Complex Analysis

This volume is an enlarged edition of a classic textbook on complex analysis. In addition to the classical material of the first edition it provides a concise and accessible treatment of Loewner theory, both in the disc and in the half-plane. Some of the new material has been described in research papers only or appears here for the first time. Each chapter ends with exercises.

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Contents:Power SeriesHolomorphic and Analytic FunctionsExponential Function, Logarithm and Winding NumberBasic Theory of Holomorphic FunctionsGlobal TheoryIsolated SingularitiesThe Picard TheoremsGeometric Aspects and the Riemann Mapping TheoremMeromorphic Functions and Runge's TheoremsRepresentations of Meromorphic FunctionsThe Prime Number TheoremHarmonic FunctionsSubharmonic FunctionsVarious ApplicationsJordan RegionsThe Dirichlet Problem and Green's FunctionsExtending Riemann Maps to the BoundaryCarathéodory ConvergenceCapacitiesLoewner Theory
Readership: Advanced undergraduates and graduate students of mathematics including mathematicians interested in Loewner theory.
Key Features:It contains an accessible introduction to the results and techniques of complex function theory with recent simplificationsIt discusses compact hulls, half-plane hulls and their capacities. Examples are computed in detailIt contains recent results in Loewner theory and proofs of them, not just in the disc, but also in the half-plane