Đang tải... (xem toàn văn)
Complex analysis is not an elementary topic, and one of the problems facing lecturers is that many of their students, particularly those with an "applied" orientation, approach the topic
Springer Undergraduate Mathematics Series John M Howie Complex Analysis With 83 Figures �Springer In memory of Katharine Preface Of all the central topics in the undergraduate mathematics syllabus, complex analysis is arguably the most attractive The huge consequences emanating from the assumption of differentiability, and the sheer power of the methods deriving from Cauchy's Theorem never fail to impress, and undergraduates actively enjoy exploring the applications of the Residue Theorem Complex analysis is not an elementary topic, and one of the problems facing lecturers is that many of their students, particularly those with an "applied" orientation, approach the topic with little or no familiarity with the E-8 argu ments that are at the core of a serious course in analysis It is, however, possible to appreciate the essence of complex analysis without delving too deeply into the fine detail of the proofs, and in the earlier part of the book I hav some of the more technical proofs that may safely be omitted Proofs' e starred are, how ever, given, since the development of more advanced analytical skills comes from imitating the techniques used in proving the major results The opening two chapters give a brief account of the preliminaries in real function theory and complex numbers that are necessary for the study of com pfex functions I have included these chapters partly with self-study in mind, but they may also be helpful to those whose lecturers airily (and wrongly) assume that students remember everything learned in previous years In what is certainly designed as a first course in complex analysis I have deemed it appropriate to make only minimal reference to the topological issues that are at the core of the subject This may be a disappointment to some pro fessionals, but I am confident that it will be appreciated by the undergraduates for whom the book is intended The general plan of the book is fairly traditional, and perhaps the only slightly unusual feature is the brief final Chapter 12, which I hope will show that the subject is very much alive In Section 12.2 I give a very brief and VIII Com plex Analysis imprecise account of Julia sets and the Mandelbrot set, and in Section 12.1 I explain the Riemann Hypothesis, arguably the most remarkable and important unsolved problem in mathematics If the eventual conqueror of the Riemann Hypothesis were to have learned the basics of complex analysis from this book, then I would rest content indeed! All too often mathematics is presented in such a way as to suggest that it was engraved in pre-history on tablets of stone The footnotes with the names and dates of the mathematicians who created complex analysis are intended to emphasise that mathematics was and is created by real people Information on these people and their achievements can be found on the St Andrews website www-hi s t ory.mc s.st-and.ac uk/hist ory/ I am grateful to my colleague John O'Connor for his help in creating the diagrams Warmest thanks are due also to Kenneth Falconer and Michael Wolfe, whose comments on the manuscript have, I hope, eliminated serious errors The responsibility for any imperfections that remain is mine alone John M Howie r University of St Andrews January, 2003 Con ten ts Preface vii Contents ix 1 What Do I Need to Know? 1 1.1 Set Theory , 2 1.2 Numbers 2 1.3 Sequences and Series 4 1.4 Functions and Continuity 7 1.5 Differentiation 10 1.6 Integration 12 1.7 Infinite Integrals 14 1.8 Calculus of Two Variables 17 2 Complex Numbers , 19 2.1 Are Complex Numbers Necessary? 19 2.2 Basic Properties of Complex Numbers 21 3 Prelude to Complex Analysis 35 3.1 Why is Complex Analysis Possible? 35 3.2 Some Useful Terminology 37 3.3 Functions and Continuity 41 3.4 The 0 and o Notations 46 4 Differentiation 51 4.1 Differentiability 51 4.2 Power Series 61