Integral Representations and Residues in Multidimensional Complex Analysis

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
..,
11HTErPAJihHhIE IlPE,llCTABJIEHl1.H 11 BhlqEThl B MHOrOMEPHOM KOMilJIEKCHOM AHAJll13E
JI. A. AM3EHJ>EPr 11 A. n. IOiKAKOB
113.llATEJibCTBO «HAYKA» CH6HPCKOE OT){EJIEHHE
HOBOCl16HPCK 1979
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 boundary 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 holomorphic functions 219
27. A basis in the space of holomorphic functions with fixed algebraic singularities · 241
Brief historical notes 249
Preface
Integral representations and the theory of residues occupy an important place in multidimensional complex analysis. At the present time these areas are continuing to develop intensively and are finding ever newer applications in function theory itself and in other areas of mathematics, as well as in theoretical physics. For example, the Khenkin-Ramirez de Arellano integral representation for functions holomorphic in strictly pseudoconvex domains has led to a number of important results in the theory of holomorphic functions of several complex variables.
We mention also a certain "nonsymmetry" in the integral representations of holomorphic functions of several complex variables that existed before 1967: functions were represented by integration of exterior differential forms. This "nonsymmetry" was removed by the Martinelli-Bochner-Koppelman formula (forms are represented by integration of forms), which has found many serious applications in multidimensional complex analysis. In particular, Khenkin used this formula to get an integral representation for a solution of the a-problem in strictly convex domains, and this led to a new proof of the main theorems of Oka without the use of sheaf theory or the theory of differential operators.
There recently appeared a Leray-Koppelman integral formula of very gen eral form that includes both all possible integral representations of holomor phic functions and formulas for the multidimensional logarithmic residue (see §3). Therefore, Chapter I contains both integral representations of general form and formulas for the logarithmic residue. We remark that integral representations and residues are closely connected, since they are based on Stokes' formula.
The theory of multidimensional residues, whose beginning goes back to Poincare, has developed in various directions and presently is far from com plete. In generalizing the theory of residues to the multidimensional case there arise serious difficulties of a topological nature, since for holomorphic func tions of several complex variables the role of isolated singular points is played by analytic sets, which can have a complicated topological structure. To
vii
viii PREFACE
overcome these difficulties it is necessary to resort to the tools of algebraic topology.
Multidimensional residues, besides having well-known applications in the investigation of Feynman integrals by homological methods and the investi gation of fundamental solutions of hyperbolic partial differential equations (Leray), have recently been applied to implicit functions, systems of nonlinear equations, and combinatorial analysis. These new applications have not yet been reflected in the existing texts of function theory. Moreover, in these texts integral representations and residues of functions of several complex variables are not presented in a sufficiently complete manner. The present book is intended to fill this gap. It arose from special courses taught by the aughors at Krasnoyarsk University in 1965-1974 (see (26) and (218)).
As already mentioned, Chapter I contains .integral representations of holo morphic functions of several complex variables and formulas for the multi dimensional logarithmic residue, presented here from a unified point of view.
In §2 we define the multiplicity of a zero of a holomorphic mapping and prove the main theorems on zeros of holomorphic mappings, using only the properties of integrals of closed forms and not using results in the local theory of analytic sets. In particular, a compact set of zeros of a holomorphic mapping is proved to be discrete. The theorem of Osgood asserting that a one-to-one holomorphic mapping is biholomorphic is obtained as a corollary in §4.
Moreover, in Chapter I we present the Andreotti-Norguet integral represen tation for the derivatives of a holomorphic function (§5) and the Martinelli Bochner-Koppelman integral representation for exterior differential forms (§7).
Chapter II deals with a special form of integral representations derived from Leray's formula. The most important of them are the Bergman-Weil formula (§9) for functions holomorphic in analytic polyhedra and the K.henkin-Ramirez de Arellano formula (§ 10) for functions holomorphic in strictly pseudoconvex domains. §12 stands somewhat isolated; there we consider an integral repre sentation for holomorphic functions with nonnegative real part (a generaliza tion of the classical Riesz-Herglotz formula).
