Residue theorem 

The statement is as follows. Suppose U is a simply connected open subset of the complex plane C, 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 rectifiable curve in U which doesn't meet any of the points a_k and whose start point equals its endpoint, then

<math>\oint_\gamma f(z) dz =

Here, Res(f,a_k) denotes the residue of f at a_k, and n(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 often 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.

In order to evaluate real integrals, the residue theorem is used in the following manner: the integrand is extended to the complex plane and its residues are computed (which is usually easy), and a part of the real axis is extended to a closed curve by attaching a half-circle in the upper or lower half-plane. The integral over this curve can then be computed using the residue theorem. Often, the half-circle part of the integral will tend towards zero if it is large enough, leaving only the real-axis part of the integral, the one we were originally interested in.

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