Preface ix

go on to study Laurent series, the Residue Theorem, Rouch´ e’s Theorem, inverse

functions and the Open Mapping Theorem. The chapter ends with a discussion of

homotopy and its relationship to integrals around closed paths. We normally do

not cover this last section in our undergraduate course, but it would certainly be

appropriate for a graduate course using this text.

Residue theory is covered in Chapter 5. We discuss techniques for computing

residues as well as a wide variety of applications of residue theory to problems

involving the calculation of integrals. We normally do not cover all of this in our

undergraduate course. We typically skip the last two sections of this chapter in

favor of covering most of Chapter 6.

Chapter 6 deals with conformal maps. We prove the Riemann Mapping The-

orem and show how it can be used to transform problems involving analytic or

harmonic functions on a simply connected subset of the plane to the analogous

problems on the unit disc. We use this technique to study the Dirichlet problem

on open, proper, simply connected subsets of C. We discuss applications to heat

flow, electrostatics, and hydrodynamics. This material is of particular interest to

students in a course on complex variables for engineering students.

The goal of Chapter 7 on analytic continuation is to prove the Picard theorems

concerning meromorphic functions at essential singularities. Along the way, we

prove the Schwarz Reflection Principle and the Monodromy Theorem, and discuss

lifting analytic functions through a covering map.

Chapters 8 and 9 are normally not covered in our one-semester undergraduate

complex variables course. Some of the topics in these chapters are, however, of-

ten covered in the graduate course. In Chapter 8 we use Weierstrass products to

construct an analytic function with a given discrete set of zeroes, with prescribed

multiplicities. This is the Weierstrass Theorem. It leads directly to the Weierstrass

Factorization Theorem for entire functions. We also prove the Mittag-Leffler Theo-

rem – which gives the existence of a meromorphic function with a prescribed set of

poles and principle parts. The final result of the chapter is the proof of Hadamard’s

Theorem characterizing entire functions of finite order. This is a key ingredient in

the proof of the Prime Number Theorem in Chapter 9.

In Chapter 9 we introduce the gamma and zeta functions, develop their basic

properties, discuss the Riemann Hypothesis, and prove the Prime Number The-

orem. Course notes by our colleague Dragan Mili´ ci´ c provided the original basis

for this material. It then went through several years of expansion and refinement

before reaching its present form. There is a great deal of technical calculation in

this material and we do not cover it in our undergraduate course. However, a few

lectures summarizing this material have proved to be a popular way to end the

graduate course.

Several standard texts in complex variables and related topics were useful guides

in preparing this text. These may also be of interest to the student who wishes to

learn more about the subject. They are listed in a short bibliography at the end of

the text.