Laplace Transform Table

Formula¹

This is table of Laplace transform.

$f(t)=\mathcal{L^{-1}}$	$F(s)=\mathcal{L} \left\{ f(t) \right\}$	Derivation
$1$	$\dfrac{1}{s}$	link
$e^{at}$	$\dfrac{1}{s-a}$	link
$t^n$	$\dfrac{n!}{s^{n+1}}$	link
$t^{p}$	$\dfrac{ \Gamma (p+1) }{ s^{p+1}}$	link
$t^{p}e^{at}$	$\dfrac{ \Gamma (p+1) }{ (s-a)^{p+1}}$	link
$\sin (at)$	$\dfrac{a}{s^2+a^2}$	link
$\cos (at)$	$\dfrac{s}{s^2+a^2}$	link
$e^{at}\sin(bt)$	$\dfrac{b}{(s-a)^2 +b^2}$	link
$e^{at}\cos(bt)$	$\dfrac{s-a}{(s-a)^2+b^2}$	link
$\sinh (at)$	$\dfrac{a}{s^2-a^2}$	link
$\cosh (at)$	$\dfrac{s}{s^2-a^2}$	link
$e^{at} \sinh (bt)$	$\dfrac{b}{(s-a)^2-b^2}$	link
$e^{at} \cosh (bt)$	$\dfrac{s-a}{(s-a)^2-b^2}$	link
$u_{c}(t)= \begin{cases} 0 & t<c \\ 1 & t\ge c\end{cases}$	$\dfrac{e^{-cs}}{s}$	link
$u_{c}(t)f(t-c)$	$e^{-cs}F(s)$	link
$f^{\prime}(t)$	$s\mathcal{L} \left\{ f(t) \right\} -f(0)$	link
$f^{(n)}$	${s^n\mathcal {L}\left\{ f(t) \right\} -s^{n-1}f(0) - \cdots -f^{(n-1)}(0) }$	link
$f(t)=f(t+T)$	$\dfrac{\displaystyle \int_{0}^T e^{-st}f(t)dt}{1-e^{-st}}$	link
$\delta (t-t_{0})$	$e^{-st_{0}}$	link
$f(ct)$	$\frac{1}{c}F \left( \frac{s}{c} \right)$	link
$\frac{1}{k}f (\frac{t}{k} )$	$F(ks)$	link
$\frac{1}{a} e^{-\frac{b} {a}t}f\left(\frac{t}{a}\right)$	$F(as+b)$	link
$t^{n}f(t)$	$(-1)^{n}F^{(n)}(s)$	link

William E. Boyce , Boyce’s Elementary Differential Equations and Boundary Value Problems (11th Edition, 2017), Chapter6 The Laplace Transform ↩︎