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Complex Analysis

by: Bak, Joseph; Newman, Donald J.

Complex Analysis

by: Bak, Joseph; Newman, Donald J.

ISBN 13:
9781441972873
ISBN 10:
1441972870
Edition: 3rd
Format: Hardcover
Copyright: 09/14/2010
Publisher: Springer Verlag

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Summary

This unusually lively textbook on complex variables introduces the theory of analytic functions, explores its diverse applications and shows the reader how to harness its powerful techniques. "Complex Analysis" offers new and interesting motivations for classical results and introduces related topics that do not appear in this form in other texts. Stressing motivation and technique, and complete with exercise sets, this volume may be used both as a basic text and as a reference. For this second edition, the authors have revised some of the existing material and have provided new exercises and solutions.

Preface to the Third Edition	p. v
Preface to the Second Edition	p. vii
The Complex Numbers	p. 1
Introduction	p. 1
The Field of Complex Numbers	p. 1
The Complex Plane	p. 4
The Solution of the Cubic Equation	p. 9
Topological Aspects of the Complex Plane	p. 12
Stereographic Projection; The Point at Infinity	p. 16
Exercises	p. 18
Functions of the Complex Variable z	p. 21
Introduction	p. 21
Analytic Polynomials	p. 21
Power Series	p. 25
Differentiability and Uniqueness of Power Series	p. 28
Exercises	p. 32
Analytic Functions	p. 35
Analyticity and the Cauchy-Riemann Equations	p. 35
The Functions e^z, sin z, cos z	p. 40
Exercises	p. 41
Line Integrals and Entire Functions	p. 45
Introduction	p. 45
Properties of the Line Integral	p. 45
The Closed Curve Theorem for Entire Functions	p. 52
Exercises	p. 56
Properties of Entire Functions	p. 59
The Cauchy Integral Formula and Taylor Expansion for Entire Functions	p. 59
Liouville Theorems and the Fundamental Theorem of Algebra; The Gauss-Lucas Theorem	p. 65
Newton's Method and Its Application to Polynomial Equations	p. 68
Exercises	p. 74
Properties of Analytic Functions	p. 77
Introduction	p. 77
The Power Series Representation for Functions Analytic in a Disc	p. 77
Analytic in an Arbitrary Open Set	p. 81
The Uniqueness, Mean-Value, and Maximum-Modulus Theorems; Critical Points and Saddle Points	p. 82
Exercises	p. 90
Further Properties of Analytic Functions	p. 93
The Open Mapping Theorem; Schwarz' Lemma	p. 93
The Converse of Cauchy's Theorem: Morera's Theorem; The Schwarz Reflection Principle and Analytic Arcs	p. 98
Exercises	p. 104
Simply Connected Domains	p. 107
The General Cauchy Closed Curve Theorem	p. 107
The Analytic Function log z	p. 113
Exercises	p. 116
Isolated Singularities of an Analytic Function	p. 117
Classification of Isolated Singularities; Riemann's Principle and the Casorati-Weierstrass Theorem	p. 117
Laurent Expansions	p. 120
Exercises	p. 126
The Residue Theorem	p. 129
Winding Numbers and the Cauchy Residue Theorem	p. 129
Applications of the Residue Theorem	p. 135
Exercises	p. 141
Applications of the Residue Theorem to the Evaluation of Integrals and Sums	p. 143
Introduction	p. 143
Evaluation of Definite Integrals by Contour Integral Techniques	p. 143
Application of Contour Integral Methods to Evaluation and Estimation of Sums	p. 151
Exercises	p. 158
Further Contour Integral Techniques	p. 161
Shifting the Contour of Integration	p. 161
An Entire Function Bounded in Every Direction	p. 164
Exercises	p. 167
Introduction to Conformal Mapping	p. 169
Conformal Equivalence	p. 169
Special Mappings	p. 175
Schwarz-Christoffel Transformations	p. 187
Exercises	p. 192
The Riemann Mapping Theorem	p. 195
Conformal Mapping and Hydrodynamics	p. 195
The Riemann Mapping Theorem	p. 200
Mapping Properties of Analytic Functions on Closed Domains	p. 204
Exercises	p. 213
Maximum-Modulus Theorems for Unbounded Domains	p. 215
A General Maximum-Modulus Theorem	p. 215
The Phragmén-Lindelöf Theorem	p. 218
Exercises	p. 223
Harmonic Functions	p. 225
Poisson Formulae and the Dirichlet Problem	p. 225
Liouville Theorems for Re f; Zeroes of Entire Functions of Finite Order	p. 233
Exercises	p. 238
Different Forms of Analytic Functions	p. 241
Introduction	p. 241
Infinite Products	p. 241
Analytic Functions Defined by Definite Integrals	p. 249
Analytic Functions Defined by Dirichlet Series	p. 251
Exercises	p. 255
Analytic Continuation; The Gamma and Zeta Functions	p. 257
Introduction	p. 257
Power Series	p. 257
Analytic Continuation of Dirichlet Series	p. 263
The Gamma and Zeta Functions	p. 265
Exercises	p. 271
Applications to Other Areas of Mathematics	p. 273
Introduction	p. 273
A Variation Problem	p. 273
The Fourier Uniqueness Theorem	p. 275
An Infinite System of Equations	p. 277
Applications to Number Theory	p. 278
An Analytic Proof of The Prime Number Theorem	p. 285
Exercises	p. 290
Answers	p. 291
References	p. 319
Appendices	p. 321
Index	p. 325
Table of Contents provided by Ingram. All Rights Reserved.

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Complex Analysis

Complex Analysis

9781441972873

1441972870

Summary

Table of Contents

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