Cauchy Residue Theorem

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.

Statement

The statement is as follows:

Let be a simply connected open subset of the complex plane containing a finite list of points ,
,
and a function defined and holomorphic on . Let be a closed rectifiable curve in , and denote the winding number of around by . The line integral of around is equal to times the sum of residues of at the points, each counted as many times as winds around the point:
$\oint_\gamma f(z)\, dz = 2\pi i \sum_^n \operatorname(\gamma, a_k) \operatorname( f, a_k ).$

If is a positively oriented simple closed curve, if is in the interior of , and 0 if not, therefore
$\oint_\gamma f(z)\, dz = 2\pi i \sum \operatorname( f, a_k )$
with the sum over those inside .

The relationship of the residue theorem to Stokes' theorem is given by the Jordan curve theorem. The general plane curve must first be reduced to a set of simple closed curves whose total is equivalent to for integration purposes; this reduces the problem to finding the integral of along a Jordan curve with interior . The requirement that be holomorphic on is equivalent to the statement that the exterior derivative on . Thus if two planar regions and of enclose the same subset of , the regions and lie entirely in , and hence
$\int_ d(f \, dz) - \int_ d(f \, dz)$
is well-defined and equal to zero. Consequently, the contour integral of along is equal to the sum of a set of integrals along paths , each enclosing an arbitrarily small region around a single — the residues of (up to the conventional factor ) at . 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.

Examples

An integral along the real axis

The integral
$\int_^\infty \frac\,dx$

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 and define the contour that goes along the real line from to and then counterclockwise along a semicircle centered at 0 from to . Take to be greater than 1, so that the imaginary unit is enclosed within the curve. Now consider the contour integral
$\int_C \,dz = \int_C \frac\,dz.$

Since is an entire function (having no singularities at any point in the complex plane), this function has singularities only where the denominator is zero. Since , that happens only where or . Only one of those points is in the region bounded by this contour. Because is
$\begin \frac & =\frac\left(\frac-\frac\right) \\ & =\frac -\frac , \end$
the residue of at is
$\operatorname_f(z)=\frac.$

According to the residue theorem, then, we have
$\int_C f(z)\,dz=2\pi i\cdot\operatorname\limits_f(z)=2\pi i \frac = \pi e^.$

The contour may be split into a straight part and a curved arc, so that
$\int_ f(z)\,dz+\int_ f(z)\,dz=\pi e^$
and thus
$\int_^a f(z)\,dz =\pi e^-\int_ f(z)\,dz.$

Using some estimations, we have
$\left, \int_\frac\,dz\ \leq \pi a \cdot \sup_ \left, \frac \ \leq \pi a \cdot \sup_ \frac \leq \frac,$
and
$\lim_ \frac = 0.$

The estimate on the numerator follows since , and for complex numbers along the arc (which lies in the upper half-plane), the argument of lies between 0 and . So,
$\left, e^\ = \left, e^\=\left, e^\=e^ \le 1.$

Therefore,
$\int_^\infty \frac\,dz=\pi e^.$

If then a similar argument with an arc that winds around rather than shows that

$\int_^\infty\frac\,dz=\pi e^t,$

and finally we have
$\int_^\infty\frac\,dz=\pi e^.$

(If then the integral yields immediately to elementary calculus methods and its value is .)

An infinite sum

The fact that has simple poles with residue 1 at each integer can be used to compute the sum
$\sum_^\infty f(n).$

Consider, for example, . Let be the rectangle that is the boundary of with positive orientation, with an integer . By the residue formula,

$\frac \int_ f(z) \pi \cot(\pi z) \, dz = \operatorname\limits_ + \sum_^N n^.$

The left-hand side goes to zero as since the integrand has order $O(n^)$. On the other hand,. Note that the Bernoulli number $B_$ is denoted by $B_$ in Whittaker & Watson's book.

$\frac \cot\left(\frac\right) = 1 - B_2 \frac + \cdots$ where the Bernoulli number $B_2 = \frac.$

(In fact, .) Thus, the residue is . We conclude:

$\sum_^\infty \frac = \frac$
which is a proof of the Basel problem.

The same trick can be used to establish the sum of the Eisenstein series:
$\pi \cot(\pi z) = \lim_ \sum_^N (z - n)^.$

We take with a non-integer and we shall show the above for . The difficulty in this case is to show the vanishing of the contour integral at infinity. We have:
$\int_ \frac \, dz = 0$
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,
$\int_ f(z) \pi \cot(\pi z) \, dz = \int_ \left(\frac + \frac\right) \pi \cot(\pi z) \, dz$
goes to zero as .

See also

* Cauchy's integral formula
* Glasser's master theorem
*Jordan's lemma
* Methods of contour integration
* Morera's theorem
* Nachbin's theorem
*Residue at infinity
* Logarithmic form

Notes

References

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External links

* {{springer, title=Cauchy integral theorem, id=p/c020900

Residue theorem
in MathWorld

Theorems in complex analysis
Analytic functions