List of the Properties of The Laplace Transform

Electrical Circuit Analysis > The Laplace Transform

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Table $ 1 $ provides a list of the properties of the Laplace transform. The last property (on convolution) will be proved later. There are other properties, but these are enough for present purposes. Table $ 2 $ summarizes the Laplace transforms of some common functions. We have omitted the factor $ u(t) $ except where it is necessary.

	Table 1: Properties of the Laplace Transform
Property	$f (t)$	$F (s)$
Linearity	$a_1f_1(t) + a_2f_2(t)$	$a_1F_1(s) + a_2F_2(s)$
Scaling	$f (at)$	${1 \over a} F \left({s \over a}\right)$
Time shift	$f (t − a)u(t − a)$	$e−asF (s)$
Frequency shift	$e^{−at}f (t)$	$F (s + a)$
Time differentiation	$df/dt$	$sF (s) − f (0_−)$
	$df^2/dt^2$	$s^2F (s) − sf (0_−) − f'(0_−)$
Time integration	$\int_0^t f (t) dt$	${1 \over s} F(s)$
Frequency differentiation	$tf (t)$	$-{d \over ds} F (s)$
Frequency integration	$f (t)/t$	$\int_s^\infty F (s) ds$
Time periodicity	$f (t) = f (t + nT )$	${F_1(s) \over 1 − e^{−sT}}$
Initial value	$f (0^+)$	$\lim_{s→∞}sF (s)$
Final value	$f (\infty)$	$\lim_{s→0}sF (s)$
Convolution	$f_1(t) ∗ f_2(t)$	$F_1(s) F_2(s)$

Table 2: Laplace transform pairs.
$f (t)$	$F (s)$
$δ(t)$	$1$
$u(t)$	${1 \over s}$
$e^{−at}$	${1 \over s+a}$
$t$	${1 \over s^2}$
$t^n$	${n! \over s^{n+1}}$
$te^{−at}$	${1 \over (s+a)^{2}}$
$t^n e^{−at}$	${n! \over (s+a)^{n+1}}$
$\sin ωt$	${ω \over s^{2}+ω^{2}}$
$\cos ωt$	${s \over s^{2}+ω^{2}}$
$\sin(ωt + θ)$	${s \sin θ + ω \cos θ \over s^{2}+ω^{2}}$
$\cos(ωt + θ)$	${s cos θ − ω sin θ \over s^{2}+ω^{2}}$
$e^{−at} \sin ωt$	${ω \over (s+a)^{2}+ω^{2}}$
$e^{−at} \cos ωt$	${s+a \over (s+a)^{2}+ω^{2}}$

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