Partial Fractions in Complex Analysis - Calculation

Calculation

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 Γ_k
• No curve passes through a pole of f
• Γ_k lies inside Γ_k+1 for all k
• , where d(Γ_k) gives the distance from the curve to the origin

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)/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 λ_k 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 λ_k is inside Γ_n, then λ_j is also inside Γ_n for all j < k.