Complex Analysis in One Variable

The original edition of this book has been out of print for some years. Since the book was first published, several people have remarked on the absence of exercises and expressed the opinion that the book would have been more useful had exercises been included. This book presents complex analysis in one variable in the context of modern mathematics, with clear connections to several complex variables, de Rham theory, real analysis, and other branches of mathematics. Thus, covering spaces are used explicitly in dealing with Cauchy's theorem, real variable methods are illustrated in the Loman-Menchoff theorem and in the corona theorem, and the algebraic structure of the ring of holomorphic functions is studied.Using the unique position of complex analysis, a field drawing on many disciplines, the book also illustrates powerful mathematical ideas and tools, and requires minimal background material. Cohomological methods are introduced, both in connection with the existence of primitives and in the study of meromorphic functionas on a compact Riemann surface. The proof of Picard's theorem given here illustrates the strong restrictions on holomorphic mappings imposed by curvature conditions.New to this second edition, a collection of over 100 pages worth of exercises, problems, and examples gives students an opportunity to consolidate their command of complex analysis and its relations to other branches of mathematics, including advanced calculus, topology, and real applications. This volume presents complex analysis in one variable in the context of modern mathematics, with connections to several complex variables, de Rham theory, real analysis, and other branches of mathematics. Covering spaces are used in dealing with Cauchy's theorem, real variable methods are illustrated in the Loman-Menchoff theorem and in the corona theorem, and the algebraic structure of the ring of holomorphic functions on a domain in C is studied. The book illustrates mathematical ideas and tools. Cohomological methods are introduced, in connection with the existence of primatives and in the study of meromorphic funtion as on a compact Riemann surface. This second edition has exercises, problems and examples for students, so they can consolidate their command of complex analysis and its relations to other branches of mathematics.