Complex Analysis

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  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 2010-09-14
  • Publisher: Springer Verlag
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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.

Table of Contents

Preface to the Third Editionp. v
Preface to the Second Editionp. vii
The Complex Numbersp. 1
Introductionp. 1
The Field of Complex Numbersp. 1
The Complex Planep. 4
The Solution of the Cubic Equationp. 9
Topological Aspects of the Complex Planep. 12
Stereographic Projection; The Point at Infinityp. 16
Exercisesp. 18
Functions of the Complex Variable zp. 21
Introductionp. 21
Analytic Polynomialsp. 21
Power Seriesp. 25
Differentiability and Uniqueness of Power Seriesp. 28
Exercisesp. 32
Analytic Functionsp. 35
Analyticity and the Cauchy-Riemann Equationsp. 35
The Functions ez, sin z, cos zp. 40
Exercisesp. 41
Line Integrals and Entire Functionsp. 45
Introductionp. 45
Properties of the Line Integralp. 45
The Closed Curve Theorem for Entire Functionsp. 52
Exercisesp. 56
Properties of Entire Functionsp. 59
The Cauchy Integral Formula and Taylor Expansion for Entire Functionsp. 59
Liouville Theorems and the Fundamental Theorem of Algebra; The Gauss-Lucas Theoremp. 65
Newton's Method and Its Application to Polynomial Equationsp. 68
Exercisesp. 74
Properties of Analytic Functionsp. 77
Introductionp. 77
The Power Series Representation for Functions Analytic in a Discp. 77
Analytic in an Arbitrary Open Setp. 81
The Uniqueness, Mean-Value, and Maximum-Modulus Theorems; Critical Points and Saddle Pointsp. 82
Exercisesp. 90
Further Properties of Analytic Functionsp. 93
The Open Mapping Theorem; Schwarz' Lemmap. 93
The Converse of Cauchy's Theorem: Morera's Theorem; The Schwarz Reflection Principle and Analytic Arcsp. 98
Exercisesp. 104
Simply Connected Domainsp. 107
The General Cauchy Closed Curve Theoremp. 107
The Analytic Function log zp. 113
Exercisesp. 116
Isolated Singularities of an Analytic Functionp. 117
Classification of Isolated Singularities; Riemann's Principle and the Casorati-Weierstrass Theoremp. 117
Laurent Expansionsp. 120
Exercisesp. 126
The Residue Theoremp. 129
Winding Numbers and the Cauchy Residue Theoremp. 129
Applications of the Residue Theoremp. 135
Exercisesp. 141
Applications of the Residue Theorem to the Evaluation of Integrals and Sumsp. 143
Introductionp. 143
Evaluation of Definite Integrals by Contour Integral Techniquesp. 143
Application of Contour Integral Methods to Evaluation and Estimation of Sumsp. 151
Exercisesp. 158
Further Contour Integral Techniquesp. 161
Shifting the Contour of Integrationp. 161
An Entire Function Bounded in Every Directionp. 164
Exercisesp. 167
Introduction to Conformal Mappingp. 169
Conformal Equivalencep. 169
Special Mappingsp. 175
Schwarz-Christoffel Transformationsp. 187
Exercisesp. 192
The Riemann Mapping Theoremp. 195
Conformal Mapping and Hydrodynamicsp. 195
The Riemann Mapping Theoremp. 200
Mapping Properties of Analytic Functions on Closed Domainsp. 204
Exercisesp. 213
Maximum-Modulus Theorems for Unbounded Domainsp. 215
A General Maximum-Modulus Theoremp. 215
The Phragmén-Lindelöf Theoremp. 218
Exercisesp. 223
Harmonic Functionsp. 225
Poisson Formulae and the Dirichlet Problemp. 225
Liouville Theorems for Re f; Zeroes of Entire Functions of Finite Orderp. 233
Exercisesp. 238
Different Forms of Analytic Functionsp. 241
Introductionp. 241
Infinite Productsp. 241
Analytic Functions Defined by Definite Integralsp. 249
Analytic Functions Defined by Dirichlet Seriesp. 251
Exercisesp. 255
Analytic Continuation; The Gamma and Zeta Functionsp. 257
Introductionp. 257
Power Seriesp. 257
Analytic Continuation of Dirichlet Seriesp. 263
The Gamma and Zeta Functionsp. 265
Exercisesp. 271
Applications to Other Areas of Mathematicsp. 273
Introductionp. 273
A Variation Problemp. 273
The Fourier Uniqueness Theoremp. 275
An Infinite System of Equationsp. 277
Applications to Number Theoryp. 278
An Analytic Proof of The Prime Number Theoremp. 285
Exercisesp. 290
Answersp. 291
Referencesp. 319
Appendicesp. 321
Indexp. 325
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