Theory of Complex Functions

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Springer Science & Business Media, Dec 6, 2012 - Mathematics - 458 pages
A lively and vivid look at the material from function theory, including the residue calculus, supported by examples and practice exercises throughout. There is also ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations - in the original language with their English translation - from their classical works. Yet the book is far from being a mere history of function theory, and even experts will find a few new or long forgotten gems here. Destined to accompany students making their way into this classical area of mathematics, the book offers quick access to the essential results for exam preparation. Teachers and interested mathematicians in finance, industry and science will profit from reading this again and again, and will refer back to it with pleasure.
 

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Contents

Historical Introduction
1
6 Connected Spaces Regions in C
39
ComplexDifferential Calculus
45
3 Holomorphic functions
56
4 Partial differentiation with respect to a y z and 2
63
Holomorphy and Conformality Biholomorphic Mappings
71
2 Biholomorphic mappings
80
Modes of Convergence in Function Theory
91
5 Special Taylor series Bernoulli numbers
220
CauchyWeierstrassRiemann Function Theory
227
3 The Cauchy estimates and inequalities for Taylor coefficients
241
4 Convergence theorems of WEIERSTRASS
248
5 The open mapping theorem and the maximum principle
256
Miscellany
265
3 Holomorphic logarithms and holomorphic roots
276
6 Asymptotic power series developments
294

2 Convergence criteria
101
Power Series
109
2 Examples of convergent power series
115
3 Holomorphy of power series
123
Elementary Transcendental Functions
133
2 The epimorphism theorem for expz and its consequences
141
3 Polar coordinates roots of unity and natural boundaries
148
4 Logarithm functions
154
5 Discussion of logarithm functions
160
Part B The Cauchy Theory
167
2 Properties of complex path integrals
178
The Integral Theorem Integral Formula and Power Series
191
2 Cauchys Integral Formula for discs
201
3 The development of holomorphic functions into power series
208
4 Discussion of the representation theorem
214
mappings 3 The local normal form 4 Geometric interpretation
303
2 Automorphisms of punctured domains
310
Convergent Series of Meromorphic Functions
321
4 The EISENSTEIN theory of the trigonometric functions
335
Laurent Series and Fourier Series
343
2 Properties of Laurent series
356
4 The theta function
365
The Residue Calculus
377
2 Consequences of the residue theorem
387
Definite Integrals and the Residue Calculus
395
Short Biographies of ABEL CAUCHY EISENSTEIN EULER RIEMANN
417
Literature
423
Symbol Inder
435
Subject Inder
443
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