The Elements of Complex Analysis,

Description

This book is intended as a first course in complex analysis for students who take their mathematics seriously. The prerequisites for reading this book consist of some basic real analysis (often as motivation) and a minimal encounter with the language of modern mathematics. The text is suit- able for students with a background of one year of analysis proper, that is one year beyond the usual preparatory informal courses on calculus and analytic geometry. There is sufficient material for a course varying from 40 to 60 lectures according to the background and needs of the students. Although the book is a development of several courses I have given to honours students of mathematics at Aberdeen University, I have tried to keep the exposition as simple as possible with the object of making some of the theory of complex analysis available to a fairly wide audience. Since there are already many books on complex analysis it is necessary to say a word about the special features of the present book. This can best be done by a simple illustration. The central theorem of complex analysis is the famous Cauchy theorem and it has long been considered a deep and difficult theorem. This is certainly true of the most general case; but the theorem admits a straightforward proof for the case in which the given domain is starlike. This latter case is adequate for almost all the results of COlnplex function theory. Since I believe that the student has every right to demand a proper treatment of the version of the Cauchy theorem which is to be used for subsequent results, I have chosen to prove the Cauchy theorem only for the starlike case. This means that I am able to give complete (and straightforward) proofs of all the subsequent theorems even though the theorems are not always the most general possible. I indicate how the theorems may be extended for the rare occasions on which this is necessary. This then describes the aim of the book--to give an introductory course in complex analysis that is rigorous and yet amenable to the serious undergraduate student. I hope that the course is also enjoyable. I have freely borrowed from other sources the ideas that I consider to be the best and the simplest. Certain sources will be transparent to the discerning reader. For example Section 9.2 is wholly inspired by the corresponding account in the more advanced book Theory of Analytic Functions of One or Several Variables, by Professor H. Cartan. On the other hand I think that occasionally some of the details are new as in parts of Chapters 5 and 10. Preface Contents 1 METRIC SPACE PRELIMINARIES 1.1 Set theoretic notation and terminology 1.2 Elementary properties of metric spaces 1.3 Continuous functions on metric spaces 1.4 Compactness 1.5 Completeness 1.6 Connectedness 2 THE COMPLEX NUMBERS 2.1 Definitions and notation 2.2 Domains in the complex plane 2.3 The extended complex plane 3 CONTINUOUS AND DIFFERENTIABLE COMPLEX FUNCTIONS 3.1 Continuous complex functions 3.2 Differentiable complex functions 3.3 The Cauchy-Riemann equations 3.4 Harmonic functions of two real variables 4 POWER SERIES FUNCTIONS 4.1 Infinite series of complex numbers 4.2 Double sequences of complex numbers 4.3 Power series functions 4.4 The exponential function 4.5 Branches-of-log 5 ARCS, CONTOURS, AND INTEGRATION 5.1 Arcs 5.2 Oriented arcs 5.3 Simple closed curves 5.4 Oriented simple closed curves 5.5 The Jordan curve theorem 5.6 Contour integration 6 CAUCHY'S THEOREM FOR STARLIKE DOMAINS 6.1 Cauchy's theorem for triangular contours 6.2 Cauchy's theorem for starlike domains 6.3 Applications 7 LOCAL ANALYSIS 7.1 Cauchy's integral formulae 7.2 Taylor expansions 7.3 The Laurent expansion 7.4 Isolated singularities 8 GLOBAL ANALYSIS 8.1 Taylor expansions revisited 8.2 Properties of zeros 8.3 Entire functions 8.4 Meromorphic functions 8.5 Convergence in d(D) 8.6 Weierstrass expansions 8.7 Topological index 8.8 Cauchy's residue theorem 8.9 Mittag-Leffler expansions 8.10 Zeros and poles revisited 8.11 The open mapping theorem 8.12 The maximum modulus principle 9 CONFORMAL MAPPING 9.1 Discussion of the Riemann mapping theorem 9.2 The automorphisms of a domain 9.3 Mappings of the boundary 9.4 Some illustrative mappings 10 ANALYTIC CONTINUATION 10.1 Direct analytic continuations 10.2 General analytic functions 10.3 Complex analytic manifolds 10.4 The gamma and zeta functions APPENDIX: RIEMANN-STIELTJES INTEGRATION SUGGESTIONS FOR FURTHER STUDY BIBLIOGRAPHY INDEX OF SPECIAL SYMBOLS SUBJECT INDEX