Complex Analysis
This unusual and lively textbook offers a clear and intuitive approach to the classical and beautiful theory of complex variables. With very little dependence on advanced concepts from several-variable calculus and topology, the text focuses on the authen
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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 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 (www.springer.com)
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 an
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