The statement is as follows. Suppose U is a simply connected open subset of the complex plane C, a_{1},...,a_{n} are finitely many points of U and f is a function which is defined and holomorphic on U \ {a_{1},...,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
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 halfcircle in the upper or lower halfplane. The integral over this curve can then be computed using the residue theorem. Often, the halfcircle part of the integral will tend towards zero if it is large enough, leaving only the realaxis part of the integral, the one we were originally interested in.
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