# Residue theorem

The **residue theorem**, sometimes called **Cauchy’s Residue Theorem**, in complex analysis is a powerful tool to evaluate line integrals of analytic 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. From a geometrical perspective, it is a special case of the generalized Stokes’ theorem.

The statement is as follows. Suppose *U* is a simply connected open subset of the complex plane, and *a*_{1},…,*a*_{n} are finitely many points of *U* and *f* is afunction which is defined and holomorphic on *U* \ {*a*_{1},…,*a*_{n}}. If γ is a rectifiable curve in *U* which bounds the *a*_{k}, but does not meet any and whose start point equals its endpoint, then

If γ is a positively oriented Jordan curve, I(γ, *a*_{k}) = 1 and so

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 integerwhich intuitively measures how many times 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.

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 *f* *dz*along γ_{i}=∂V is equal to the sum of a set of integrals along paths λ_{j}, each enclosing an arbitrarily small region around a single *a*_{j}—the residues of *f* (up to the conventional factor 2π*i*) at {*a*_{j}}. Summing over {γ_{i}}, we recover the final expression of the contour integral in terms of the winding numbers {I(γ, *a*_{k})}.

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 as the radius of the half-circle grows, leaving only the real-axis part of the integral, the one we were originally interested in.

## Example

The integral

arises in probability theory when calculating the characteristic function of the Cauchy distribution. It resists the techniques of elementarycalculus 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. 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

Using some estimations, we have

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.)

## References

- General references

- Ahlfors, Lars (1979),
*Complex Analysis*, McGraw Hill, ISBN 0-07-085008-9 - Mitronivić, Dragoslav; Kečkić, Jovan (1984),
*The Cauchy method of residues: Theory and applications*, D. Reidel Publishing Company, ISBN 90-277-1623-4 - Lindelöf, Ernst (1905),
*Le calcul des résidus et ses applications à la théorie des fonctions*, Editions Jacques Gabay (published 1989), ISBN 2-87647-060-8