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Full Description
Complex analysis is one of the most beautiful subjects that we learn as graduate students. Part of the joy comes from being able to arrive quickly at some "real theorems". The fundamental techniques of complex variables are also used to solve real problems in neighboring subjects, such as number theory or PDEs. This book is a text for a first-year graduate course in complex analysis. It is an engaging and modern introduction to the subject, reflecting the authors' expertise both as mathematicians and as expositors. All the material usually treated in such a course is covered here, but following somewhat different principles. To begin with, the authors emphasize how this subject is a natural outgrowth of multivariable real analysis. Complex function theory has long been a flourishing independent field. However, an efficient path into the subject is to observe how its rudiments arise directly from familiar ideas in calculus. The authors pursue this point of view by comparing and contrasting complex analysis with its real variable counterpart. Explanations of certain topics in complex analysis can sometimes become complicated by the intermingling of the analysis and the topology.
Here, the authors have collected the primary topological issues in a separate chapter, leaving the way open for a more direct and less ambiguous approach to the analytic material. The book concludes with several chapters on special topics, including full treatments of special functions, the prime number theorem, and the Bergman kernel. The authors also treat $Hp$ spaces and Painleve's theorem on smoothness to the boundary for conformal maps. A large number of exercises are included. Some are simply drills to hone the students' skills, but many others are further developments of the ideas in the main text. The exercises are also used to explore the striking interconnectedness of the topics that constitute complex analysis.
Contents
Fundamental concepts; Complex line integrals; Applications of the Cauchy integral; Meromorphic functions and residues; The zeros of a holomorphic function; Holomorphic functions as geometric mappings; Harmonic functions; Infinite series and products; Applications of infinite sums and products; Analytic continuation; Topology; Rational approximation theory; Special classes of holomorphic functions; Hilbert spaces of holomorphic functions, the Bergman kernel, and biholomorphic mappings; Special functions; The prime number theorem; Real analysis; The statement and proof of Goursat's theorem; References; Index
