A Primer of Real Analytic Functions
It is a pleasure and a privilege to write this new edition of A Primer 0/ Real Ana lytic Functions. The theory of real analytic functions is the wellspring of mathe matical analysis. It is remarkable that this is the first book on the subject, and we wa
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Edited by Herbert Amann, University of Zürich Steven G. Krantz, Washington University, St. Louis Shrawan Kumar, University of North Carolina at Chapel Hili
Steven G. Krantz Harold R. Parks
APrimerof Real Analytic Functions Second Edition
Springer Science+Business Media, LLC
Steven G. Krantz Washington University Department of Mathematics St. Louis, MO 63130-4899 U.S.A.
Harold R. Parks Oregon State University Department of Mathematics Corvallis, OR 97331-4605 U.S.A.
Library of Congress CataIoging-in-Pubücation Data
A CIP catalogue record for Ibis book is available from the Library of Congress, Washington D.C., USA.
AMS Subject Classifications: Primary: 26E05 , 30BIO, 32C05; Secondary: 14PI5, 26A99. 26BIO, 26B40, 26EIO, 30B40, 32C09, 35AIO, 54C30
Printed on acid-free paper 1t>2002 Springer Science+Business Media New York Birkhäuser Originally published by Birkhäuser Boston in 2002 Softcover reprint ofthe hardcover 2nd edition 2002 All rights reserved. This work may not be translated or copied in whole or in part without the wriUen permission of the 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 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. ISBN 978-1-4612-6412-5 ISBN 978-0-8176-8134-0 (eBook) DOI 10.1007/978-0-8176-8134-0 Reformatted from the author's files by TEXniques, Inc., Carnbridge. MA.
9 876 54 3 2 I
SPIN 10846987
To the memory of Frederick J. Almgren, Jr. (1933-1997), teacher and friend
Contents
Preface to the Second Edition
ix
Preface to the First Edition
xi
1
1
2
Elementary Properties 1.1 Basic Properties of Power Series 1.2 Analytic Continuation. . . . .. 1.3 The Formula of Faa di Bruno .. 1.4 Composition of Real Analytic Functions 1.5 Inverse Functions . . . . . . . . . . . . Multivariable Calculus of Real Analytic Functions 2.1 Power Series in Several Variables. . . . . . . 2.2 Real Analytic Functions of Several Variables . 2.3 The Implicit Function Theorem. . . . . . . . . 2.4 A Special Case of the Cauchy-Kowalewsky Theorem 2.5 The Inverse Function Theorem . . . . . . . . . . . . 2.6 Topologies on the Space of Real Analytic Functions . 2.7 Real Analytic Submanifolds . . . . . . . . . . . . 2.7.1 Bundles over aReal Analytic Submanifold 2.8 The General Cauchy-Kowalewsky Theorem . . . .
11 16 18 20
25 25
29 35
42 47 50 54 56
61
viii
3
Contents
Classical Topics 3.0 Introductory Remarks . . . . . . . . . 3.1 The Theorem ofPringsheim and Boas 3.2 Besicovitch's Theorem . . . . . . . . 3.3 Whitney's Extension and Approximation Theorems 3.4 The Theorem of S. Bernstein . . . . . . . . . . . .
67 67 68 72 75 79
4 Some
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