Mary Attenborough, in Mathematics for Electrical Engineering and Computing, 2003
Laplace and z transforms
A quick formula for partial fractions
Let's see if we can learn a thing or two about partial fraction expansion, or sometimes it's called partial fraction decomposition. The whole idea is to take rational functions- and a rational function is just a function or expression where it's one expression divided by another- and to essentially expand them or decompose them into simpler parts.
There is a quick way of getting the partial fractions expansion, called the âcover upâ rule, which works in the case where all the roots of the denominator of F(s) are distinct. If F(s) = P(s)/ Q(s) then write Q(s) in terms of its factors Q (s)
where sl ⦠snare its distinct roots. Then we can find the constant Ar, etc., for the partial fraction expansion from covering up each of the factors of Q in term and substituting s = s r in the rest of the expression:
then
where
In this case
Then A is found by substituting s =1 into
This gives the partial fraction expansion, as before, as
This method can also be used for complex poles; for instance,
The roots of the denominator are â3, j2 and âj2, so we get for the partial fraction expansion
The inverse transform can be found directly or the last two terms can be combined to give the expansion as