Inverse Laplace Transform

Inverse Laplace Transform
If L{f(t) }= F(s), then we call f(t) the inverse Laplace transform of F(s) and write:

L^?(-1@) {F(s) } =f(t)

The symbol L^?(-1@)is called theinverse Laplace transform operator and it is a linear operator:
L^?(-1@) {aF_1 (s)+bF_2 (s) }=af_1 (t)+bf_2 (t)

The following table list the inverse Laplace transform of some elementary functions:

F(s) L^?(-1@) {F(s) }=f(t)
1/s 1
1/s^?(2@) t
2!/s^?(3@) t^2
n!/s^?(n+1@) t^n , n=1,2,3…
1/(s-a) e^?(at@)
b/(s^?(2@)+b^?(2@) ) sin?bt
s/(s^?(2@)+b^?(2@) ) cos?bt
b/(s^?(2@)-b^?(2@) ) sinh?bt
s/(s^?(2@)-b^?(2@) ) cosh?bt




Because of the relationship between Laplace transform and inverse Laplace transform, any theorem or property involving Laplace transform will have a corresponding inverse Laplace transform. Here we will list some of the important results involving inverse Laplace transform:

If L^?(-1@) {F(s) }=f(t), then:

L^?(-1@) {F(s-a) }=e^?(at@) f(t)( from L.T. first shift property)

L^?(-1@) {e^?(-as@) F(s) }=u(t-a)f(t-a)( from L.T. second shift property)