Complex Analysis

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Complex Analysis

Some Applications of the Residue Theorem∗ Supplementary Lecture Notes MATH 322, Complex Analysis Winter 2005
Pawel Hitczenko Department of Mathematics Drexel University Philadelphia, PA 19104, U.S.A. email: phitczenko@math.drexel.edu

∗I

would like to thank Frederick Akalin for pointing out a couple of typos.

1

1

Introduction

These notes supplement a freely downloadable book Complex Analysis by George Cain (henceforth referred to as Cain’s notes), that I served as a primary text for an undergraduate level course in complex analysis. Throughout these notes I will make occasional references to results stated in these notes. The aim of my notes is to provide a few examples of applications of the residue theorem. The main goal is to illustrate how this theorem can be used to evaluate various types of integrals of real valued functions of real variable. Following Sec. 10.1 of Cain’s notes, let us recall that if C is a simple, closed contour and f is analytic within the region bounded by C except for finitely many points z0 , z1 , . . . , zk then
k

f (z)dz = 2πi
C j=0

Resz=zj f (z),

where Resz=a f (z) is the residue of f at a.

2
2.1

Evaluation of Real-Valued Integrals.
Definite integrals involving trigonometric functions


We begin by briefly discussing integrals of the form F (sin at, cos bt)dt.
0

(1)

Our method is easily adaptable for integrals over a different range, for example between 0 and π or between ±π. Given the form of an integrand in (1) one can reasonably hope that the integral results from the usual parameterization of the unit circle z = eit , 0 ≤ t ≤ 2π. So, let’s try z = eit . Then (see Sec. 3.3 of Cain’s notes), cos bt = z b + 1/z b eibt + e−ibt = , 2 2 sin at = eiat − e−iat z a − 1/z a = . 2i 2i

Moreover, dz = ieit dt, so that dt = Putting all of this into (1) yields


dz . iz

F (sin at, cos bt)dt =
0 C

F

z a − 1/z a z b + 1/z b , 2i 2

dz , iz

where C is the unit circle. This...

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