In this book integral representations are considered only for bounded domains, a reflection of the personal preferences of the authors (see, for example, (52)-(54), [60)-(62), (224] and (236) about integral representations in unbounded domains).
The theory of multidimensional residues is presented in Chapter III. We formulate the problem of this theory (§ 13) and indicate some ways of solving it that follow from Stokes' formula, the Cauchy-Poincare integral theorem, the
PREFACE ix
Alexander-Pontryagin topological duality (§14), and the de Rham duality (§15). Then (§§16 and 17) we give the main results in the theory of residues on a complex analytic manifold (the Leray theory of residues). The chapter concludes with ·an application of the Alexander-Pontryagin duality and the de Rham duality to the computation of residues of meromorphic and rational forms (§§ 18 and 19).
This book does not reflect results on extension of the Leray theory of residues to forms having singularities on submanifolds of codimension greater than 1, nor residues of currents, nor the abstract theory of residues which studies dual homomorphisms of the homology and cohomology groups of a closed subset and its complement in a manifold, nor Grothendieck residues, nor Dolbeault residues of semimeromorphic forms.
Chapter IV deals with so-called "external" applications (found during the last few years mainly by Krasnoyarsk mathematicians) of various formulas for the multidimensional logarithmic residue and simply for residues.
In §20 we consider applications of the multiple logarithmic residue for representing implicit functions of several variables determined by systems of equations, in particular, for inverting a holomorphic mapping in the form of a multiple power series or a function series (multidimensional analogues of the Lagrange expansion and the Biirmann-Lagrange expansion), with estimates given of the domain of convergence for these series. We consider some cases in which single-valued branches of multi-valued implicit functions are de termined, and obtain effective formulas for the coefficients of a Weierstrass pseudopol'ynomial.
In §21 the formulas for the multiple logarithmic residue are applied to systems of nonlinear equations, in particular, to the solution of the problem of eliminating unknowns in systems of algebraic equations. We obtain multidi mensional analogues of the Waring formula which expresses power sums of the roots of equations in terms of their coefficients.
The multiplicity of a zero of a system of holomorphic functions is de termined from the Taylor expansions of these functions in §22. Here the main tool of the investigation is again the theory of multiple residues.
In §23 multidimensional residues are used to find generating functions and to compute various combinatorial sums in closed form.
Chapter V contains some "internal" applications, mainly of integral repre sentations to problems in multidimensional function theory. There are discus sions of holomorphic extension from the boundary of a domain to the whole domain and of expansion of holomorphic functions in series of polynomials or "partial fractions" (§24). In §25 we prove the main theorems of Oka without using sheaf theory or the theory of differential operators. The text in this
x PREFACE
section was written by Sh. A. Dautov on the basis of an unpublished manuscript which G. M. Khenkin kindly made available to the authors. In general, Khenkin's ideas had a large influence on the contents of Chapters I, II, and V. §26 contains information about the description of forms orthogonal to holo morphic functions when integrated over the boundary of a domain, about the general form of formulas for integral representation of holomorphic functions and for the logarithmic residue, and about the solvability of the Lewy equation. §27 deals with bases in spaces of holomorphic functions with fixed algebraic singularities.
The limited space in the book did not permit the authors to present many other applications of integral representations and residues. Here we should mention first of all the work of Leray on differential equations, as well as the applications of residues to the investigation of Feynman integrals. A number of other references are mentioned in the section Brief historical comments.
We note that §§1, 3, 5-12, 21, and 24 were written by L.A. Aizenberg, §§2, 4, 13-20, 23, and 27 by A. P. Yuzhakov, §§25 and 26 by Sh. A. Dautov, and §22 by A. K. Tsikh. The authors thank G. M. Khenkin and the editor, Sh. A. Dautov, for numerous useful improvements and changes in the text.
