Problems and Solutions for Complex Analysis

By: Rami Shakarchi, R. Shakarchi, Serge Lang

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Paperback | 1 November 1999

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260 Pages

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All the exercises plus their solutions for Serge Lang's fourth edition of "Complex Analysis," ISBN 0-387-98592-1. The problems in the first 8 chapters are suitable for an introductory course at undergraduate level and cover power series, Cauchy's theorem, Laurent series, singularities and meromorphic functions, the calculus of residues, conformal mappings, and harmonic functions. The material in the remaining 8 chapters is more advanced, with problems on Schwartz reflection, analytic continuation, Jensen's formula, the Phragmen-Lindeloef theorem, entire functions, Weierstrass products and meromorphic functions, the Gamma function and Zeta function. Also beneficial for anyone interested in learning complex analysis.

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Preface	p. vii
Complex Numbers and Functions	p. 1
Definition	p. 1
Polar Form	p. 3
Complex Valued Functions	p. 8
Limits and Compact Sets	p. 9
The Cauchy-Riemann Equations	p. 12
Power Series	p. 14
Formal Power Series	p. 14
Convergent Power Series	p. 19
Relations Between Formal and Convergent Series	p. 24
Analytic Functions	p. 28
Differentiation of Power Series	p. 29
The Inverse and Open Mapping Theorems	p. 31
Cauchy's Theorem, First Part	p. 36
Holomorphic Functions on Connected Sets	p. 36
Integrals over Paths	p. 37
The Homotopy Form of Cauchy's Theorem	p. 42
Existence of Global Primitives. Definition of the Logarithm	p. 43
The Local Cauchy Formula	p. 45
Winding Numbers and Cauchy's Theorem	p. 48
The Global Cauchy Theorem	p. 48
Applications of Cauchy's Integral Formula	p. 51
Uniform Limits of Analytic Functions	p. 51
Laurent Series	p. 60
Isolated Singularities	p. 66
Calculus of Residues	p. 76
The Residue Formula	p. 76
Evaluation of Definite Integrals	p. 93
Conformal Mappings	p. 119
Analytic Automorphisms of the Disc	p. 119
The Upper Half Plane	p. 122
Other Examples	p. 126
Fractional Linear Transformations	p. 137
Harmonic Functions	p. 146
Definition	p. 146
Examples	p. 153
Basic Properties of Harmonic Functions	p. 159
The Poisson Formula	p. 165
Construction of Harmonic Functions	p. 167
Schwarz Reflection	p. 175
Reflection Across Analytic Arcs	p. 175
The Riemann Mapping Theorem	p. 179
Statement of the Theorem	p. 179
Compact Sets in Function Spaces	p. 181
Analytic Continuation along Curves	p. 185
Continuation Along a Curve	p. 185
The Dilogarithm	p. 187
Applications of the Maximum Modulus Principle and Jensen's Formula	p. 191
Jensen's Formula	p. 191
The Picard-Borel Theorem	p. 198
The Phragmen-Lindelof and Hadamard Theorems	p. 201
Entire and Meromorphic Functions	p. 206
Infinite Products	p. 206
Weierstrass Products	p. 211
Functions of Finite Order	p. 213
Meromorphic Functions, Mittag-Leffler Theorem	p. 214
The Gamma and Zeta Functions	p. 219
The Differentiation Lemma	p. 219
The Gamma Function	p. 223
The Lerch Formula	p. 235
Zeta Functions	p. 238
The Prime Number Theorem	p. 241
Basic Analytic Properties of the Zeta Function	p. 241
The Main Lemma and its Application	p. 245
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Problems and Solutions for Complex Analysis

At a Glance

Paperback

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