Preface

This book developed from a course given in the Campus Honors Program at the

University of Illinois Urbana-Champaign in the fall semester of 2008. The aims

of the course were to introduce bright students, most of whom were freshmen,

to complex numbers in a friendly, elegant fashion and to develop reasoning skills

belonging to the realm of elementary complex geometry. In the spring semester

of 2010 I taught another version of the course, in which a draft of this book was

available online. I therefore wish to acknowledge the Campus Honors Program at

UIUC for allowing me to teach these courses and to thank the 27 students who

participated in them.

Many elementary mathematics and physics problems seem to simplify magically

when viewed from the perspective of complex analysis. My own research interests

in functions of several complex variables and CR geometry have allowed me to

witness this magic daily. I continue the preface by mentioning some of the speciﬁc

topics discussed in the book and by indicating how they ﬁt into this theme.

Every discussion of complex analysis must spend considerable time with power

series expansions. We include enough basic analysis to study power series rigorously

and to solidify the backgrounds of the typical students in the course. In some sense

two speciﬁc power series dominate the subject: the geometric and exponential

series.

The geometric series appears all throughout mathematics and physics and even

in basic economics. The Cauchy integral formula provides a way of deriving from

the geometric series the power series expansion of an arbitrary complex analytic

function. Applications of the geometric series appear throughout the book.

The exponential series is of course also crucial. We deﬁne the exponential

function via its power series, and we deﬁne the trigonometric functions by way of

the exponential function. This approach reveals the striking connections between

the functional equation

ez+w

=

ezew

and the profusion of trigonometric identities.

vii