Furthermore, the authors use this opportunity to thank V. A. Bolotov, A. M. Kytmanov, E. S. Mkrtchyan, M.A. Mkrtchyan, V. A. Stepanenko, and A. K. Tsikh for help in preparing the manuscript for printing.
The authors
Preliminaries
1°. Integration of differential forms. Stokes' formula. We assume that the reader is acquainted with the concepts of a differentiable manifold and a complex analytic manifold (see, for example, [133], [144] or [189]}. A differen tial form w of degree p (a p-f orm) on a di// erentiable man if old ( 1) X can in a neighborhood of each point a E X be represented in local coordinates x = ( x 1' ... 'x n ) in the form
w(x)=LaAx)dx; /\···/\dx;, (O.l) I P
J
where J = (ip ... , ip), l .;;; i1 .;;; n, the a Ax) are real or complex functions of x, and /\ is the symbol for exterior multiplication, which satisfies the anticommu tation condition
dx; /\ dx; = 0. (0.2)
The representation (0.1) of a p-form w as given is unique if 1 .;;; i 1 < i 2 < · · · < i P .;;; n. In passing to other local coordinates y = ( y 1, ••• ,yn) the expres sion (0.1) is transformed to
[ a(x;, ... ,x;)l L LaAx(y)) 1
p dy11 /\ • • • /\dy1_. 1.;;;;,<··· <Jp<;;;n J a(Y;,,.···Y;.)
Differential forms can be added and multiplied by functions. The operation of exterior multiplication is defined for them; in local coordinates it is realized according to the rule for multiplying polynomials, with the condition (0.2) taken into account. This condition implies the following property of the exterior product: w /\ cp = (- l)Pqcp /\ w, where p and q are the degrees of w and cp. We say that w is a differential form of class c<r> if its coefficients aAx) in the given local representation (0.1) are r times continuously differentiable.
( 1) The exact definition of a differential form, along with a more detailed presentation of the material in this section, can be found in the books [102], [133], [158], [161] and [218].
1
2 PRELIMINARIES
The form w is said to be regular if r = oo. The linear space of p-forms of class c<r> on the manifold Xis denoted by c;'>(X). Following Leray's book [113], we often denote the space c;00 >( X) of regular p-forms also by QP( X).
The operator d of exterior differentiation assigns to each form w E c;'>(X) the form dw E c;'.;.( l)( X) determined in local coordinates by the formula
dw = ~da1(x) /\ dx1 , (0.3)
i=l X;
dx = dx. /\ · · · /\dx .. J 11 Ip
The exterior differential has the following properties: 1) d( w 1 + w2 ) = d w 1
+ dw2 ; 2) d(w /\ cp) = dw /\ cp + (-l)Pw /\ dcp, where pis the degree of the form w; 3) d dw = 0. A differential form w is said to be closed if dw = 0, and exact if w = dcp for some form cp. The set of closed regular p-forms is denoted by ZP( X), and the set of exact regular p-forms by BP( X). The properties 1)-3) of the exterior differential imply the inclusions of groups (vector spaces) BP(X) c ZP(X) c QP(X).
The factor group HP(X) = ZP(Z)/BP(X) is called the p-dimensional de Rham cohomology group of the man if old X. Its elements are called cohomology classes. Two forms w1 and w2 that belong to the same cohomology class are said to be cohomologous ( w 1 ~ w2 ).
An infinitely differentiable mapping of manifolds
f: X-+ y (0.4)
induces group homomorphisms
j: QP(Y) .... QP(X),
defined as follows. Let x = (x1, ••• ,xn) and y = (y1, ••• ,yn) be local coordi nates in neighborhoods U0 and Vb of the points a EX and b = f(a) E Y, and suppose that the form w E QP(Y) is represented as w(y) = ~,a;(y) dy;,
/\ · · · /\dy; . Then in U0 p
f{w){x) = w{/{x)) = ~a;(y(x)) dy;(x) /\ · · · /\dy;(x), I p
J
where dy;(x) = ~}= 1 (ay;/ax1)dy1. Since do j = j 0 d, jinduces a factor homo morphism f*.
