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