Properties of Laplace Transform

Property name	Time domain function	Laplace transform	Comment
	f (t)	F(s)
Linearity	a f (t)+bg(t)	aF(s) + bG(s)	a,b are constant
Scale change	f (at)	$\frac{1}{a}F\left ( \frac{s}{a} \right )$	a>0
Shift	e^-at f (t)	F(s + a)
Delay	f (t-a)	e^-asF(s)
Derivation	$\frac{df(t)}{dt}$	sF(s) - f (0)
N-th derivation	$\frac{d^nf(t)}{dt^n}$	sⁿf (s) - s^n-1f (0) - s^n-2f '(0)-...-f^(n-1)(0)
Power	tⁿ f (t)	$(-1)^n\frac{d^nF(s)}{ds^n}$
Integration	$\int_{0}^{t}f(x)dx$	$\frac{1}{s}F(s)$
Reciprocal	$\frac{1}{t}f(t)$	$\int_{s}^{\infty }F(x)dx$
Convolution	f (t) * g (t)	F(s) ⋅ G(s)	* is the convolution operator
Periodic function	f (t) = f (t+T)	$\frac{1}{1-e^{-sT}}\int_{0}^{T}e^{-sx}f(x)dx$

See also