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.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Historical Introduction
1
of continuity
9
2 Fundamental topological concepts
17
6 Connected spaces Regions in C
39
ComplexDifferential Calculus
45
3 Holomorphic functions
56
4 Partial differentiation with respect to x y z and z
65
Holomorphy and Conformality Biholomorphic Mappings
71
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
Isolated Singularities Meromorphic Functions
303

2 Biholomorphic mappings
80
Modes of Convergence in Function Theory
91
2 Convergence criteria
101
Chapter 4 Power Series
109
2 Examples of convergent power series
115
3 Holomorphy of power series
123
4 Logarithm functions
154
5 Discussion of logarithm functions
160
Part B The Cauchy Theory
167
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
5 Special Taylor series Bernoulli numbers
220
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
Chapter 14 Definite Integrals and the Residue Calculus
395
Short Biographies oAbel Cauchy Eisenstein Euler Riemann
417
Literature
423
Symbol Index
435
Subject Index
443
Copyright

Other editions - View all

Common terms and phrases

Bibliographic information