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,
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