# 7. The Inverse Laplace Transform

## Definition

Partial Fraction Types

Integral and Periodic Types

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)))