Geometric Function Theory Explorations in Complex Analysis
Complex variables is a precise, elegant, and captivating subject. Presented from the point of view of modern work in the field, this new book addresses advanced topics in complex analysis that verge on current areas of research, including invariant geomet
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Advisory Board Anthony W. Knapp, State University of New York at Stony Brook, Emeritus
Steven G. Krantz
Geometric Function Theory Explorations in Complex Analysis
Birkh¨auser Boston • Basel • Berlin
Steven G. Krantz Washington University Department of Mathematics St. Louis, MO 63130 U.S.A. e-mail to: [email protected] http://www.math.wustl.edu/˜sk/
Cover design by Mary Burgess. Mathematics Subject Classicification (2000): 30Axx, 30Bxx, 30Cxx, 30Dxx, 30Exx, 30Fxx, 30H05, 32Axx, 32Bxx, 32Dxx, 32Fxx, 31Axx, 35N15 Library of Congress Cataloging-in-Publication Data Krantz, Steven G. (Steven George), 1951Geometric function theory : explorations in complex analysis / Steven G. Krantz. p. cm. – (Cornerstones) Includes bibliographical references and index. ISBN 0-8176-4339-7 (alk. paper) 1. Geometric function theory. 2. Functions of complex variables. I. Title. II. Cornerstones (Birkh¨auser) QA331.7.K733 2005 515’.9–dc22
ISBN-10 0-8176-4339-7 ISBN-13 978-0-8176-4339-3
2005050071
eISBN 0-8176-4440-7
Printed on acid-free paper.
c 2006 Birkh¨auser Boston
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA) and the author, 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 in the United States of America. 987654321 www.birkhauser.com
(TXQ/MP)
To the memory of Don Spencer
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Part I Classical Function Theory Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Invariant Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Conformality and Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Bergman’s Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Calculation of the Bergman Kernel for the Disk . . . . . . . . . . . . . 1.3.1 Construction of the Bergman Kernel for the Disk by Conformal Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Construction of the Bergman Kernel by means of an Orthonormal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Construction of the Bergman Kernel by way of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 A New Application . . . . . . . . . . . . . . . . . . .
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