Preface xi

who invited me to speak at California State University, Fresno on my experiences

teaching this course, also made useful comments. My colleague Jeremy Tyson made

many valuable suggestions on both the mathematics and the exposition. I asked

several friends to look at the N-th draft for various large N. Bob Vanderbei, Rock

Rodini, and Mike Bolt all made many useful comments which I have incorporated.

I thank Sergei Gelfand and Ed Dunne of the American Mathematical Society for

encouragement; Ed Dunne provided me marked-up versions of two drafts and shared

with me, in a lengthy phone conversation, his insights on how to improve and

complete the project. Cristin Zannella and Arlene O’Sean of the AMS oversaw the

ﬁnal editing and other ﬁnishing touches. Finally I thank my wife Annette and our

four children for their love.

Preface for the student

I hope that this book reveals the beauty and usefulness of complex numbers to

you. I want you to enjoy both reading it and solving the problems in it. Perhaps you

will spot something in your own area of interest and beneﬁt from applying complex

numbers to it. Students in my classes have found applications of ideas from this

book to physics, music, engineering, and linguistics. Several students have become

interested in historical and philosophical aspects of complex numbers. I have not

yet seen anyone get excited about the hysterical aspects of complex numbers.

At the very least you should see many places where complex numbers shed a

new light on things you have learned before. One of my favorite examples is trig

identities. I found them rather boring in high school and later I delighted in proving

them more easily using the complex exponential function. I hope you have the same

experience. A second example concerns certain deﬁnite integrals. The techniques

of complex analysis allow for stunningly easy evaluations of many calculus integrals

and seem to lie within the realm of science ﬁction.

This book is meant to be readable, but at the same time it is precise and rigor-

ous. Sometimes mathematicians include details that others feel are unnecessary or

obvious, but do not be alarmed. If you do many of the exercises and work through

the examples, then you should learn plenty and enjoy doing it. I cannot stress

enough two things I have learned from years of teaching mathematics. First, stu-

dents make too few sketches. You should strive to merge geometric and algebraic

reasoning. Second, deﬁnitions are your friends. If a theorem says something about

a concept, then you should develop both an intuitive sense of the concept and the

discipline to learn the precise deﬁnition. When asked to verify something on an

exam, start by writing down the deﬁnition of that something. Often the deﬁnition

suggests exactly what you should do!

Some sections and paragraphs introduce more sophisticated terminology than

is necessary at the time, in order to prepare for later parts of the book and even

for subsequent courses. I have tried to indicate all such places and to revisit the

crucial ideas. In case you are struggling with any material in this book, remain

calm. The magician will reveal his secrets in due time.