An Algorithm for the Machine Computation of Partial-Fractions Expansion of Functions Having Multiple Poles

The partial-fractions expansion of a function F(s)/(s-a)^m, m > 1, involves the computation of m coefficients, namely (1 /i!)(dⁱF(a)/dsⁱ), 0 ≤ i ≤ m-1. Wehrhahn [1] and Karni [3] have provided a method for computing these coefficients algebraically. A new approach is taken here which involves approximating a multiple pole by neighboring simple poles. The theory developed turns out to have a very interesting resemblance to the FFT algorithm. The algorithm is illustrated by several examples. Typical applications are finding the inverse Laplace transform of a function having multiple poles and the evaluation of higher order derivatives of an arbitrary function H(z) at some arbitrary z=z0.