Laplace Transform Inversion by Inverse Integral

Cuthbert Nyack
In many cases of interest, the inverse Laplace Transform can be found from partial fraction expansions. However in some cases it may be necessary to use the inverse integral defined below
The limits of integration mean that the integral is carried out along a line parallel to the imaginary axis and distant c from it. c is chosen so any singularities are to the left of the line.If F(s) has simple poles then the integral can be represented as follows. It is assumed that the integral is closed by a semicircle in the left of the s plane.

The following oversimplified example illustrates using the inverse integral for the case where F(s) has simple poles.
This transform has simple poles at - a and - b. The residue at s = - a is
and at s = - b it is
Putting these results together gives the inverse transform f(t).

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Copyright 1996 Cuthbert A. Nyack.