x Preface

Chapter 8 introduces additional appealing topics such as the fundamental the-

orem of algebra (for which we give three proofs), winding numbers, Rouche’s theo-

rem, Pythagorean triples, conformal mappings, the quaternions, and (a brief men-

tion of) complex analysis in higher dimensions. The section on conformal mappings

includes a brief discussion of non-Euclidean geometry. The section on quaternions

includes the observation that there are many quaternionic square roots of -1, and

hence it illuminates the earliest material used in deﬁning C. The ﬁnal result proved

concerns polarization; it justiﬁes treating z and z as independent variables, and

hence it also uniﬁes much of the material in the book.

Our bibliography includes many excellent books on complex analysis in one

variable. One naturally asks how this book diﬀers from those. The primary diﬀer-

ence is that this book begins at a more elementary level. We start at the logical

beginning, by discussing the natural numbers, the rational numbers, and the real

numbers. We include detailed discussion of some truly basic things, such as the

existence of square roots of positive real numbers, the irrationality of

√

2, and

several diﬀerent deﬁnitions of C itself. Hence most of the book can be read by

a smart freshman who has had some calculus, but not necessarily any real anal-

ysis. A second diﬀerence arises from the desire to engage an audience of bright

freshmen. I therefore include discussion, examples, and exercises on many topics

known to this audience via real variables, but which become more transparent us-

ing complex variables. My ninth grade mathematics class (more than forty years

ago) was tested on being able to write word-for-word the deﬁnitions of hyperbola,

ellipse, and parabola. Most current college freshmen know only vaguely what these

objects are, and I found myself reciting those deﬁnitions when I taught the course.

During class I also paused to carefully prove that .999... really equals 1. Hence

the book contains various basic topics, and as a result it enables spiral learning.

Several concepts are revisited with high multiplicity throughout the book. A third

diﬀerence from the other books arises from the inclusion of several unusual topics,

as described throughout this preface.

I hope, with some conﬁdence, that the text conveys my deep appreciation for

complex analysis and geometry. I hope, but with more caution, that I have purged

all errors from it. Most of all I hope that many will enjoy reading it and solving

the exercises in it.

I began expanding the sketchy notes from the course into this book during

the spring 2009 semester, during which I was partially supported by the Kenneth

D. Schmidt Professorial Scholar award. I therefore wish to thank Dr. Kenneth

Schmidt and also the College of Arts and Sciences at UIUC for awarding me this

prize. I have received considerable research support from the NSF for my work

in complex analysis; in particular I acknowledge support from NSF grant DMS-

0753978. The students in the ﬁrst version of the course survived without a text;

their enthusiasm and interest merit praise. Over the years many other students

have inspired me to think carefully how to present complex analysis and geometry

with elegance. Another positive influence on the evolution from sketchy notes to

this book was working through some of the material with Bill Heiles, Professor

of Piano at UIUC and one who appreciates the art of mathematics. Jing Zou,

computer science student at UIUC, prepared the ﬁgures in the book. Tom Forgacs,