If Z is a complex analytic manifold with local coordinates z1 = x1 + iy1,
j = l, ... , n, then instead of dx 1 and dy1 we take the linear combinations dz1 = dx1 + idy1 and dz1 = dx1 - idy1 of them as basis differentials. A form w E c;~~(Z) is said to be a form of type ( p, q) if in an arbitrary local system
PRELIMINARIES 3
of coordinates z = (z1, ... ,zn) it looks like w = }:1.~ a1.~(z) dz1 /\dz~. where dz1 =dz; /\ · · · /\dz; and dz. = dz1. /\ • • • /\dz1 .. The class of these forms is
I p tf' I q
denoted by C,!~>(Z). The class C,!~J(Z) is introduced similarly. If r = O. then we write simply Cp.q(Z). A form is said to be holomorphic if it is of type ( p. 0) and its coefficients are holomorphic functions.
We shall consider integrals of differential forms over piecewise smooth oriented surfaces on a manifold X. In order that the apparatus of algebraic topology can be applied, the surface is assumed to be broken up into parts: singular simplexes. A p-dimensional smooth singular simplex (a p-simplex) on a differential manifold X is defined to be a pair "P = (dp• g), where aP is a rectilinear simplex in RP (for example, aP = {t = (t1, ... ,tp) ERP: t1 ;;;;. 0, t 1 + · · · + t P .,.;; I}), and g: a P __. X is a continuously differentiable mapping. Moreover, the simplex is assigned some orientation, determined by the order of the coordinates t1, ... ,tr If other coordinates ,.1, ... ,,.Pare chosen in a neigh borhood of ap, then they determine the same orientation if at/(h = a(t I• ••• ,tp)/a( '1"1, ... ,'l"p) > 0, and the Opposite Orientation if at/a'I" < 0.
The simplex "P taken with the opposite orientation is denoted by -aP. A finite linear combination
c = ""m.a<i> p ,,(,,. I p (0.5)
of oriented smooth singular p-simplexes, where them; are integers(2). is called a smooth singular p-chain. The set of p-chains on a manifold X forms an abelian group C/ X). The carrier of the simplex "P = (dp, g) is defined to be the set I "PI= g( a P ), and the carrier of the chain (0.5) is the set
lcpl= LJ la,!ill. m;.PO
The ( p - I )-simplex a,!0.1 = (a~>_ I• g b~>_ I), where a~>_ I is a ( p - I) dimensional face of the rectilinear simplex aP' is called a face of the singular p-simplex "r We choose a system of coordinates t1, ... ,tp in RP :J ap determin ing the orientation of "P in such a way that the face a~>_ 1 lies in the plane t 1 = 0, and t 1 .,.;; 0 at the points t E d p" Then the parameters t 2 , ••• , t P de termine the orientation of the face a,!0. 1 coherent with that of "r The boundary of an oriented simplex "P is defined to be the ( p - 1 )-chain formed by summing its coherently oriented faces: aaP = }:f =O a,!0. 1. The boundary of the chain (0.5) is defined by the formula acp - }:; m;aa;0 . The following property holds:
aacp = 0. (0.6)
( 2 ) It is sometimes convenient to consider chains with real or complex coefficients (see § 15).
