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:
- The origin lies inside each curve Γk
- No curve passes through a pole of f
- Γk lies inside Γk+1 for all k
- , where d(Γk) gives the distance from the curve to the origin
Suppose also that there exists an integer p such that
Writing PP(f(z); z = λk) for the principal part of the Laurent expansion of f about the point λk, we have
if p = -1, and if p > -1,
where the coefficients cj,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)/zj+1 at z = 0 must still be included in the sum.
Note that in the case of λ0 = 0, we can use the Laurent expansion of f(z) about the origin to get
so that the polynomial terms contributed are exactly the regular part of the Laurent series up to zp.
For the other poles λk where k ≥ 1, 1/zj+1 can be pulled out of the residue calculations:
To avoid issues with convergence, the poles should be ordered so that if λk is inside Γn, then λj is also inside Γn for all j < k.
Read more about this topic: Partial Fractions In Complex Analysis
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