Common Laplace Transform Pairs#

Transforms of Elementary Signals#

	$f(t)$	$F(s)$	ROC
1	${\displaystyle \delta (t)}$	${\displaystyle 1}$	All $s$
2	${\displaystyle \delta (t-a)}$	${\displaystyle e^{-as}}$	All $s$
3	${\displaystyle u_0(t)}$	${\displaystyle \frac{1}{s}}$	Re($s$) > 0
4	${\displaystyle -u_0(-t)}$	${\displaystyle \frac{1}{s}}$	Re($s$) < 0
5	${\displaystyle t{u}_0(t)}$	${\displaystyle \frac{1}{s^2}}$	Re($s$) > 0
6	${\displaystyle t^n{u}_0(t)}$	${\displaystyle \frac{n!}{s^{n+1}}}$	Re($s$) > 0
7	${\displaystyle e^{-at}{u}_0(t)}$	${\displaystyle \frac{1}{s+a}}$	Re($s$) > $-$Re($a$)
8	${\displaystyle -e^{-at}{u}_0(-t)}$	${\displaystyle \frac{1}{s+a}}$	Re($s$)< $-$Re($a$)
9	${\displaystyle t^n{e}^{-at}{u}_0(t)}$	${\displaystyle \frac{n!}{(s+a{)}^{n+1}}}$	Re($s$) > $-$Re($a$)
10	${\displaystyle -t^n{e}^{-at}{u}_0(-t)}$	${\displaystyle \frac{n!}{(s+a{)}^{n+1}}}$	Re($s$) < $-$Re($a$)
11	${\displaystyle \sin (\omega t)u_0(t)}$	${\displaystyle \frac{\omega }{s^2+{\omega }^2}}$	Re($s$) > 0
12	${\displaystyle \cos (\omega t)u_0(t)}$	${\displaystyle \frac{s}{s^2+{\omega }^2}}$	Re($s$) > 0
13	${\displaystyle e^{-at}\sin (\omega t)u_0(t)}$	${\displaystyle \frac{\omega }{(s+a{)}^2+{\omega }^2}}$	Re($s$) > $-$Re($a$)
14	${\displaystyle e^{-at}\cos (\omega t)u_0(t)}$	${\displaystyle \frac{s+a}{(s+a{)}^2+{\omega }^2}}$	Re($s$) > $-$Re($a$)

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