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elements 'he of Duncan J. JOHN WILEY & SONS London New York Sydney . . Digitized by the Internet Archive in 2016 with funding from Kahle/Austin Foundation https://archive.org/details/elementsofcompleOOdunc T). T. Soaker The elements of COMPLEX ANALYSIS The elements of COMPLEX ANALYSIS Duncan J. Department of Mathematics , King's College University of , Aberdeen Scotland , JOHN WILEY & SONS London New York Sydney © & Copyright 1968 by John Wiley Sons, Ltd. All rights reserved. No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine language without the written permission of the publisher. Library of Congress catalog card number 68-29702 SBN 471 22565 7 Cloth bound SBN 471 22566 5 Paper bound & Printed in Great Britain by Spottiswoode, Ballantyne Co. Ltd., London and Colchester PREFACE 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 mini- mal 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 complex 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 , vi Preface 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 dis- cerning reader. For example Section 9.2 is wholly inspired by the corre- sponding account in the more advanced book Theory ofAnalytic 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. It is a pleasure to record several personal acknowledgements. I owe much indirectly to Professor F. F. Bonsall; any lucidity of style that may occur in this book arises largely from his personal example as a supervisor and friend. Some of the work for this book was carried out while I was visiting Yale University where I was fortunate to have many fruitful con- versations with Professor E. Lee Stout. I am grateful to my Aberdeen colleagues Eric E. Morrison and Alan J. White who read parts of the manuscript and gave encouragement and constructive criticisms. I am also indebted to the secretaries and typists of the mathematics departments both at Aberdeen and Yale who prepared various editions of the manu- my script. Finally I wish to express thanks to the editorial staff of John & Wiley Sons for their advice and encouragement. Aberdeen University 1968 John Duncan , CONTENTS METRIC SPACE PRELIMINARIES 1 ... 1.1 Set theoretic notation and terminology 1 1 .2 Elementary propert.ies o.f met.ric s.paces.... 3 .13 1.3 Continuous functi.ons o.n met.ric s.paces.... . . . 1.4 Compactness ........ 19 1.5 Completeness 28 1.6 Connectedness 30 THE COMPLEX NUMBERS ...... 2 ..... 2.1 Definitions and notation ...... 35 2.2 Domains in the complex plane 43 2.3 The extended complex plane 50 CONTINUOUS AND DIFFERENTIABLE COMPLEX 3 FUNCTIONS ..... ..... 3.1 Continuous complex functions 54 ..... 3.2 Differentiable complex functions 60 3.3 The Cauchy-Riemann equations 67 .71 3.4 Harmonic functions of two real variables . . 4 POWER SERIES FUNCTIONS .... .... 4.1 Infinite series of comple.x num.bers..... 75 . 4.2 Double sequences of complex numbers 79 ...... 4.3 Power series funct.ions ....... 84 4.4 The exponential function 90 4.5 Branches-of-log 95 vii Contents Vlll .......... ARCS, CONTOURS, AND INTEGRATION 5 ........ 5.1 Arcs 101 5.2 Oriented arcs 105 5.3 Simple closed curves 108 .114 5.4 Oriented simple closed curves . . . . .119 5.5 The Jordan curve theorem . . . . . .122 5.6 Contour integration . . . . . . CAUCHY’S THEOREM FOR STARLIKE DOMAINS 6 .131 6.1 Cauchy’s theorem for triangular contours . . .135 6.2 Cauchy’s theorem for starlike domains . . . .138 6.3 Applications . . . . . . . LOCAL ANALYSIS 7 .147 7.1 Cauchy’s integral formulae . . . . . .152 7.2 Taylor expansions . . . . . . .158 7.3 The Laurent expansion . . . . . . .163 7.4 Isolated singularities . . . . . . GLOBAL ANALYSIS 8 .174 8.1 Taylor expansions revisited . . . . . .176 8.2 Properties of zeros ...... . . . . . . .181 8.3 Entire functions ....... . . . . . . . 8.4 Meromorphic functions 185 8.5 Convergence in stf(D) ....... 189 .197 8.6 Weierstrass expansions ...... . . . . . . 8.7 Topological index ...... 203 8.8 Cauchy’s residue theorem ...... 209 8.9 Mittag-Leffler expansions ...... 214 8.10 Zeros and poles revisited 222 ..... 8.11 The open mapping theorem 230 8.12 The maximum modulus principle 234 CONFORMAL MAPPING 9 ..... 9.1 Discussion of the Riemann ma.ppin.g th.eore.m . .. .. 243 9.2 The automorphisms of a domain 247 9.3 Mappings of the boundary 255 9.4 Some illustrative mappings .261 . . . . .