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A first course in complex analysis with applications PDF

517 Pages·2003·2.987 MB·English
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A First Course in Complex Analysis with Applications Dennis G. Zill Loyola Marymount University Patrick D. Shanahan Loyola Marymount University World Headquarters Jones and Bartlett Publishers Jones and Bartlett Publishers Jones and Bartlett Publishers 40 Tall Pine Drive Canada International Sudbury, MA01776 2406 Nikanna Road Barb House, Barb Mews 978-443-5000 Mississauga, ON L5C 2W6 London W6 7PA [email protected] CANADA UK www.jbpub.com Copyright ©2003 by Jones and Bartlett Publishers, Inc. Library of Congress Cataloging-in-Publication Data Zill, Dennis G., 1940- Afirst course in complex analysis with applications / Dennis G. Zill, Patrick D. Shanahan. p. cm. Includes indexes. ISBN 0-7637-1437-2 1. Functions of complex variables. I. Shanahan, Patrick, 1931- II. Title. QA331.7 .Z55 2003 515’.9—dc21 2002034160 All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or any information storage or retrieval system, without written permission from the copyright owner. Chief Executive Officer: Clayton Jones Chief Operating Officer: Don W. Jones, Jr. Executive V.P. and Publisher: Robert W. Holland, Jr. V.P., Design and Production: Anne Spencer V.P., Manufacturing and Inventory Control: Therese Bräuer Director, Sales and Marketing: William Kane Editor-in-Chief, College: J. Michael Stranz Production Manager: Amy Rose Marketing Manager: Nathan Schultz Associate Production Editor: Karen Ferreira Editorial Assistant: Theresa DiDonato Production Assistant: Jenny McIsaac Cover Design: Night & Day Design Composition: Northeast Compositors Printing and Binding: Courier Westford Cover Printing: John Pow Company This book was typeset with Textures on a Macintosh G4. The font families used were Computer Modern and Caslon. The first printing was printed on 50# Finch opaque. Printed in the United States of America 06 05 04 03 02 10 9 8 7 6 5 4 3 2 1 For Dana, Kasey, and Cody Contents 7.1 Contents Preface ix Chapter 1. Complex Numbers and the Complex Plane 1 1.1 Complex Numbers and Their Properties 2 1.2 Complex Plane 10 1.3 Polar Form of Complex Numbers 16 1.4 Powers and Roots 23 1.5 Sets of Points in the Complex Plane 29 1.6 Applications 36 Chapter 1 Review Quiz 45 Chapter 2. Complex Functions and Mappings 49 2.1 Complex Functions 50 2.2 Complex Functions as Mappings 58 2.3 Linear Mappings 68 2.4 Special Power Functions 80 2.4.1 The Power Function zn 81 2.4.2 The Power Function z1/n 86 2.5 Reciprocal Function 100 2.6 Limits and Continuity 110 2.6.1 Limits 110 2.6.2 Continuity 119 2.7 Applications 132 Chapter 2 Review Quiz 138 Chapter 3. Analytic Functions 141 3.1 Differentiability and Analyticity 142 3.2 Cauchy-Riemann Equations 152 3.3 Harmonic Functions 159 3.4 Applications 164 Chapter 3 Review Quiz 172 v vi Contents Chapter 4. Elementary Functions 175 4.1 Exponential and Logarithmic Functions 176 4.1.1 Complex Exponential Function 176 4.1.2 Complex Logarithmic Function 182 4.2 Complex Powers 194 4.3 Trigonometric and Hyperbolic Functions 200 4.3.1 Complex Trigonometric Functions 200 4.3.2 Complex Hyperbolic Functions 209 4.4 Inverse Trigonometric and Hyperbolic Functions 214 4.5 Applications 222 Chapter 4 Review Quiz 232 Chapter 5. Integration in the Complex Plane 235 5.1 Real Integrals 236 5.2 Complex Integrals 245 5.3 Cauchy-Goursat Theorem 256 5.4 Independence of Path 264 5.5 Cauchy’s Integral Formulas and Their Consequences 272 5.5.1 Cauchy’s Two Integral Formulas 273 5.5.2 Some Consequences of the Integral Formulas 277 5.6 Applications 284 Chapter 5 Review Quiz 297 Chapter 6. Series and Residues 301 6.1 Sequences and Series 302 6.2 Taylor Series 313 6.3 Laurent Series 324 6.4 Zeros and Poles 335 6.5 Residues and Residue Theorem 342 6.6 Some Consequences of the Residue Theorem 352 6.6.1 Evaluation of Real Trigonometric Integrals 352 6.6.2 Evaluation of Real Improper Integrals 354 6.6.3 Integration along a Branch Cut 361 6.6.4 The Argument Principle and Rouch´e’s Theorem 363 6.6.5 Summing Infinite Series 367 6.7 Applications 374 Chapter 6 Review Quiz 386 Contents vii Chapter 7. Conformal Mappings 389 7.1 Conformal Mapping 390 7.2 Linear Fractional Transformations 399 7.3 Schwarz-Christoffel Transformations 410 7.4 Poisson Integral Formulas 420 7.5 Applications 429 7.5.1 Boundary-Value Problems 429 7.5.2 Fluid Flow 437 Chapter 7 Review Quiz 448 Appendixes: I Proof of Theorem 2.1 APP-2 II Proof of the Cauchy-Goursat Theorem APP-4 III Table of Conformal Mappings APP-9 Answers for Selected Odd-Numbered Problems ANS-1 Index IND-1 Preface 7.2 Preface Philosophy Thistextgrewoutofchapters17-20inAdvancedEngineer- ing Mathematics, Second Edition (Jones and Bartlett Publishers), by Dennis G. Zill and the late Michael R. Cullen. This present work represents an