4 PRELIMINARIES
A chain y E C/ X) is called a cycle if ay = 0. The property (0.6) means that the boundary of a chain is a cycle. A cycle y is said to be homologous to zero (y - 0) if there exists a chain c such that ac = y. Let Z/X) = {YE C/X): ay = O} andB/X) = {ac: c E Cp+ 1(X)}. The factor group Hp(X) = Z/ X)/ B/ X) is called the p-dimensional smooth singular homology group of the manifold X. For differentiable manifolds the smooth singular homology groups are isomorphic to homology groups defined in other ways (see [l 14], Appendix A). The definition of smooth singular homology given here is also suitable for a piecewise smooth manifold. A cycle y is said to be weakly homologous to zero ( y ~ 0) if ky - 0 for some integer k. The factor group of Z/ X) by the subgroup of cycles weakly homologous to zero is called a weak homology group. It coincides with HP( X) in the cases of interest to us. We denote it too by Hp( X). If Hp( X) is a group with finitely many generators, then its dimension is called the p-dimensional Betti number of the manifold X. A system { y1} C
ZP( X) of cycles is called a p-dimensional homology basis if any cycle y E Z/ X) can be uniquely represented in the form y ~ ~ 1 m1 y1, where the m 1 are integers. A mapping (0.4) of manifolds assigns to each simplex "P =(AP, g) in X a simplex/(ap) =(AP' fog) in Y. This determines a homomorphism/: Cp(X) --+ Cp( Y f and, since f 0 a = a 0 f, also a homomorphism f * : Hp( x) --+ Hp( Y).
The integral of a i =form w Of class c<0> over a smooth p-simplex aP = (AP' g) is defined by the equality
(0.7)
where A(t) dt 1 /\ • • • /\dtP = g(w) is a form on AP C RP. If the form is represented as in (0.1 ), then
a(x; , ... ,xi) A(t) = ~aAx(t)) ( ' )
J a t 1, ... ,tp
The last integeral in (0.7) is understood as an ordinary p-fold integral in RP. Obviously, the definition (0.7) is invariant with respect to the choice of local coordinates on the manifold and the parametrization of the simplex "r The integral of a form w over the chain (0.5) is defined by the equality
(0.8)
We remark that every piecewise smooth oriented compact surface on X can be broken up into smooth simplexes, i.e., can be represented as a chain c P E C/ X).
PRELIMINARIES 5
From the definition of the homomorphism j and f and formulas (0.7) and (0.8) we get -
THEOREM O.l (Change-of-variables formula). If f: X ..... Y is a mapping of manifolds and y E C/ X), then for every form w of class c<0l on Y
ji<w)=j w. y /(y)
Thus, the homomorphisms j and fare dual with respect to integration. The duality of the homomorphisms d and a is established by
THEOREM 0.2 (Stokes' formula). If w is a p-form of class C(I) on the manifold X and c E Cp+ 1(X), then
( w =jdw. lac c
COROLLARY 0.3. The integral of an exact form over a cycle is equal to zero.
COROLLARY 0.4. The integral of a closed form over a cycle that is weakly homologous to zero is equal to zero.
COROLLARY 0.5. If y1 :::::::: y2 and w1 = w2 + dcp, then fy, w1 = fy 2 w2.
Corollary 0.5 enables us to define the integral of a cohomology class h* = { w} E HP( X) over a homology class h* = { y} = H/ X) by the equality
In what follows we shall often have to consider bounded domains D in the complex space en of variables z = (z1,. .. ,zn), where zk = xk + iYk, k = l, ... ,n. We assume that en is oriented in such a way that
f/x 1 /\ • • • /\dxn /\ dy1 /\ · · · /\dyn
= (- .!_) nf, dz /\ ... /\dz /\ dz /\ ... /\dz > 0 2 D I n I n •
The boundary aD of D is assigned the orientation induced by that of D. If D = { z : p( z) < 0}, where p is a real-valued function of class c< r) in a neighborhood of <lD such that grad p = (p: , ... ,p:) =I= 0 on aD, then we write ... ...,, <lD E c<rl. According to Stokes' formula,
11
6 PRELIMINARIES
We remark that the orientation of en adopted in [51] and [172] is such that
J dx 1 /\ dy1 /\ • • • /\dxn /\ dyn D
= (-~} n f/z1 /\ dz 1 /\ • • • /\d'in /\ dzn > 0.
