List of the Properties of The Laplace Transform

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)$$F (s)$
$u(t)$${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}}$