\displaystyle e^{\lambda t} \sin \omega t$	$\displaystyle{ \omega \over (s-\lambda)^2+\omega^2 }$	$\displaystyle (s > \lambda)$
$\displaystyle \cosh bt$	$\displaystyle{ s \over s^2-b^2 }$	$\displaystyle (s > |b|)$
$\displaystyle \sinh bt$	$\displaystyle{ b \over s^2-b^2 }$	$\displaystyle (s > |b|)$
$\displaystyle t \cos \omega t$	$\displaystyle{ s^2-\omega^2 \over (s^2+\omega^2)^2 }$	$\displaystyle (s>0)$
$\displaystyle t \sin \omega t$	$\displaystyle{ 2\omega s \over (s^2+\omega^2)^2 }$	$\displaystyle (s>0)$
$\displaystyle \sin \omega t -\omega t\cos \omega t$	$\displaystyle{ 2\omega^3\over (s^2+\omega^2)^2 }$	$\displaystyle (s>0)$
$\displaystyle \omega t - \sin \omega t$	$\displaystyle{ \omega^3 \over s^2(s^2+\omega^2) }$	$\displaystyle (s>0)$
$\displaystyle \frac{1}{t}\sin\omega t$	$\displaystyle{ \arctan \left({\omega \over s}\right) }$	$\displaystyle (s>0)$
$\displaystyle e^{at}f(t)$	$\displaystyle{ F(s-a) }$
$\displaystyle t^kf(t)$	$\displaystyle (-1)^{k}F^{(k)}(s)$
$\displaystyle f(\omega t)$	$\displaystyle{ \frac{1}{\omega}F\left(\frac{s}{\omega } \right), \quad \omega >0 }$
$\displaystyle u(t-\tau)$	$\displaystyle{ e^{-\tau s} \over s }$	$\displaystyle (s>0)$
$\displaystyle u(t-\tau)f(t-\tau)\, (\tau > 0)$	$\displaystyle{ e^{-\tau s}F(s) }$
$\displaystyle \displaystyle {\int^t_o f(\tau)g(t-\tau)\, d\tau}$	$\displaystyle{ F(s) \cdot G(s) }$
$\displaystyle \delta(t-a)$	$\displaystyle{ e^{-as} }$	$\displaystyle (s>0)$