Preface During their first year at the University of Chicago, graduate students in mathematics take classes in algebra, analysis, and geometry, one of each every quarter. The analysis courses typically cover real analysis and measure theory, functional analysis and applications, and complex analysis. This book grew out of the author’s notes for the complex analysis classes which he taught during the Spring quarters of 2007 and 2008. These courses covered elementary aspects of complex analysis such as the Cauchy integral theorem, the residue theorem, Laurent series, and the Riemann mapping theorem, but also more advanced material selected from Riemann surface theory. Needless to say, all of these topics have been covered in excellent text- books as well as classic treatises. This book does not try to compete with the works of the old masters such as Ahlfors [1, 2], Hurwitz–Courant [44], Titchmarsh [80], Ahlfors–Sario [3], Nevanlinna [67], and Weyl [88]. Rather, it is intended as a fairly detailed, yet fast-paced introduction to those parts of the theory of one complex variable that seem most useful in other ar- eas of mathematics (geometric group theory, dynamics, algebraic geometry, number theory, functional analysis). There is no question that complex analysis is a cornerstone of a math- ematics specialization at every university and each area of mathematics re- quires at least some knowledge of it. However, many mathematicians never take more than an introductory class in complex variables which ends up being awkward and slightly outmoded. Often this is due to the omission of Riemann surfaces and the assumption of a computational, rather than a geometric point of view. The author has therefore tried to emphasize the intuitive geometric un- derpinnings of elementary complex analysis that naturally lead to Riemann vii
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