Partial fractions in complex analysis

In complex analysis, a partial fraction expansion is a way of writing a meromorphic function f(z) as an infinite sum of rational functions and polynomials. When f(z) is a rational function, this reduces to the usual method of partial fractions.

By using polynomial long division and the partial fraction technique from algebra, any rational function can be written as a sum of terms of the form 1 / (az + b)^k + p(z), where a and b are complex, k is an integer, and p(z) is a polynomial. Just as polynomial factorization can be generalized to the Weierstrass factorization theorem, there is an analogy to partial fraction expansions for certain meromorphic functions.

A proper rational function, i.e. one for which the degree of the denominator is greater than the degree of the numerator, has a partial fraction expansion with no polynomial terms. Similarly, a meromorphic function f(z) for which |f(z)| goes to 0 as z goes to infinity at least as quickly as |1/z|, has an expansion with no polynomial terms.

Let f(z) be a function meromorphic in the finite complex plane with poles at λ₁, λ₂, ..., and let (Γ₁, Γ₂, ...) be a sequence of simple closed curves such that:

Suppose also that there exists an integer p such that

Writing PP(f(z); z = λ_k) for the principal part of the Laurent expansion of f about the point λ_k, we have

if p = -1, and if p > -1,

where the coefficients c_j,k are given by

λ₀ should be set to 0, because even if f(z) itself does not have a pole at 0, the residues of f(z)/z^j+1 at z = 0 must still be included in the sum.

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