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vv AND LOCAL PROPERTIES
Dedicado h minha filha L ~ NACHBIN A
NORTH-HOLLAND MATHEMATICS STUDIES
1
Holomorphic Functions, Domains of Holomorphy and Local Properties LEOPOLDO NACHBIN George Eastman Professor University of Rochester Rochester, New York, USA
Notes prepared by Richard M. Aron
i 970 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
. LONDON
0 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM - 1970 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 the prior permission of the. Copyright owner.
Library of Congress Catalog Card Number 78-134642 ISBN North-Holland 07204 2041 5 ISBN American Elsevier 07444 10054 7
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NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NORTH-HOLLAND PUBLISHING COMPANY, LTD. - LONDON SOLE DISTRIBUTORS FOR THE U.S.A. AND CANADA:
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FOREWORD
This t e x t i s p a r t l y based on courses t h a t I taught a t t h e I n s t i t u t o de Matematica Pura e Aplicada, Rio de J a n e i r o i n 1958 and 1967, a t t h e Universidade do Recife i n 1959, and a t t h e Universidade de Sa"o Paul0 i n 1967.
It reproduces l i t e r a l l y
a course t h a t I taught a t t h e University o f Rochester during t h e F a l l of 1967. These l e c t u r e s were intended as an elementary i n t r o d u c t i o n t o s e v e r a l complex v a r i a b l e s , arranged t o lead t o c e r t a i n f e a t u r e s of t h e s u b j e c t having no c o u n t e r p a r t i n a standard Domains of holomorphy and
treatment of one complex v a r i a b l e .
l o c a l p r o p e r t i e s seem t o be e s p e c i a l l y s u i t a b l e i n giving a beginner i n t h e f i e l d a f l a v o r o f how s e v e r a l complex v a r i a b l e s d i f f e r from one complex v a r i a b l e .
Limitation i n time and t h e
d e s i r e t o present t h e m a t e r i a l i n an e x p l i c i t form had t h e i r n a t u r a l i m p l i c a t i o n i n t h e s t y l e of t h e c o u r s e .
of t h i s volume are encouraged t o use t h e materia
The r e a d e r s quoted i n
t h e Bibliography i n o r d e r t o f u r t h e r t h e i r study i n t h e d i r e c t i o n of l o c a l and g l o b a l complex a n a l y s i s as needed i n present-day mathematics and physics. Thanks a r e due t o M r . Richard M. Aron, of t h e University
of Rochester, f o r preparing t h i s t e x t . LEOPOLDO NACHBIN
Rochester December 1967 V
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TABLE OF CONTENTS
Chapter 1: Holomorphic Functions
1 2 3 4
. . . .
Section 5
.
Sec t i o n Section Section Section
Section 6 Section 7 Section 8 Chapter 2 :
. .
Domains of Holomorphy
Section Section Section Section Section Chapter 3 :
.
............ 1 ..... 4 ........................................ 9 ................................. 12 ...........................................16 ......... 25 ................................30 ................................... 31
Ho lomorphic Functions of 1 Complex Variable Holomorphic Functions of Several Complex Variables Cauchy I n t e g r a l D i f f e r e n t i a t i o n of Holomorphic Functions and the Cauchy I n e q u a l i t i e s The Natural Topology on t h e Spaces of Holomorphic Functions Taylor S e r i e s and Unique Analytic Continuation Maximum Modulus Theorem Holomorphic Mappings
.
.
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