This book is an ideal text for an advanced course in the theory of complex functions. The author leads the reader to experience function theory personally and to participate in the work of the creative mathematician. The book contains numerous glimpses of the function theory of several complex variables, which illustrate how autonomous this discipline has become. Topics covered include Weierstrass's product theorem, Mittag-Leffler's theorem, the Riemann mapping theorem, and Runge's theorems on approximation of analytic functions. In addition to these standard topics, the reader will find Eisenstein's proof of Euler's product formula for the sine function; Wielandt's uniqueness theorem for the gamma function; a detailed discussion of Stirling's formula; Iss'sa's theorem; Besse's proof that all domains in C are domains of holomorphy; Wedderburn's lemma and the ideal theory of rings of holomorphic functions; Estermann's proofs of the overconvergence theorem and Bloch's theorem; a holomorphic imbedding of the unit disc in C3; and Gauss's expert opinion of November 1851 on Riemann's dissertation. Remmert elegantly presents the material in short clear sections, with compact proofs and historical comments interwoven throughout the text. The abundance of examples, exercises, and historical remarks, as well as the extensive bibliography, will make this book an invaluable source for students and teachers. TOC:Contents: Infinite Products and Partial Fraction Series: Infinite Poducts of Holomorphic Functions.- The Gamma Function.- Entire Functions with Prescribed Zeros.- Holomorphic Functions with Prescribed Zeros.- Iss'sa's Theorem. Regions of Holomorphy.- Functions with Prescribed Principal Parts. Mapping Theory: The Theorems of Montel and Vitali..- The Riemann Mapping Theorem.- Appendix to Chapter 8: Caratheodory-Koebe Theory.- Automorphisms and Finite Inner Maps. Selecta: The Theorems of Bloch, Picard, and Schottky.- Boundary Behavior of Power Series.- Runge Theory for Compact Sets.- Runge Theory for Domains.- Invariance of the Number of Holes. Short Biographies. Name Index. Subject Index.
