**Calculation**

Let *f(z)* be a function meromorphic in the finite complex plane with poles at *λ _{1}*,

*λ*, ..., and let (

_{2}*Γ*,

_{1}*Γ*, ...) be a sequence of simple closed curves such that:

_{2}- The origin lies inside each curve
*Γ*_{k} - No curve passes through a pole of
*f* *Γ*lies inside_{k}*Γ*for all_{k+1}*k*- , where
*d(Γ*gives the distance from the curve to the origin_{k})

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*, we have

_{k}if *p = -1*, and if *p > -1*,

where the coefficients *c _{j,k}* are given by

λ_{0} 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 λ_{0} = 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*.

Read more about this topic: Partial Fractions In Complex Analysis

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