7. The Inverse Laplace Transform

Definition

If `G(s)=Lap{g(t)}`, then the inverse transform of `G(s)` is defined as:

`Lap^{:-1:}G(s) = g(t)`

Some Properties of the Inverse Laplace Transform

We first saw these properties in the Table of Laplace Transforms.

Property 1: Linearity Property

`Lap^{:-1:}{a\ G_1(s) + b\ G_2(s)}` ` = a\ g_1(t) + b\ g_2(t)`

Property 2: Shifting Property

If `Lap^{:-1:}G(s) = g(t)`, then `Lap^{:-1:}G(s - a) = e^(at)g(t)`.

Property 3

If `Lap^{:-1:}G(s) = g(t)`, then `Lap^{:-1:}{(G(s))/s}=int_0^tg(t)dt`.

Property 4

If `Lap^{:-1:}G(s) = g(t),` then

`Lap^{:-1:}{e^(-as)G(s)} = u(t - a) * g(t - a)`.

Continues below

Examples

Find the inverse of the following transforms and sketch the functions so obtained.

(a) `G(s)=2/s(e^(-3s)-e^(-4s))`

(b) `G(s)=(2s+1)/s^2e^(-2s)-(3s+1)/s^2e^(-3s)`

(c) `G(s)=1/(s^2+9)e^(-pis//2)`

(d) `G(s)=1/((s-5)^2)e^(-s)`

(e) `G(s)=(s+4)/(s^2+9)`

(f) `G(s)=3/(s^2+4s+13)`

(g) `G(s)=(1-e^((1-s)T))/((s-1)(1-e^(-sT))` (where T is a constant)

Examples Involving Partial Fractions

We first met Partial Fractions in the Methods of Integration section. You may wish to revise partial fractions before attacking this section.

Obtain the inverse Laplace transforms of the following functions:

(h) `G(s)=(2s^2-16)/(s^3-16s)`

(i) `G(s)=3/(s^2(s+2))`

Integral and Periodic Types

(j) `G(s)=omega_0/(s(s^2+(omega_0)^2))`

(k) `G(s)=(s+b)/(s(s^2+2bs+b^2+a^2))`

(l) `G(s)=(1-e^(-sT))/(s(1+e^(-sT)))`