Usually, we can find the inverse of the Laplace transform by looking it up in a table. In this entry, we show an alternative method that inverts Laplace transforms through the powerful method of contour integration.

Consider the piece-wise differentiable function that vanishes for . We can express the function by the complex Fourier representation of

for any value of the real constant , where the integral

exists. By multiplying both sides of first equation by and bringing it inside the first integral

.

With the substitution , where is a new, complex variable of integration,

.

The quantity inside the square brackets is the Laplace transform . Therefore, we can express in terms of its transform by the complex contour integral

**Theorem **(Bromwich’s integral). The following line integral runs along the line parallel to the imaginary axis and units to the right of it, the so-called Bromwich contour

.

**Example**. Let us invert

.

*Solution*. From Bromwich’s integral,

where is a semicircle of infinite radius in either the right or left half of the -plane and is the closed contour that includes and Bromwich’s contour.

Our first task is to choose an appropriate contour so that the integral along vanishes. By Jordan’s lemma, this requires a semicircle in the right half-plane if and a semicircle in the left half-plane if . Consequently, by considering these two separate cases, we force the second integral to zero and the inversion simply equals the closed contour.

Consider the case first. Because Bromwich’s contour lies to the right of any singularities, there are no singularities within the closed contour and .

Consider now the case . Within the closed contour in the left half-plane, there is a second-order pole at and a simple pole at . Therefore,

where

and

.

Taking our earlier results into account, the inverse equals

.

Source: *Green’s Functions with Applications* by Dean G. Duffy

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Excuse me, I have a question.

the first equality on the solution of the example, I think z have to be put instead of s.

like this

Am I right?

Comment by zariski — September 10, 2010 @ 19:41

Hi zariski.

You are right, it also appears that in the second line need to be changed, i.e., the whole stuff reads as follows

Thanks for visiting my blog and also pointing out the misprint.

Comment by Ngô Quốc Anh — September 10, 2010 @ 21:27

Thanks for this post. It was a useful overview.

Comment by Sunil — August 12, 2012 @ 9:10