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Unformatted text preview: Joseph Bak Donald J. Newman Complex Analysis Third Edition 13 Undergraduate Texts in Mathematics Editorial Board S. Axler K.A. Ribet For other titles Published in this series, go to Joseph Bak • Donald J. Newman Complex Analysis Third Edition 1C Joseph Bak City College of New York Department of Mathematics 138th St. & Convent Ave. New York, New York 10031 USA [email protected] Editorial Board: S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected] Donald J. Newman (1930–2007) K. A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720 USA [email protected] ISSN 0172-6056 ISBN 978-1-4419-7287-3 e-ISBN 978-1-4419-7288-0 DOI 10.1007/978-1-4419-7288-0 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010932037 Mathematics Subject Classification (2010): 30-xx, 30-01, 30Exx © Springer Science+Business Media, LLC 1991, 1997, 2010, Corrected at 2nd printing 2017 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media ( ) Preface to the Third Edition Beginning with the first edition of Complex Analysis, we have attempted to present the classical and beautiful theory of complex variables in the clearest and most intuitive form possible. The changes in this edition, which include additions to ten of the nineteen chapters, are intended to provide the additional insights that can be obtained by seeing a little more of the “big picture”. This includes additional related results and occasional generalizations that place the results in a slightly broader context. The Fundamental Theorem of Algebra is enhanced by three related results. Section 1.3 offers a detailed look at the solution of the cubic equation and its role in the acceptance of complex numbers. While there is no formula for determining the roots of a general polynomial, we added a section on Newton’s Method, a numerical technique for approximating the zeroes of any polynomial. And the Gauss-Lucas Theorem provides an insight into the location of the zeroes of a polynomial and those of its derivative. A series of new results relate to the mapping properties of analytic functions. A revised proof of Theorem 6.15 leads naturally to a discussion of the connection between critical points and saddle points in the complex plane. The proof of the Schwarz Reflection Principle has been expanded to include reflection across analytic arcs, which plays a key role in a new section (14.3) on the mapping properties of analytic functions on closed domains. And our treatment of special mappings has been enhanced by the inclusion of Schwarz-Christoffel transformations. A single interesting application to number theory in the earlier editions has been expanded into a new section (19.4) which includes four examples from additive number theory, all united in their use of generating functions. Perhaps the most significant changes in this edition revolve around the proof of the prime number theorem. There are two new sections (17.3 and 18.2) on Dirichlet series. With that background, a pivotal result on the Zeta function (18.10), which seemed to “come out of the blue”, is now seen in the context of the analytic continuation of Dirichlet series. Finally the actual proof of the prime number theorem has been considerably revised. The original independent proofs by Hadamard and de la Vallée Poussin were both long and intricate. Donald Newman’s 1980 article v vi Preface to the Third Edition presented a dramatically simplified approach. Still the proof relied on several nontrivial number-theoretic results, due to Chebychev, which formed a separate appendix in the earlier editions. Over the years, further refinements of Newman’s approach have been offered, the most recent of which is the award-winning 1997 article by Zagier. We followed Zagier’s approach, thereby eliminating the need for a separate appendix, as the proof relies now on only one relatively straightforward result due of Chebychev. The first edition contained no solutions to the exercises. In the second edition, responding to many requests, we included solutions to all exercises. This edition contains 66 new exercises, so that there are now a total of 300 exercises. Once again, in response to instructors’ requests, while solutions are given for the majority of the problems, each chapter contains at least a few for which the solutions are not included. These are denoted with an asterisk. Although Donald Newman passed away in 2007, most of the changes in this edition were anticipated by him and carry his imprimatur. I can only hope that all of the changes and additions approach the high standard he set for presenting mathematics in a lively and “simple” manner. In an earlier edition of this text, it was my pleasure to thank my former student, Pisheng Ding, for his careful work in reviewing the exercises. In this edition, it is an even greater pleasure to acknowledge his contribution to many of the new results, especially those relating to the mapping properties of analytic functions on closed domains. This edition also benefited from the input of a new generation of students at City College, especially Maxwell Musser, Matthew Smedberg, and Edger Sterjo. Finally, it is a pleasure to acknowledge the careful work and infinite patience of Elizabeth Loew and the entire editorial staff at Springer. Joseph Bak City College of NY April 2010 Preface to the Second Edition One of our goals in writing this book has been to present the theory of analytic functions with as little dependence as possible on advanced concepts from topology and several-variable calculus. This was done not only to make the book more accessible to a student in the early stages of his/her mathematical studies, but also to highlight the authentic complex-variable methods and arguments as opposed to those of other mathematical areas. The minimum amount of background material required is presented, along with an introduction to complex numbers and functions, in Chapter 1. Chapter 2 offers a somewhat novel, yet highly intuitive, definition of analyticity as it applies specifically to polynomials. This definition is related, in Chapter 3, to the Cauchy-Riemann equations and the concept of differentiability. In Chapters 4 and 5, the reader is introduced to a sequence of theorems on entire functions, which are later developed in greater generality in Chapters 6–8. This two-step approach, it is hoped, will enable the student to follow the sequence of arguments more easily. Chapter 5 also contains several results which pertain exclusively to entire functions. The key result of Chapters 9 and 10 is the famous Residue Theorem, which is followed by many standard and some not-so-standard applications in Chapters 11 and 12. Chapter 13 introduces conformal mapping, which is interesting in its own right and also necessary for a proper appreciation of the subsequent three chapters. Hydrodynamics is studied in Chapter 14 as a bridge between Chapter 13 and the Riemann Mapping Theorem. On the one hand, it serves as a nice application of the theory developed in the previous chapters, specifically in Chapter 13. On the other hand, it offers a physical insight into both the statement and the proof of the Riemann Mapping Theorem. In Chapter 15, we use “mapping” methods to generalize some earlier results. Chapter 