Common Laplace Transform Pairs#

Transforms of Elementary Signals#

	$f(t)$	$F(s)$
1	${\displaystyle \delta (t)}$	${\displaystyle 1}$
2	${\displaystyle \delta (t-a)}$	${\displaystyle e^{-as}}$
3	${\displaystyle u_0(t)}$	${\displaystyle \frac{1}{s}}$
4	${\displaystyle t{u}_0(t)}$	${\displaystyle \frac{1}{s^2}}$
5	${\displaystyle t^n{u}_0(t)}$	${\displaystyle \frac{n!}{s^{n+1}}}$
6	${\displaystyle e^{-at}{u}_0(t)}$	${\displaystyle \frac{1}{s+a}}$
7	${\displaystyle t^n{e}^{-at}{u}_0(t)}$	${\displaystyle \frac{n!}{(s+a{)}^{n+1}}}$
8	${\displaystyle \sin (\omega t)u_0(t)}$	${\displaystyle \frac{\omega }{s^2+{\omega }^2}}$
9	${\displaystyle \cos (\omega t)u_0(t)}$	${\displaystyle \frac{s}{s^2+{\omega }^2}}$
10	${\displaystyle e^{-at}\sin (\omega t)u_0(t)}$	${\displaystyle \frac{\omega }{(s+a{)}^2+{\omega }^2}}$
11	${\displaystyle e^{-at}\cos (\omega t)u_0(t)}$	${\displaystyle \frac{s+a}{(s+a{)}^2+{\omega }^2}}$

See also: Wikibooks: Engineering_Tables/Laplace_Transform_Table and Laplace Transform—WolframMathworld for more complete references.