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Problems and Solutions for Complex Analysis - Rami Shakarchi

Problems and Solutions for Complex Analysis


Published: 14th October 1999
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This book contains all the exercises and solutions of Serge Lang's Complex Analy- sis. Chapters I through VITI of Lang's book contain the material of an introductory course at the undergraduate level and the reader will find exercises in all of the fol- lowing topics: power series, Cauchy's theorem, Laurent series, singularities and meromorphic functions, the calculus of residues, conformal mappings and har- monic functions. Chapters IX through XVI, which are suitable for a more advanced course at the graduate level, offer exercises in the following subjects: Schwarz re- flection, analytic continuation, Jensen's formula, the Phragmen-LindelOf theorem, entire functions, Weierstrass products and meromorphic functions, the Gamma function and the Zeta function. This solutions manual offers a large number of worked out exercises of varying difficulty. I thank Serge Lang for teaching me complex analysis with so much enthusiasm and passion, and for giving me the opportunity to work on this answer book. Without his patience and help, this project would be far from complete. I thank my brother Karim for always being an infinite source of inspiration and wisdom. Finally, I want to thank Mark McKee for his help on some problems and Jennifer Baltzell for the many years of support, friendship and complicity. Rami Shakarchi Princeton, New Jersey 1999 Contents Preface vii I Complex Numbers and Functions 1 1. 1 Definition . . . . . . . . . . 1 1. 2 Polar Form . . . . . . . . . 3 1. 3 Complex Valued Functions . 8 1. 4 Limits and Compact Sets . . 9 1. 6 The Cauchy-Riemann Equations .

Prefacep. vii
Complex Numbers and Functionsp. 1
Definitionp. 1
Polar Formp. 3
Complex Valued Functionsp. 8
Limits and Compact Setsp. 9
The Cauchy-Riemann Equationsp. 12
Power Seriesp. 14
Formal Power Seriesp. 14
Convergent Power Seriesp. 19
Relations Between Formal and Convergent Seriesp. 24
Analytic Functionsp. 28
Differentiation of Power Seriesp. 29
The Inverse and Open Mapping Theoremsp. 31
Cauchy's Theorem, First Partp. 36
Holomorphic Functions on Connected Setsp. 36
Integrals over Pathsp. 37
The Homotopy Form of Cauchy's Theoremp. 42
Existence of Global Primitives. Definition of the Logarithmp. 43
The Local Cauchy Formulap. 45
Winding Numbers and Cauchy's Theoremp. 48
The Global Cauchy Theoremp. 48
Applications of Cauchy's Integral Formulap. 51
Uniform Limits of Analytic Functionsp. 51
Laurent Seriesp. 60
Isolated Singularitiesp. 66
Calculus of Residuesp. 76
The Residue Formulap. 76
Evaluation of Definite Integralsp. 93
Conformal Mappingsp. 119
Analytic Automorphisms of the Discp. 119
The Upper Half Planep. 122
Other Examplesp. 126
Fractional Linear Transformationsp. 137
Harmonic Functionsp. 146
Definitionp. 146
Examplesp. 153
Basic Properties of Harmonic Functionsp. 159
The Poisson Formulap. 165
Construction of Harmonic Functionsp. 167
Schwarz Reflectionp. 175
Reflection Across Analytic Arcsp. 175
The Riemann Mapping Theoremp. 179
Statement of the Theoremp. 179
Compact Sets in Function Spacesp. 181
Analytic Continuation along Curvesp. 185
Continuation Along a Curvep. 185
The Dilogarithmp. 187
Applications of the Maximum Modulus Principle and Jensen's Formulap. 191
Jensen's Formulap. 191
The Picard-Borel Theoremp. 198
The Phragmen-Lindelof and Hadamard Theoremsp. 201
Entire and Meromorphic Functionsp. 206
Infinite Productsp. 206
Weierstrass Productsp. 211
Functions of Finite Orderp. 213
Meromorphic Functions, Mittag-Leffler Theoremp. 214
The Gamma and Zeta Functionsp. 219
The Differentiation Lemmap. 219
The Gamma Functionp. 223
The Lerch Formulap. 235
Zeta Functionsp. 238
The Prime Number Theoremp. 241
Basic Analytic Properties of the Zeta Functionp. 241
The Main Lemma and its Applicationp. 245
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780387988313
ISBN-10: 0387988319
Audience: General
Format: Paperback
Language: English
Number Of Pages: 246
Published: 14th October 1999
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.39 x 15.6  x 1.4
Weight (kg): 0.37