A First Course in
Complex Analysis with Applications
Dennis G. Zill Loyola Marymount University
Patrick D. Shanahan Loyola Marymount University
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Copyright © 2003 by Jones and Bartlett Publishers, Inc. Library of Congress Cataloging-in-Publication Data Zill, Dennis G., 1940A 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. 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 z n 81 2.4.2 The Power Function z 1/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 2