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
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