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.
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