Problems and Solutions for Complex Analysis,9780387988313

Problems and Solutions for Complex Analysis

by ;
ISBN13: 9780387988313
ISBN10: 0387988319
Format: Paperback
Pub. Date: 1999-09-01
Publisher(s): Springer Verlag
List Price: $73.49

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Summary

This volume contains all the exercises, and 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 the undergraduate level and cover the following topics: power series, Cauchy's theorem, Laurent series, singularities and meromorphic functions, the calculus of residues, conformal mappings, and harmonic functions. The material in Chapters 9-16 is more advanced. The reader will find 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. This volume also serves as an independent source of problems with detailed answers beneficial for anyone interested in learning complex analysis.

Table of Contents

Preface vii
Complex Numbers and Functions
1(13)
Definition
1(2)
Polar Form
3(5)
Complex Valued Functions
8(1)
Limits and Compact Sets
9(3)
The Cauchy-Riemann Equations
12(2)
Power Series
14(22)
Formal Power Series
14(5)
Convergent Power Series
19(5)
Relations Between Formal and Convergent Series
24(4)
Analytic Functions
28(1)
Differentiation of Power Series
29(2)
The Inverse and Open Mapping Theorems
31(5)
Cauchy's Theorem, First Part
36(12)
Holomorphic Functions on Connected Sets
36(1)
Integrals over Paths
37(5)
The Homotopy Form of Cauchy's Theorem
42(1)
Existence of Global Primitives. Definition of the Logarithm
43(2)
The Local Cauchy Formula
45(3)
Winding Numbers and Cauchy's Theorem
48(3)
The Global Cauchy Theorem
48(3)
Applications of Cauchy's Integral Formula
51(25)
Uniform Limits of Analytic Functions
51(9)
Laurent Series
60(6)
Isolated Singularities
66(10)
Calculus of Residues
76(43)
The Residue Formula
76(17)
Evaluation of Definite Integrals
93(26)
Conformal Mappings
119(27)
Analytic Automorphisms of the Disc
119(3)
The Upper Half Plane
122(4)
Other Examples
126(11)
Fractional Linear Transformations
137(9)
Harmonic Functions
146(29)
Definition
146(7)
Examples
153(6)
Basic Properties of Harmonic Functions
159(6)
The Poisson Formula
165(2)
Construction of Harmonic Functions
167(8)
Schwarz Reflection
175(4)
Reflection Across Analytic Arcs
175(4)
The Riemann Mapping Theorem
179(6)
Statement of the Theorem
179(2)
Compact Sets in Function Spaces
181(4)
Analytic Continuation along Curves
185(6)
Continuation Along a Curve
185(2)
The Dilogarithm
187(4)
Applications of the Maximum Modulus Principle and Jensen's Formula
191(15)
Jensen's Formula
191(7)
The Picard-Borel Theorem
198(3)
The Phragmen-Lindel of and Hadamard Theorems
201(5)
Entire and Meromorphic Functions
206(13)
Infinite Products
206(5)
Weierstrass Products
211(2)
Functions of Finite Order
213(1)
Meromorphic Functions, Mittag-Leffler Theorem
214(5)
The Gamma and Zeta Functions
219(22)
The Differentiation Lemma
219(4)
The Gamma Function
223(12)
The Lerch Formula
235(3)
Zeta Functions
238(3)
The Prime Number Theorem
241(1)
Basic Analytic Properties of the Zeta Function
241(4)
The Main Lemma and its Application
245

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