Residue

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Residue theorem

In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it is a special case of the generalized Stokes' theorem.

The statement is as follows:

Suppose U is a simply connected open subset of the complex plane, and a₁,...,a_n are finitely many points of U and f is a function which is defined and holomorphic on U \ {a₁,...,a_n}. If γ is a closed rectifiable curve in U which does not meet any of the a_k,

If γ is a positively oriented simple closed curve, I(γ, a_k) = 1 if a_k is in the interior of γ, and 0 if not, so

with the sum over those k for which a_k is inside γ.

Here, Res(f, a_k) denotes the residue of f at a_k, and I(γ, a_k) is the winding number of the curve γ about the point a_k. This winding number is an integer which intuitively measures how many times the curve γ winds around the point a_k; it is positive if γ moves in a counter clockwise ("mathematically positive") manner around a_k and 0 if γ doesn't move around a_k at all.