16 deals with the properties of harmonic functions and the related theory of heat conduction. A second goal of this book is to give the student a feeling for the wide applicability of complex-variable techniques even to questions which initially do not seem to belong to the complex domain. Thus, we try to impart some of the enthusiasm vii viii Preface to the Second Edition apparent in the famous statement of Hadamard that "the shortest route between two truths in the real domain passes through the complex domain." The physical applications of Chapters 14 and 16 are good examples of this, as are the results of Chapter 11. The material in the last three chapters is designed to offer an even greater appreciation of the breadth of possible applications. Chapter 17 deals with the different forms an analytic function may take. This leads directly to the Gamma and Zeta functions discussed in Chapter 18. Finally, in Chapter 19, a potpourri of problems–again, some classical and some novel–is presented and studied with the techniques of complex analysis. The material in the book is most easily divided into two parts: a first course covering the materials of Chapters 1–11 (perhaps including parts of Chapter 13), and a second course dealing with the later material. Alternatively, one seeking to cover the physical applications of Chapters 14 and 16 in a one-semester course could omit some of the more theoretical aspects of Chapters 8, 12, 14, and 15, and include them, with the later material, in a second-semester course. The authors express their thanks to the many colleagues and students whose comments were incorporated into this second edition. Special appreciation is due to Mr. Pi-Sheng Ding for his thorough review of the exercises and their solutions. We are also indebted to the staff of Springer-Verlag Inc. for their careful and patient work in bringing the manuscript to its present form. Joseph Bak Donald J. Newmann Contents Preface to the Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 The Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Field of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The Solution of the Cubic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Topological Aspects of the Complex Plane . . . . . . . . . . . . . . . . . . . . . 12 1.5 Stereographic Projection; The Point at Infinity . . . . . . . . . . . . . . . . . . 16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2 Functions of the Complex Variable z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Analytic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Differentiability and Uniqueness of Power Series . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 21 21 25 28 32 3 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Analyticity and the Cauchy-Riemann Equations . . . . . . . . . . . . . . . . . 3.2 The Functions e z , sin z, cos z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 35 40 41 4 Line Integrals and Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Properties of the Line Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Closed Curve Theorem for Entire Functions . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 45 45 52 56 ix x 5 6 Contents Properties of Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Cauchy Integral Formula and Taylor Expansion for Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Liouville Theorems and the Fundamental Theorem of Algebra; The Gauss-Lucas Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Newton’s Method and Its Application to Polynomial Equations . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 59 65 68 74 Properties of Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Power Series Representation for Functions Analytic in a Disc . . 6.2 Analytic in an Arbitrary Open Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Uniqueness, Mean-Value, and Maximum-Modulus Theorems; Critical Points and Saddle Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 77 77 81 Further Properties of Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Open Mapping Theorem; Schwarz’ Lemma . . . . . . . . . . . . . . . . . 7.2 The Converse of Cauchy’s Theorem: Morera’s Theorem; The Schwarz Reflection Principle and Analytic Arcs . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 93 98 104 8 Simply Connected Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The General Cauchy Closed Curve Theorem . . . . . . . . . . . . . . . . . . . . 8.2 The Analytic Function log z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 107 113 116 9 Isolated Singularities of an Analytic Function . . . . . . . . . . . . . . . . . . . . . 9.1 Classification of Isolated Singularities; Riemann’s Principle and the Casorati-Weierstrass Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Laurent Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 117 120 126 10 The Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Winding Numbers and the Cauchy Residue Theorem . . . . . . . . . . . . . 10.2 Applications of the Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 129 135 141 7 11 Applications of the Residue Theorem to the Evaluation of Integrals and Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Evaluation of Definite Integrals by Contour Integral Techniques . . . 11.2 Application of Contour Integral Methods to Evaluation and Estimation of Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 90 143 143 143 151 158 Contents xi 12 Further Contour Integral Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Shifting the Contour of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 An Entire Function Bounded in Every Direction . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 161 164 167 13 Introduction to Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Conformal Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Special Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Schwarz-Christoffel Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 169 175 187 192 14 The Riemann Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Conformal Mapping and Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . 14.2 The Riemann Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Mapping Properties of Analytic Functions on Closed Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 195 200 15 Maximum-Modulus Theorems for Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 A General Maximum-Modulus Theorem . . . . . . . . . . . . . . . . . . . . . . . 15.2 The Phragmén-Lindelöf Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 213 215 215 218 223 16 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Poisson Formulae and the Dirichlet Problem . . . . . . . . . . . . . . . . . . . . 16.2 Liouville Theorems for Re f ; Zeroes of Entire Functions of Finite Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 225 17 Different Forms of Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Analytic Functions Defined by Definite Integrals . . . . . . . . . . . . . . . . 17.3 Analytic Functions Defined by Dirichlet Series . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 241 241 249 251 255 18 Analytic Continuation; The Gamma and Zeta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Analytic Continuation of Dirichlet Series . . ...
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