ex- pansionandrevisionofthatoriginalmaterialandisintendedforuseineither aone-semesteroraone-quartercourse. Itsaimistointroducethebasicprin- ciples and applications of complex analysis to undergraduates who have no prior knowledge of this subject. The motivation to adapt the material from Advanced Engineering Math- ematics into a stand-alone text sprang from our dissatisfaction with the suc- cession of textbooks that we have used over the years in our departmental undergraduatecourseofferingincomplexanalysis. Ithasbeenourexperience thatbooksclaimingtobeaccessibletoundergraduateswereoftenwrittenata level that was too advanced for our audience. The “audience” for our junior- level course consists of some majors in mathematics, some majors in physics, but mostly majors from electrical engineering and computer science. At our institution, atypicalstudentmajoringinscienceorengineeringdoesnottake theory-oriented mathematics courses in methods of proof, linear algebra, ab- stract algebra, advanced calculus, or introductory real analysis. Moreover, the only prerequisite for our undergraduate course in complex variables is the completion of the third semester of the calculus sequence. For the most part, then, calculus is all that we assume by way of preparation for a student to use this text, although some working knowledge of differential equations would be helpful in the sections devoted to applications. We have kept the theory in this introductory text to what we hope is a manageable level, con- centrating only on what we feel is necessary. Many concepts are conveyed in an informal and conceptual style and driven by examples, rather than the formal definition/theorem/proof. We think it would be fair to characterize this text as a continuation of the study of calculus, but also the study of the calculusoffunctionsofacomplexvariable. Donotmisinterpretthepreceding words; we have not abandoned theory in favor of “cookbook recipes”; proofs of major results are presented and much of the standard terminology is used. Indeed, there are many problems in the exercise sets in which a student is asked to prove something. We freely admit that any student—not just ma- jors in mathematics—can gain some mathematical maturity and insight by attempting a proof. But we know, too, that most students have no idea how to start a proof. Thus, in some of our “proof” problems, either the reader ix x Preface is guided through the starting steps or a strong hint on how to proceed is provided. The writing herein is straightforward and reflects the no-nonsense style of Advanced Engineering Mathematics. Content We have purposely limited the number of chapters in this text toseven. Thiswasdonefortwo“reasons”: toprovideanappropriatequantity of material so that most of it can reasonably be covered in a one-term course, and at the same time to keep the cost of the text within reason. Here is a brief description of the topics covered in the seven chapters. • Chapter 1 The complex number system and the complex plane are examined in detail. • Chapter 2 Functions of a complex variable, limits, continuity, and mappings are introduced. • Chapter 3 The all-important concepts of the derivative of a complex function and analyticity of a function are presented. • Chapter 4 The trigonometric, exponential, hyperbolic, and logarith- mic functions are covered. The subtle notions of multiple-valued func- tions and branches are also discussed. • Chapter 5 The chapter begins with a review of real integrals (in- cluding line integrals). The definitions of real line integrals are used to motivate the definition of the complex integral. The famous Cauchy- Goursat theorem and the Cauchy integral formulas are introduced in this chapter. Although we use Green’s theorem to prove Cauchy’s the- orem, a sketch of the proof of Goursat’s version of this same theorem is given in an appendix. • Chapter6 Thischapterintroducestheconceptsofcomplexsequences andinfiniteseries. ThefocusofthechapterisonLaurentseries,residues, andtheresiduetheorem. Evaluationofcomplexaswellasrealintegrals, summation of infinite series, and calculation of inverse Laplace and in- verse Fourier transforms are some of the applications of residue theory that are covered. • Chapter 7 Complex mappings that are conformal are defined and used to solve certain problems involving Laplace’s partial differential equation. Features Eachchapterbeginswithitsownopeningpagethatincludesa tableofcontentsandabriefintroductiondescribingthematerialtobecovered in the chapter. Moreover, each section in a chapter starts with introduc- tory comments on the specifics covered in that section. Almost every section ends with a feature called Remarks in which we talk to the students about areas where real and complex calculus differ or discuss additional interesting topics (such as the Riemann sphere and Riemann surfaces) that are related

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