Integral Representations and Residues in Multidimensional Complex Analysis

298 Pages·1984·19.45 MB·English

by  I. A. Aizenberg, A. P. Yuzhakov

Integral Representations
and Residues <~ ---.~~-. · -~--
in Multidimensional
Complex Analysis
"'
by LvA. AIZENBERG
A.P. YUZHAKOV
Volume 58
American Mathematical Society
P.O. Box 6248, Providence, Rhode Island 02940
Telephone (401) 272-9500
CORRECTION NOTICE
L. A. Aizenberg, one of the authors of "Integral Representations
and Residues in Multidimensional Complex Analysis", is incorrectly
identified on the cover and in the front matter as I.. A. Aizenberg.
Our apologies for this error.
location: 201 Charles Street at Randall Square
Integral Representations and Residues
in
Multidimensional Complex Analysis
TRANSLATIONS OF MATHEMATICAL MONOGRAPHS
VOLUME 58
Integral Representations
and Residues
in Multidimensional
Complex Analysis
..,
by L.A. AIZENBERG
A.P.YUZHAKOV
American Mathematical Society · Providence · Rhode Island
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Translated from the Russian by H. H. McFaden
Translation edited by Lev J. Leifman
1980 Mathematics Subject Classification. Primary 32-02, 32A25, 32A27; Secondary 05AI5,
30CI5, 30C40, 30E20, 32A05, 32A30, 32CIO, 32C25, 32C30, 32C37, 32C40, 32005, 32E05, 32EIO,
32FI5, 32F20, 32HIO.
ABSTRACT. This book deals with integral representations of holomorphic functions of several
complex variables, the multidimensional logarithmic residue, and the theory of multidimensional
residues. Applications are given to implicit function theory, systems of nonlinear equations,
computation of the multiplicity of a zero of a mapping, and computation of combinatorial sums in
closed form. Certain applications in multidimensional complex analysis are considered.
The monograph is intended for specialists in theoretical and applied mathematics and theoreti-
cal physics, and for post graduate and graduate students interested in multidimensional complex
analysis or its applications.
Library of Congress Cataloging in Publication Data
Aizenberg, Lev Abramovich, 1937-
Integral representations and residues in multidimensional complex analysis.
(Translations of mathematical monographs; v. 58)
Translation of: Integral'nye predstavlenha i vychety v mnogomernom
kompleksnom analize.
Bibliography: p.
Includes indexes.
1. Holomorphic functions. 2. Integral representations. 3. Congruences
and residues. I. fuzhakov, .Aleksandr Petrovich. II. Title. III. Series.
QA331.A46413 1983 515.9'8 83-15549
ISBN 0-8218-4511-X
All rights reserved except those granted to the United States Government
This book may not be reproduced in any form without permission of the publisher.
Copyright © 1983 by the American Mathematical Society
Table of Contents
Preface vu
Preliminaries 1
CHAPTER I. Integral Representations and the Logarithmic Residue 15
1. The Martinell!-Bochner integral representation 15
2. Multiplicity of a zero of a holomorphic mapping. The Rouche
principle 18
3. The fundamental integral formula of Leray and Koppelman 25
4. The Cauchy formula. The logarithmic residue with respect to
the skeleton 30
5. The Andreotti-Norguet formula and generalizations of it 38
6. The Bergman kernel function, the Szego kernel, and integral
representations with a holomorphic kernel on the Shilov bound-
ary 42
7. The Martinelli-Bochner-Koppelman integral representation for
exterior differential forms 49
CHAPTER II. Integral Representations of Special Form for Holomor-
phic Functions 59
8. Modifications and the simplest particular cases of the Leray
formula 59
9. The Bergman-Weil formula 65
10. Integral representation for strictly pseudoconvex domains 66
11. Integral formulas for functions holomorphic in n-circular do-
mains 73
12. The Schwarz kernel and integral representations of holomorphic
functions with nonnegative real part 80
CHAPTER ID. The Theory of Residues 93
13. Statement of the problem 93
14. Application of Alexander-Pontryagin duality 94
15. Application of de Rham duality 98
v
vi CONTENTS
16. The Leray theory of residues 101
17. Cohomological reduction of certain semimeromorphic and ra-
tional forms 113
18. Residues of rational functions of two variables 119
19. Local residues of certain meromorphic and rational functions in
~ 1~
CHAPTER IV. Applications to Implicit FUnctions, Systems of Non-
linear Equations, Computation of the Multiplicity of a Zero, and
Combinatorics 137
20. Expansion of implicit functions in power series and function
series 137
21. Application of the multidimensional logarithmic residue to sys-
tems of nonlinear equations 157
22. Computation of the multiplicity of a zero of a system of holomor-
phic functions from their Taylor series 175
23. Application of multiple residues for finding generating functions
and computing combinatorial sums 188
CHAPTER V. Some Applications in Multidimensional Complex
Analysis 195
24. The Hartogs-Bochner theorem on necessary analytic extension.
Approximation of holomorphic functions on Weil polyhedra and
linearly convex compact sets 195
25. The <3-problem and Oka's theorems 204
26. Forms orthogonal to holomorphic functions. Lewy's equation.
General form of integral representation formulas for holomor-
phic functions 219
27. A basis in the space of holomorphic functions with fixed al-
gebraic singularities · 241
Brief historical notes 249
Bibliography 259
Subject index 279
Index of notation 283