# 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.