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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. From a geometrical perspective, it can be seen as a special case of the generalized Stokes' theorem.

The statement is as follows:

Let U be a simply connected open subset of the complex plane containing a finite list of points *a*_{1}, ..., *a*_{n},
*U*_{0} = *U* \ {*a*_{1}, …, *a*_{n}},
and a function f defined and holomorphic on *U*_{0}. Let γ be a closed rectifiable curve in *U*_{0}, and denote the winding number of γ around *a*_{k} by I(*γ*, *a*_{k}). The line integral of f around γ is equal to 2*πi* times the sum of residues of f at the points, each counted as many times as γ winds around the point:

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, therefore

with the sum over those

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 {*γ*_{i}} whose total is equivalent to γ for integration purposes; this reduces the problem to finding the integral of *f* *dz* along a Jordan curve *γ*_{i} with interior V. The requirement that f be holomorphic on *U*_{0} = *U* \ {*a*_{k}} is equivalent to the statement that the exterior derivative *d*(*f* *dz*) = 0 on *U*_{0}. Thus if two planar regions V and W of U enclose the same subset {*a*_{j}} of {*a*_{k}}, the regions *V* \ *W* and *W* \ *V* lie entirely in *U*_{0}, and hence

is well-defined and equal to zero. Consequently, the contour integral of

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.

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 *e*^{itz} is an entire function (having no singularities at any point in the complex plane), this function has singularities only where the denominator *z*^{2} + 1 is zero. Since *z*^{2} + 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

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 *C*′ that winds around −*i* rather than *i* shows that

and finally we have

(If *t* = 0 then the integral yields immediately to elementary calculus methods and its value is π.)

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 Γ_{N} 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 the integrand has order . On the other hand,^{[2]}

where the Bernoulli number

(In fact, *z*/2 cot(*z*/2) = *iz*/1 − *e*^{−iz} − *iz*/2.) Thus, the residue Res_{z=0} is −*π*^{2}/3. We conclude:

which is a proof of the Basel problem.

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

We take *f*(*z*) = (*w* − *z*)^{−1} with w a non-integer and we shall show the above for w. The difficulty in this case is to show the vanishing of the contour integral at infinity. We have:

since the integrand is an even function and so the contributions from the contour in the left-half plane and the contour in the right cancel each other out. Thus,

goes to zero as