A First Course in Complex Analysis.pdf

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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 40 Tall Pine Drive Sudbury, MA 01776 978-443-5000 info@jbpub.com www.jbpub.com Jones and Bartlett Publishers Canada 2406 Nikanna Road Mississauga, ON L5C 2W6 CANADA Jones and Bartlett Publishers International Barb House, Barb Mews London W6 7PA UK Copyright © 2003 by Jones and Bartlett Publishers, Inc. Library of Congress Cataloging-in-Publication Data Zill, Dennis G., 1940- A first 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. QA331.7 .Z55 2003 515’.9—dc21 I. Shanahan, Patrick, 1931- II. Title. 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. Chapter 2. Chapter 3. 1 2 10 Complex Numbers and the Complex Plane Complex Numbers and Their Properties 1.1 Complex Plane 1.2 Polar Form of Complex Numbers 1.3 Powers and Roots 1.4 Sets of Points in the Complex Plane 1.5 Applications 1.6 Chapter 1 Review Quiz 23 36 16 29 45 Complex Functions and Mappings 49 2.1 2.2 2.3 2.4 Complex Functions Complex Functions as Mappings Linear Mappings Special Power Functions 50 68 80 58 2.4.1 The Power Function zn 2.4.2 The Power Function z1/n 81 86 2.5 2.6 2.7 Reciprocal Function 100 Limits and Continuity 110 2.6.1 Limits 2.6.2 Continuity 119 110 Applications Chapter 2 Review Quiz 132 138 141 Analytic Functions 3.1 3.2 3.3 3.4 Diﬀerentiability and Analyticity 142 Cauchy-Riemann Equations Harmonic Functions Applications Chapter 3 Review Quiz 152 159 164 172 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 Complex Powers Trigonometric and Hyperbolic Functions 194 200 4.3.1 Complex Trigonometric Functions 4.3.2 Complex Hyperbolic Functions 209 200 Inverse Trigonometric and Hyperbolic Functions Applications Chapter 4 Review Quiz 232 214 222 4.2 4.3 4.4 4.5 235 Integration in the Complex Plane 5.1 5.2 5.3 5.4 5.5 Real Integrals 236 Complex Integrals Cauchy-Goursat Theorem 256 Independence of Path 264 Cauchy’s Integral Formulas and Their Consequences 245 272 5.5.1 Cauchy’s Two Integral Formulas 5.5.2 Some Consequences of the Integral Formulas 284 277 Applications Chapter 5 Review Quiz 297 273 5.6 313 302 Series and Residues 6.1 6.2 6.3 6.4 6.5 6.6 301 Sequences and Series Taylor Series Laurent Series Zeros and Poles Residues and Residue Theorem 342 Some Consequences of the Residue Theorem 352 324 335 6.6.1 Evaluation of Real Trigonometric Integrals 352 Chapter 5. Chapter 6. 6.6.2 Evaluation of Real Improper Integrals Integration along a Branch Cut 361 6.6.3 6.6.4 The Argument Principle and Rouch´e’s 354 Theorem 363 Summing Inﬁnite Series 367 6.6.5 6.7 Applications Chapter 6 Review Quiz 374 386

Contents vii Chapter 7. Conformal Mappings 389 7.1 7.2 7.3 7.4 7.5 390 Conformal Mapping Linear Fractional Transformations Schwarz-Christoﬀel Transformations Poisson Integral Formulas Applications 420 429 399 410 7.5.1 Boundary-Value Problems 7.5.2 Fluid Flow 437 429 Chapter 7 Review Quiz 448 Appendixes: I II III Proof of Theorem 2.1 APP-2 Proof of the Cauchy-Goursat Theorem APP-4 Table of Conformal Mappings APP-9 Answers for Selected Odd-Numbered Problems ANS-1 Index IND-1

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