viii Preface

computational examples and exercises. The applications in Chapter 6 are given

special emphasis.

When taught as a one-semester graduate course, we assume students have some

knowledge of complex numbers and so Chapter 1 is given just a brief review. Chap-

ters 2, 3, and 4 are covered completely, parts of Chapters 5 and 6 are covered, along

with Chapter 7. If time allows, Chapters 8 and 9 are summarized at the end of the

course. In the homework, the emphasis is on the theoretical exercises.

We have tried to present this material in a fashion which is both rigorous

and concise, with simple, straightforward explanations. We feel that the modern

tendency to expand textbooks with ever more material, excessively verbose ex-

planations, and more and more bells and whistles, simply gets in the way of the

student’s understanding of the material.

The exercises differ widely in level of abstraction and level of diﬃculty. They

vary from the simple to the quite diﬃcult and from the computational to the the-

oretical. There are exercises that ask students to prove something or to construct

an example with certain properties. There are exercises that ask students to apply

theoretical material to help do a computation or to solve a practical problem. Each

section contains a number of examples designed to illustrate the material of the

section and to teach students how to approach the exercises for that section.

This text, in its various incarnations, has been used by the author and his

colleagues for several years at the University of Utah. Each use has led to improve-

ments, additions, and corrections.

The text begins, in Chapter 1, with a discussion of the fact that the real number

system is insuﬃcient for some purposes (solving polynomial equations). The system

of complex numbers is developed in an attempt to remedy this problem. We then

study basic arithmetic of complex numbers, convergence of sequences and series of

complex numbers, power series, the exponential function, polar form for complex

numbers and the complex logarithm.

The core material of a complex variables course is the material covered here

in Chapters 2 and 3. Analytic functions are introduced in Chapter 2 as functions

which have a complex derivative. This leads to a discussion of the Cauchy-Riemann

equations and harmonic functions. We then introduce contour integrals and the

index function (winding number) for closed paths. We prove the Cauchy-integral

theorem for triangles and then for convex sets. We believe that this approach leads

to the simplest and quickest rigorous proof of a form of Cauchy’s theorem that can

be used to prove the existence of power series expansions for analytic functions.

The proof that analytic functions have power series expansions occurs early in

Chapter 3. This is followed by a wide range of powerful applications with simple

proofs – Morera’s Theorem, Liouville’s Theorem, the Fundamental Theorem of

Algebra, the characterization of zeroes and singularities of analytic functions, the

Maximum Modulus Principle, and Schwarz’s Lemma.

Chapter 4 begins with a proof of the general form of Cauchy’s Theorem and

Cauchy’s Formula. These theorems involve functions which are analytic on a general

open set. The integration takes place around a cycle (a generalization of a closed

path) which is required to have index zero about any point not in the set. We