Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

Ims

Res

L

λ

Figure 25.6 The integration path of the inverse Laplace transform is along

the infinite lineL.Thequantityλmust be positive and large enough for all

poles of the integrand to lie to the left ofL.

25.5 Inverse Laplace transform

As a further example of the use of contour integration we now discuss a method

whereby the process of Laplace transformation, discussed in chapter 13, can be

inverted.

It will be recalled that the Laplace transform ̄f(s)ofafunctionf(x),x≥0, is

given by

̄f(s)=

∫∞

0

e−sxf(x)dx, Res>s 0. (25.24)

In chapter 13, functionsf(x) were deduced from the transforms by means of a

prepared dictionary. However, an explicit formula for an unknown inverse may

be written in the form of an integral. It is known as theBromwich integraland is

given by

f(x)=

1

2 πi

∫λ+i∞

λ−i∞

esxf ̄(s)ds, λ > 0 , (25.25)

wheresis treated as a complex variable and the integration is along the lineL

indicated in figure 25.6. The position of the line is dictated by the requirements

thatλis positive and that all singularities of ̄f(s) lie to the left of the line.

That (25.25) really is the unique inverse of (25.24) is difficult to show for general

functions and transforms, but the following verification should at least make it