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Table of Laplace transforms: common transforms and properties

The transforms you need to solve ODEs and control problems, on one page. The function of time is on the left, its transform on the right.

Basic transforms

f(t)	F(s)
$1$	$\frac{1}{s}$
$t$	$\frac{1}{s ^{2}}$
$t^{n}$	$\frac{n !}{s ^{n + 1}}$
$e^{a t}$	$\frac{1}{s - a}$
$t^{n} e^{a t}$	$\frac{n !}{( s - a ) ^{n + 1}}$
$u (t - a)$	$\frac{e ^{- a s}}{s}$
$δ (t - a)$	$e^{- a s}$

Sines, cosines and hyperbolic functions

f(t)	F(s)
$sin ω t$	$\frac{ω}{s ^{2} + ω ^{2}}$
$cos ω t$	$\frac{s}{s ^{2} + ω ^{2}}$
$e^{a t} sin ω t$	$\frac{ω}{( s - a ) ^{2} + ω ^{2}}$
$e^{a t} cos ω t$	$\frac{s - a}{( s - a ) ^{2} + ω ^{2}}$
$t sin ω t$	$\frac{2 ω s}{( s ^{2} + ω ^{2} ) ^{2}}$
$t cos ω t$	$\frac{s ^{2} - ω ^{2}}{( s ^{2} + ω ^{2} ) ^{2}}$
$sinh a t$	$\frac{a}{s ^{2} - a ^{2}}$
$cosh a t$	$\frac{s}{s ^{2} - a ^{2}}$

Properties

f(t)	F(s)
$f^{'} (t)$	$s F (s) - f (0)$
$f^{''} (t)$	$s^{2} F (s) - s f (0) - f^{'} (0)$
$\int_{0}^{t} f (τ) d τ$	$\frac{F ( s )}{s}$
$e^{a t} f (t)$	$F (s - a)$
$f (t - a) u (t - a)$	$e^{- a s} F (s)$
$t f (t)$	$- F^{'} (s)$
$(f * g) (t)$	$F (s) G (s)$

Notes

u(t − a) is the unit step, switching from 0 to 1 at time a, and δ(t − a) is the Dirac delta.

To invert a transform, split F(s) into partial fractions that each appear in the table. The explainer page has a calculator for it.