Handbook of Complex Variables
This book is written to be a convenient reference for the working scientist, student, or engineer who needs to know and use basic concepts in complex analysis. It is not a book of mathematical theory. It is instead a book of mathematical practice. All the
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Steven G. Krantz
Handbook of Complex Variables With 102 Figures
Springer Science+Business Media, LLC
Steven G. Krantz Department of Mathematics Washington University in St. Louis St. Louis, MO 63130 USA
Library of Congress Cataloging.in·Publication Data Krantz, Steven G. (Steven George), 1951Handbook of complex variab1es / Steven G. Krantz. p. cm. Includes bib1iographica1 references and index. ISBN 978-1-4612-7206-9
ISBN 978-1-4612-1588-2 (eBook)
DOI 10.1007/978-1-4612-1588-2 1. Functions of complex variab1es. 2. Mathematical analysis. 1. Title. QA331.7.K744 1999 515'.9-dc21 99.20156 CIP AMS Subject C1assifications: 30-00, 32-00, 33-00 Printed on acid-free paper. © 1999 Springer Science+Business Media New York Originally published by Birkhăuser Boston in 1999 Softcover reprint of the hardcover 1st edition 1999
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AII rights reserved. This work may not be translated or copied in whole or in part without the written permission ofthe publisher (Springer Science+Business Media, LLC ), 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 of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 978-1-4612-7206-9 SPIN 19954551 Typeset by the author in
987654321
J.t.'IEX.
To the memory of Lars Valerian Ahlfors, 1907-1996.
Contents
Preface
xix
List of Figures
xxi
1 The Complex Plane 1.1 Complex Arithmetic . . . . . . . 1.1.1 The Real Numbers. . . 1.1.2 The Complex Numbers 1.1.3 Complex Conjugate . . 1.1.4 Modulus of a Complex Number 1.1.5 The Topology of the Complex Plane. 1.1.6 The Complex Numbers as a Field . . 1.1.7 The Fundamental Theorem of Algebra. 1.2 The Exponential and Applications . . . . . . . 1.2.1 The Exponential Function 1.2.2 The Exponential Using Power Series. 1.2.3 Laws of Exponentiation . . . . . . 1.2.4 Polar Form of a Complex Number . . 1.2.5 Roots of Complex Numbers. . . . . . 1.2.6 The Argument of a Complex Number 1.2.7 Fundamental Inequalities . . . . . . . 1.3 Holomorphic Functions. . . . . . . . . . . . . . 1.3.1 Continuously Differentiable and C k Functions 1.3.2 The Cauchy-Riemann Equations. . 1.3.3 Derivatives.. . . . . . . . . . . . . 1.3.4 Definition of Holomorphic Function 1.3.5 The Complex Derivative
1 1 1 1 2 2 3 6 7 7 7 8 8 8 10 11 12 12 12 13 13 14 15
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Contents
viii 1.3.6
Alternative Terminology for Holomorphic Functions . . . 1.4 The Relationship of Holomorphic and Harmonic Functions . . . . . . . . . . 1.4.1 Harmonic Functions. . . . . 1.4.2 Holomorphic and Harmonic Functions . 2 Complex Line Integrals 2.1 Real and Complex Line Integrals 2.1.1 Curves....... . . 2.1.2 Closed Curves . . . . . 2.1.3 Diffe
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