Complex Analysis

by Beardon, Alan F.

• ISBN13: 9780486837185
• ISBN10: 0486837181
• Format: Paperback
• Copyright: 2019-12-18
• Publisher: Dover Pubns

Summary

With its emphasis on the argument principle in analysis and topology, this book represents a different approach to the teaching of complex analysis. The three-part treatment provides geometrical insights by covering angles, basic complex analysis, and interactions with plane topology while focusing on the concepts of angle and winding numbers.
Part I takes a critical look at the concept of an angle, illustrating that because a nonzero complex number varies continuously, one may select a continuously changing value of its argument. Part II builds upon this material, using the argument and its continuous variation as a tool in further studies and clarifying the complementary aspects of complex analysis and plane topology. Part III explores the link between the two subjects to their mutual benefit. The first two sections are intended for advanced undergraduates and graduate students in mathematics and contain sufficient material for a single course. The final section is geared toward the complex analyst and is intended to provide a foundation for further study.

Author Biography

Alan F. Beardon received his PhD from the University of London in 1964 and was Professor of Mathematics at the University of Cambridge from 1970 until he became Emeritus in 2007. His many books include A Primer on Riemann Surfaces, The Geometry of Discrete Groups, and Limits: A New Approach to Real Analysis.

Table of Contents

Contents

Part I Angles

Chapter 1
1.1 Sets
1.2 Complex numbers
1.3 Upper bounds
1.4 Square roots
1.5 Distance

Chapter 2
2.1 Infinite series
2.2 Tests for convergence
2.3 The Cauchy project

Chapter 3
3.1 Continuity
3.2 Real continuous functions

Chapter 4
4.1 The exponential function
4.2 The trigonometric functions
4.3 Periodicity
4.4 The hyperbolic functions

Chapter 5
5.1 The argument of a complex number
5.2 Logarithms
5.3 Exponents
5.4 Continuity of the logarithm

Part II Basic Complex Analysis

Chapter 6
6.1 Open and closed sets
6.2 Connected sets
6.3 Limits
6.4 Compact sets
6.5 Homeomorphisms
6.6 Uniform convergence

Chapter 7
7.1 Plane curves
7.2 The index of a curve
7.3 Properties of the index

Chapter 8
8.1 Polynomials
8.2 Power series
8.3 Analytic functions
8.4 Inequalities
8.5 The zeros of analytic functions

Chapter 9
9.1 Derivatives
9.2 Line integrals
9.3 Inequalities
9.4 Chains and cycles
9.5 Evaluation of integrals
9.6 Cauchy's Theorem
9.7 Applications

Chapter 10
10.1 Conformal mapping
10.2 Stereographic projection
103. Mobius transformations

Part III Interactions with Plane Topology

Chapter 11
11.1 Simply connected domains
11.2 The Riemann Mapping Theorem
11.3 Branches of the argument
11.4 The Jordan Curve Theorem
11.5 Conformal mapping of a Jordan domain

Appendix
Bibliography
Index