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

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.