Laplace Transform Properties

Laplace Transform Properties#

No.	Name	Time Domain $f (t)$	Complex Frequency Domain $F (s)$
1.	Linearity	$a_{1} f_{1} (t) + a_{2} f_{2} (t) + \dots + a_{n} f_{n} (t)$	$a_{1} F_{1} (s) + a_{2} F_{2} (s) + \dots + a_{n} F_{n} (s)$
2.	Time shifting	$f (t - a) u_{0} (t - a)$	$e^{- a s} F (s)$
3.	Frequency shifting	$e^{- a s} f (t)$	$F (s + a)$
4.	Time scaling	$f (a t)$	$\frac{1}{a} F (\frac{s}{a})$
5.	Time differentiation	$\frac{d}{d t} f (t)$	$s F (s) - f (0^{-})$
6.	Frequency differentiation	$t f (t)$	$- \frac{d}{d s} F (s)$
7.	Time integration	$\int_{- \infty}^{t} f (τ) d τ$	$\frac{F (s)}{s} + \frac{f (0^{-})}{s}$
8.	Frequency integration	$\frac{f (t)}{t}$	$\int_{s}^{\infty} F (σ) d σ$
9.	Time Periodicity	$f (t + n T)$	$\frac{\int_{0}^{T} f (t) e^{- s t} d t}{1 - e^{- s T}}$
10.	Initial value theorem	$lim_{t \to 0} f (t)$	$lim_{s \to \infty} s F (s) = f (0^{-})$
11.	Final value theorem	$lim_{t \to \infty} f (t)$	$lim_{s \to 0} s F (s) = f (\infty)$
12.	Time convolution	$f_{1} (t) * f_{2} (t)$	$F_{1} (j s) F_{2} (s)$
13.	Frequency convolution	$f_{1} (t) f_{2} (t)$	$\frac{1}{j 2 π} F_{1} (s) * F_{2} (s)$