Complex Analysis Fundamentals of the Classical Theory of Functions
This clear, concise introduction to the classical theory of one complex variable is based on the premise that "anything worth doing is worth doing with interesting examples." The content is driven by techniques and examples rather than definiti
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For Milada
John Stalker
Complex Analysis Fundamentals of the Classical Theory of Functions
Springer Science+Business Media, LLC
John Stalker Department of Mathematics Princeton University Princeton, New Jersey 08544
Ubrary of Congress Cataloging-in-Publication Data John Stalker Complex analysis : fundamentals of the classical theory of functions I John Stalker. p. em. Includes bibliographical references and index. ISBN 978-0-8176-4918-0 ISBN978-0-8176-4919-7 (eBook) DOI 10.1007/978-0-8176-4919-7 1. Functions of complex variables. 2. Mathematical analysis. I. Title. QA331.7S73 1998 97-52126 515'.93--21 CIP
AMS codes: 30, 33 Printed on acid-free paper
© 1998 John Stalker
Originally published by Birkhiiuser Boston in 1998 Softcover reprint of the hardcover lst edition 1998 Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying. recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhlluser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Springer Science+Business Media, LLC. ISBN 978-0-8176-4918-0 Typeset by the author in Latex
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Contents Preface
vii
Outline
ix
1 Special Functions 1.1 The Gamma Function . 1.2 The Distribution of Primes I 1.3 Stirling's Series . . . . . 1.4 The Beta Integral 1.5 The Whittaker Function . . . 1.6 The Hypergeometric Function 1.7 Euler-MacLaurin Summation . 1.8 The Zeta Function . . . . . . . 1.9 The Distribution of Primes II . • • • •
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2 Analytic Functions 2.1 Contour Integration ...... 2.2 Analytic Functions ...... 2.3 The Cauchy Integral Formula . 2.4 Power Series and Rigidity . . . 2.5 The Distribution of Primes III 2.6 Meromorphic Functions . . . . 2.7 Bernoulli Polynomials Revisited 2.8 Mellin-Bames Integrals I . 2.9 Mellin-Bames Integrals II . .
3 Elliptic and Modular Functions 3.1 Theta Functions . 3.2 Eisenstein Series 3.3 Lattices 3.4 Elliptic Functions 3.5 Complex Multiplication . 3.6 Quadratic Reciprocity . . 3.7 Biquadratic Reciprocity . . . •
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• • •
1
2 11
19 26 30 42 53 59 70 77
77 85 89 96 102 107 115 124 134 149
149 153 164 177 183 188 194
vi
CONTENTS
A Quick Review of Real Analysis
201
Bibliography
221
Index
226
Preface All modem introductions to complex analysis follow, more or less explicitly, the pattern laid down in Whittaker and Watson [75]. In "part I'' we find the foundational material, the basic definitions and theorems. In "part II" we find the examples and applications. Slowly we begin to understand why we read part I. Historically this is an anachronism. Pedagogically it is a disaster. Part I
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