Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.13 Definite integrals using contour integration

We note that result (24.60) is a special case of (24.63) in whichθ 2 is equal to

θ 1 +2π.

24.13 Definite integrals using contour integration

The remainder of this chapter is devoted to methods of applying contour integra-

tion and the residue theorem to various types of definite integral. However, three

applications of contour integration, in which obtaining a value for the integral is

not the prime purpose of the exercise, have been postponed until chapter 25. They

are the location of the zeros of a complex polynomial, the evaluation of the sums

of certain infinite series and the determination of inverse Laplace transforms.

For the integral evalations considered here, not much preamble is given since,

for this material, the simplest explanation is felt to be via a series of worked

examples that can be used as models.

24.13.1 Integrals of sinusoidal functions

Suppose that an integral of the form

∫ 2 π

0

F(cosθ,sinθ)dθ (24.64)

is to be evaluated. It can be made into a contour integral around the unit circle

Cby writingz=expiθ, and hence

cosθ=^12 (z+z−^1 ), sinθ=−^12 i(z−z−^1 ),dθ=−iz−^1 dz. (24.65)

This contour integral can then be evaluated using the residue theorem, provided

the transformed integrand has only a finite number of poles inside the unit circle

andnoneonit.

Evaluate

I=

∫ 2 π

0

cos 2θ

a^2 +b^2 − 2 abcosθ

dθ, b > a > 0. (24.66)

By de Moivre’s theorem (section 3.4),

cosnθ=^12 (zn+z−n). (24.67)

Usingn= 2 in (24.67) and straightforward substitution for the other functions ofθin
(24.66) gives

I=

i

2 ab

∮

C

z^4 +1

z^2 (z−a/b)(z−b/a)

dz.

Thus there are two poles insideC, a double pole atz= 0 and a simple pole atz=a/b
(recall thatb>a).
We could find the residue of the integrand atz= 0 by expanding the integrand as a
Laurent series inzand identifying the coefficient ofz−^1. Alternatively, we may use the