Complex analysis in one variable / Raghavan Narasimhan

This book provides an alternative for a first-year graduate course in the classical theory of functions of one complex variable. A theme of the book is to relate classical complex analysis to other branches of mathematics. It includes many of the standard topics for a basic graduate course, but the exposition and viewpoint are strongly influenced by the theory of several complex variables. In fact, there is even a brief chapter dealing with functions of several complex variables; this is used to show that their behavior is sometimes quite different from functions of one complex variable. One pleasant feature of the text is an early and elementary treatment of the theorems of Picard, Landau and Schottky via Ahlfors' extension of Schwarz's lemma in Chapter 4. In addition to covering many of the standard topics, the author also provides a treatment of covering spaces, the inhomogeneous Cauchy-Riemann equation, compact Riemann surfaces and Wolff's proof of the corona theorem. Overall, the author's approach is analytic rather than geometric. Some classical topics not covered include Möbius transformations, elliptic functions and entire functions. The lack of exercise sets will hinder the use of the book as a text. Also, there is a substantial (for a typical first-year graduate course) list of prerequisites: multivariable calculus, point set topology, elementary Lebesgue integration and elementary functionalanalysis. (MathSciNet)

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