# Numerical Inversion of the Laplace Transform: The Fourier Series Approximation

Numerical Inversion of the Laplace Transform: The Fourier Series Approximation

This Demonstration shows how you can numerically compute the inverse of the Laplace transform of a simple function : and . The selected method is the Fourier series approximation. This method uses the following formula in order to perform the inversion of :

F(p)

f(t)

ℒ(f(t))(p)=F(p)=

1

p+1

2

f(t)=sin(t)

F(p)

f(t)=F(σ)+ΣReFσ+cos-ImFσ+sin

e

σt

T

1

2

∞

k=1

kπi

T

kπt

T

kπi

T

kπt

T

You can select the appropriate values of and that give the correct inverse. This choice must be such that and , where is a measure of the maximum relative error and is the exponential order of .

σ

T

2T>t

max

σ=σ-

0

ln(E)

2T

E

σ

0

f(t)

The red curve is the sine function and the blue dots are the selected numerical values of the inverse of .

F(p)

You can clearly see how this method may fail to give an accurate inverse if the values of and are not correctly selected. The first snapshot presents a correct inversion result. The next two snapshots show situations where the method gives erroneous data.

σ

T