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 and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. The residue theorem should not be confused with special cases of the generalized Stokes' theorem; however, the latter can be used as an ingredient of its proof.

Statement of Cauchy's residue theorem

The statement is as follows: Residue theorem: Let U be a simply connected open subset of the complex plane containing a finite list of points and a function f holomorphic on U_0. Letting \gamma be a closed rectifiable curve in U_0, and denoting the residue of f at each point a_k by and the winding number of \gamma around a_k by the line integral of f around \gamma is equal to 2\pi i times the sum of residues, each counted as many times as \gamma winds around the respective point: If \gamma is a positively oriented simple closed curve, is 1 if a_k is in the interior of \gamma and 0 if not, therefore with the sum over those a_k inside \gamma. The relationship of the residue theorem to Stokes' theorem is given by the Jordan curve theorem. The general plane curve γ must first be reduced to a set of simple closed curves whose total is equivalent to \gamma for integration purposes; this reduces the problem to finding the integral of f, dz along a Jordan curve \gamma_i with interior V. The requirement that f be holomorphic on is equivalent to the statement that the exterior derivative on U_0. Thus if two planar regions V and W of U enclose the same subset {a_j} of {a_k}, the regions and lie entirely in U_0, hence is well-defined and equal to zero. Consequently, the contour integral of f, dz along is equal to the sum of a set of integrals along paths \gamma_j, each enclosing an arbitrarily small region around a single a_j — the residues of f (up to the conventional factor 2\pi i at {a_j}. Summing over we recover the final expression of the contour integral in terms of the winding numbers 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, forming a semicircle. 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 as the radius of the half-circle grows, leaving only the real-axis part of the integral, the one we were originally interested in.

Calculation of residues

Examples

An integral along the real axis

The integral arises in probability theory when calculating the characteristic function of the Cauchy distribution. It resists the techniques of elementary calculus but can be evaluated by expressing it as a limit of contour integrals. Suppose t > 0 and define the contour C that goes along the real line from −a to a and then counterclockwise along a semicircle centered at 0 from a to −a . Take a to be greater than 1, so that the imaginary unit i is enclosed within the curve. Now consider the contour integral Since eitz is an entire function (having no singularities at any point in the complex plane), this function has singularities only where the denominator z2 + 1 is zero. Since z2 + 1 = (z + i)(z − i) , that happens only where z = i or z = −i . Only one of those points is in the region bounded by this contour. Because f(z) is the residue of f(z) at z = i is According to the residue theorem, then, we have The contour C may be split into a straight part and a curved arc, so that and thus Using some estimations, we have and The estimate on the numerator follows since t > 0 , and for complex numbers z along the arc (which lies in the upper half-plane), the argument φ of z lies between 0 and π. So, Therefore, If t < 0 then a similar argument with an arc that winds around −i rather than i shows that [[Image:Contour example 2.svg|right|300px|thumb|The contour .]] and finally we have (If t = 0 then the integral yields immediately to elementary calculus methods and its value is π.)

Evaluating zeta functions

The fact that π cot(πz) has simple poles with residue 1 at each integer can be used to compute the sum Consider, for example, f(z) = z−2 . Let ΓN be the rectangle that is the boundary of [−N − 1⁄2, N + 1⁄2]2 with positive orientation, with an integer N. By the residue formula, The left-hand side goes to zero as N → ∞ since is uniformly bounded on the contour, thanks to using on the left and right side of the contour, and so the integrand has order O(N^{-2}) over the entire contour. On the other hand, where the Bernoulli number (In fact, z⁄2 cot(z⁄2) = iz⁄1 − e−iz − iz⁄2 .) Thus, the residue is −π2⁄3 . We conclude: which is a proof of the Basel problem. The same argument works for all where n is a positive integer, giving usThe trick does not work when, since in this case, the residue at zero vanishes, and we obtain the useless identity.

Evaluating Eisenstein series

The same trick can be used to establish the sum of the Eisenstein series: