Theory of Complex Functions

Front Cover
Springer Science & Business Media, 1991 - Mathematics - 453 pages
The material from function theory, up to the residue calculus, is developed in a lively and vivid style, well motivated throughout by examples and practice exercises. Additionally, there is ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations (original language together with English translation) from their classical works. Yet the book is far from being a mere history of function theory. Even experts will find here few new or long forgotten gems, like Eisenstein's novel approach to the circular functions. This book is destined to accompany many students making their way into a classical area of mathematics which represents the most fruitful example to date of the intimate connection between algebra and analysis. For exam preparation it offers quick access to the essential results and an abundance of interesting inducements. Teachers and interested mathematicians in finance, industry and science will also find reading it profitable, again and again referring to it with pleasure.
 

Contents

Historical Introduction
1
Part A Elements of Function Theory
9
2 Fundamental topological concepts
17
4 Convergent and absolutely convergent series
26
6 Connected spaces Regions in C
39
ComplexDifferential Calculus
45
3 Holomorphic functions
56
4 Partial differentiation with respect to x y z and ž
65
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
253
Miscellany
265
3 Holomorphic logarithms and holomorphic roots
276
Logarithmic derivative Existence lemma 2 Homologically simply
283
6 Asymptotic power series developments
297

Holomorphy and Conformality Biholomorphic Mappings
71
2 Biholomorphic mappings
80
n for automorphisms of E4 Homogeneity of E and
91
2 Convergence criteria
98
3 Normal convergence of series
104
2 Examples of convergent power series
117
Formal termwise differentiation and integration
123
4 Logarithm functions
154
5 Discussion of logarithm functions
160
Complex Integral Calculus
167
2 Properties of complex path integrals
179
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
Isolated Singularities Meromorphic Functions
303
281
309
2 Automorphisms of punctured domains
310
Convergent Series of Meromorphic Functions
321
2 The partial fraction development of π сot
329
4 The EISENSTEIN theory of the trigonometric functions
335
Laurent Series and Fourier Series
343
3 Periodic holomorphic functions and Fourier series
361
The Residue Calculus
377
2 Consequences of the residue theorem
387
2 Further evaluation of integrals
401
Short Biographies of ABEL CAUCHY EISENSTEIN EULER RIEMANN
417
Symbol Index
435
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