Classical Topics in Complex Function TheoryAn ideal text for an advanced course in the theory of complex functions, this book leads readers to experience function theory personally and to participate in the work of the creative mathematician. The author includes numerous glimpses of the function theory of several complex variables, which illustrate how autonomous this discipline has become. In addition to standard topics, readers will find Eisenstein's proof of Euler's product formula for the sine function; Wielandts uniqueness theorem for the gamma function; Stirlings formula; Isssas theorem; Besses proof that all domains in C are domains of holomorphy; Wedderburns lemma and the ideal theory of rings of holomorphic functions; Estermanns proofs of the overconvergence theorem and Blochs theorem; a holomorphic imbedding of the unit disc in C3; and Gausss expert opinion on Riemanns 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, combine to make an invaluable source for students and teachers alike |
Contents
| 3 | |
2 The Gamma Function | 33 |
3 Entire Functions with Prescribed Zeros | 73 |
4 Holomorphic Functions with Prescribed Zeros | 88 |
5 Isssas Theorem Domainsof Holomorphy | 107 |
6 Functions with Prescribed Principal Parts | 125 |
Part B Mapping Theory | 144 |
7 The Theorems of Montel and Vitali | 147 |
10 The Theorems of Bloch Picard and Schottky | 225 |
11 Boundary Behavior of Power Series | 243 |
12 Runge Theory for Compact Sets | 267 |
13 Runge Theory for Regions | 289 |
14 In variance of the Number of Holes | 308 |
Classical Topics in Complex Function Theory | 319 |
Short Biographies | 322 |
Symbol Index | 329 |
8 The Riemann Mapping Theorem | 166 |
9 Automorphisms and Finite Inner Maps | 203 |
Part C Selecta | 223 |
| 331 | |
| 336 | |
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Common terms and phrases
Algebraic approximated uniformly arbitrary Aut G biholomorphic boundary point Carathéodory Cauchy closed path coefficients compact set compactly in G construction converges compactly converges normally Corollary denote domain G domain of holomorphy E O(G entire function equation Euler example existence theorem expansion follows immediately formula function f function holomorphic function theory Funktionen hence holes holomorphic functions ideal integral Journ Koebe domain lemma Let f Let G little theorem locally finite logarithmic map f mathematician meromorphic functions Mittag-Leffler series Mittag-Leffler's theorem Montel Montel's theorem neighborhood nonconstant nonvanishing normal convergence null homologous Picard's Picard's little theorem polynomial positive divisor power series principal part distribution product theorem Proof Proposition prove radius region Riemann mapping theorem Runge pair Runge's Section sequence fn simply connected simply connected domains Springer statement Subsection Taylor series topological Vitali's theorem Weierstrass product Werke zero