It is easy to see that the integrals computed with this orientation differ from those computed with out orientation by the factor (-1r<n- 1>12• A particular case of Corollary 0.4 is
THEOREM 0.6 (Cauchy-Poincare). Suppose that the function f(z) is holomor phic in a domain D c en. Then for any ( n + 1 )-chain a in D
1 f(z) dz= 1 f(z 1, ••• ,zn) dz1 /\ • • • /\dzn = 0. 00 00
From Stokes' formula and a property of the exterior differential we get the following formulas for integration by parts.
PROPOSITION 0. 7. Suppose that cp E OP( X), if; E Oq( X), and y E Cp+q+ 1( X). Then
PROPOSITION 0.8. Suppose that the functions f( z) and cp( z) are holomorphic in a domain DC en andy E Zn(D). Then
where a= (a1····•an), jaj= al+ ... +an, and aialf/aza = aa1+···+a•f/azj1 ... az:·.
Let D be a bounded domain in e" with piecewise smooth boundary aD, and let the domains D 1 and D; be such that D; <£ D 1 <£ D (see 5° for the notation). OnD1\D; we are given an exterior differential form a of dimension 2n - 1 that is continuous in D\D; and closed in D\D;. The domain D can be approximated from the inside by polyhedrons Dm (m = 2,3, ... ), D 1 C Dm, such that
lim f /3 = f /3, m ... 00 JoDm JoD
PRELIMINARIES 7
for every form /3 of degree 2n - 1 that is continuous in D\D;. On the other hand, fao a = f ao a by Corollary 0.5, and so we have
m I
COROLLARY 0.9. Under the above assumptions, fao, a = f ao a.
This corollary is frequently used below for proving the validity of various integral representations.
We shall sometimes encounter double forms c.J(t, z), i.e., forms in z whose coefficients are (ordinary) forms in r. or forms in r whose coefficients are forms in z. In this connection it is assumed that the differentials of the variables z commute with those of the variables r. See [144] for more details on double forms.
2°. Some propositions from analysis and algebra. Let c<m,>..) be the class of m
times continuously differentiable functions whose derivatives of order m all satisfy a Holder condition with exponent ;\. The condition aD E c<m,>..) is defined analogously, where aD is the boundary of a bounded domain in en.
PROPOSITION 0.10. Suppose that dD E c<m+I,>..) and that f E c<m,>..>aD, m ;;a. 0 and 0 < ;\ .;;;.; 1. Then the functions defined by the integral
1 f(O df1 /\ · · · /\dfk-1 /\ dfk+1 /\ · · · /\dfn /\ dr1 /\ · · · /\drn
avr (lr1 -z1i2+ ··· +lrn-zni2)n-i
inside and outside the domain can be extended to the closures of the corresponding open Sets as functions of class C(m+ I,>.'), and the functions defined by the integral
j f(r}df1 /\ · · · /\dfn /\ dr1 /\ · · · /\drn
Dr (1r1 - Z112 + ... +1rn - znl2r-I can be extended as functions of class c<m+ 2.>..'>, where 0 < ;\' < ;\ (see, for example, [69], Chapter II, §19, Theorem 4, §20, Theorem 2).
Let X and Y be differentiable manifolds and f: X --+ Y a mapping of class C(I). A point p EX is called a critical point off if min(dim X,dim Y) > rank( df)P, where ( df)P is the differential off at p. The range off on the critical points is called the wrinkle and denoted by wr( f ).
THEOREM 0.11 (Sard). If dim X = dim Y, then the Lebesgue measure of wr(f) in Y is zero. The Lebesgue measure of wr( f) in Y is also zero if X, Y E C rx> and f E c<rx>>.
