Table of Laplace Transform

Function name	Time domain function	Laplace transform
Function name	f (t)	F(s) = L{f (t)}
Constant	1	$\frac{1}{s}$
Linear	t	$\frac{1}{s^2}$
Power	tⁿ	$\frac{n!}{s^{n+1}}$
Power	t^a	Γ(a+1) ⋅ s ^-(a+1)
Exponent	e^at	$\frac{1}{s-a}$
Sine	sin at	$\frac{a}{s^2+a^2}$
Cosine	cos at	$\frac{s}{s^2+a^2}$
Hyperbolic sine	sinh at	$\frac{a}{s^2-a^2}$
Hyperbolic cosine	cosh at	$\frac{s}{s^2-a^2}$
Growing sine	t sin at	$\frac{2as}{(s^2+a^2)^2}$
Growing cosine	t cos at	$\frac{s^2-a^2}{(s^2+a^2)^2}$
Decaying sine	e^-atsin ωt	$\frac{\omega }{\left ( s+a \right )^2+\omega ^2}$
Decaying cosine	e^-atcos ωt	$\frac{s+a }{\left ( s+a \right )^2+\omega ^2}$
Delta function	δ(t)	1
Delayed delta	δ(t-a)	e^-as

See also