# Laplace Transform Inverse by
Partial Fraction Expansion

Cuthbert Nyack

Consider a Laplace Transform F(s) which can be represented as a
ratio of 2 polynomials as shown below.
If the denominator can be factorised with simple roots at r_{1},
r_{2} etc. then F(s) can be expanded as:-
With coefficients
If Q(s) has a multiple zero of order m
(F(s) has a multiple pole of order m)
then the expansion of
F(s) must be written
with coefficients

## Example

Consider finding the inverse f(t) of F(s) below
Since F(s) has a second order pole at the origin and complex
conjugate poles at -1/2 ± j0.866, then it can be expanded to
The coefficient a_{1} is calculated
a_{2} is
a_{3} is
and a_{4} is
The Expansion for F(s) now becomes
with inverse
and simplifies to

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Copyright 1996 © Cuthbert A. Nyack.