# Residue theorem

The **residue theorem** in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula.

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 I(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 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.

## Example

The integral

(which arises in probability theory as (a scalar multiple
of) the characteristic function of the Cauchy distribution)
resists the techniques of elementary calculus. We will
evaluate it by expressing it as a limit of contour integrals
along 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. The contour integral is

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
*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

It can be shown that **if t > 0 then**

Therefore **if t > 0 then**

A similar argument with an arc that winds around −*i*
rather than *i* shows that **if t < 0 then**

and finally we have

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

## Humor

Q: What's the value of a contour integral around Western Europe?

A: Zero, because all the Poles are in Eastern Europe. de:Residuensatz fr:Théorème des résidus he:משפט השאריות