MTH632 Handouts

COMPLEX VARIABLES AND APPLICATIONS, EIGHTH EDITION

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ISBN 978–0–07–305194–9 MHID 0–07–305194–2

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Library of Congress Cataloging-in-Publication Data

Brown, James Ward. Complex variables and applications / James Ward Brown, Ruel V. Churchill.— 8th ed. p. cm. Includes bibliographical references and index. ISBN 978–0–07–305194–9— ISBN 0–07–305194–2 (hard copy : acid-free paper) 1. Functions of complex variables. I. Churchill, Ruel Vance, 1899II. Title. QA331.7.C524 2009 515r.9— dc22 2007043490

ABOUT THE AUTHORS

JAMES WARD BROWN is Professor of Mathematics at The University of Michigan– Dearborn. He earned his A.B. in physics from Harvard University and his

A.M. and Ph.D. in mathematics from The University of Michigan in Ann Arbor, where he was an Institute of Science and Technology Predoctoral Fellow. He is coauthor with Dr. Churchill of Fourier Series and Boundary Value Problems, now in its seventh edition. He has received a research grant from the National Science Foundation as well as a Distinguished Faculty Award from the Michigan Association of Governing Boards of Colleges and Universities. Dr. Brown is listed in Who’s

Who in the World.

RUEL V. CHURCHILL was, at the time of his death in 1987, Professor Emeritus of Mathematics at The University of Michigan, where he began teaching in 1922. He received his B.S. in physics from the University of Chicago and his M.S. in physics and Ph.D. in mathematics from The University of Michigan. He was coauthor with Dr. Brown of Fourier Series and Boundary Value Problems, a classic text that he first wrote almost 70 years ago. He was also the author of Operational Mathematics. Dr. Churchill held various offices in the Mathematical Association of America and in other mathematical societies and councils.

iii · CONTENTS · Preface · x · Complex Numbers

Sums and Products Basic Algebraic Properties Further Properties Vectors and Moduli Complex Conjugates Exponential Form Products and Powers in Exponential Form Arguments of Products and Quotients Roots of Complex Numbers Examples Regions in the Complex Plane

Functions of a Complex Variable Mappings Mappings by the Exponential Function Limits Theorems on Limits