Classical Topics in Complex Function Theory

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Springer Science & Business Media, Mar 14, 2013 - Mathematics - 352 pages
An 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
 

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Contents

1 Infinite Products of Holomorphic Functions
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
144
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

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