Publisher Description

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: 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

Long Description

Intended for advanced undergraduates and graduate students in mathematics, this book represented a new approach to the teaching of complex analysis with emphasis on the argument principle in analysis and topology. The 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.

Details