Only the operations of addition and multiplication appear in a numerical determinant; therefore, we can introduce a determinant whose elements are members of an arbitrary ring. Such a determinant can be expanded according
8 PRELIMINARIES
to the usual rules, with the place of each factor in the terms of the sum fixed by the number of the column to which this factor belongs. It is convenient to introduce the following notation in connection with the way in which a determinant is expanded: if 01, ... ,om are n-dimensional vectors, then D., ...... J(Jl, ... ,Om) is the nth-order determinant with the vector 01 in the first P1 columns, the Vector 02 in the next P2 columns, ... ,and the vector om in the last Pm columns, where P1 +···+Pm= n.
We give some properties of determinants needed for what follows.
PROPERTY 0.12. If two rows are interchanged, then the determinant changes sign. If all the elements in some column commute with all the elements in a neighboring column, then the determinant changes sign when these columns are interchanged, but if the corresponding elements anticommute, then the determi nant does not change.
Suppose now that the determinant is composed of elements of a ring of double exterior differential forms. Its rows are said to be linearly dependent if some linear combination of them with function coefficients vanishes, and for each point of the domain in which the determinant is defined at least one coefficient does not vanish.
PROPERTY 0.13. If all the elements of some column (row) are multiplied by one and the same function, then the determinant is multiplied by this function. If the rows of the determinant are linearly dependent, then it is equal to 0.
If the forms a.; (i = 1, ... ,m} are of class c<1> and degree qi in z, then
a (a. /\ ... /\a. > z I m m
- "' ( )q,+···+q,_, "5 - ~ -1 <X.1 /\ · · · /\a.i-1 /\ uza.i /\ a.i+l /\ · · · /\a.m. i=l
A consequence of this is (here and below, azO = (az01, ... ,azOn))
PROPERTY 0.14. If the elements of the columns Oi (i = 1, ... ,n) of a determi nant are double forms Of class C(l) and degree qi in Z, then
azDi .... A oi , ... ,on) n
= °"' (-l}q,+···+q,_,D (01 0;-1 a Oi 0i+1 on) """"' 1 •...• 1 , ••• , ' z ' , ••• , • i=l
In what follows we shall use elementary concepts from measure theory without explaining, for example, the words "finite Borel measure" or "sup with respect to a given measureµ." (denoted by esssup,.). We also use the concept of a partition of unity subordinate to a given cover [133].
PRELIMINARIES 9
3°. Some facts from function theory. It is assume that the reader is familiar with the elementary properties of holomorphic functions of several complex variables. They can be found in the first chapters of the textbooks [51], [57], [148], [173], [189], [201] and [26]. For example, it is assumed known that a function holomorphic in a domain can be expanded in a multiple power series in a neighborhood of each point of the domain, and that a holomorphic mapping with nonzero Jacobian is locally invertible. It is assumed also that the reader is acquainted with such concepts as the envelope of holomorphy X( D) of a domain D, a complex line, a complex hyperplane, a circular domain, and an n-circular domain.
Recall that if f( z 1, ••• , z n) is a holomorphic mapping of a domain D c en into a space of the same complex dimension, f= (f1, ••• ,fn), fk = uk + ivk,
zk = xk + iyk, k = 1, ... ,n, then
I 0(/1,. .. ,/n) 12 - 0(u1,···,Un,V1, ... ,Vn) a(z1,· .. ,zn) a(xp ... ,xn,yl,. .. ,yn).
If the boundary oD is piecewise smooth, the form a E Cn n-i(oD) is a-exact, i.e., a = ap on oD for some fJ E C~~~- 2(oD), and f is holo~oq,hic on D, then by Stokes' formula (see Corollary 0.3)
1 fa= 1 fa{J = 1 a(ffJ) = 1 d(ffJ) = 0. av av av av
Suppose that f is holomorphic in a neighborhood of a point a. The order da(f) of a zero off at a is defined to be the smallest degree (in the variables jointly) of the terms with nonzero coefficients in the Taylor series of this function at the given point.
4°. Some concepts and facts from algebraic topology(3) will be needed in §§ 14, 16, and 18 of Chapter III. The singular homology groups are defined in topology for arbitrary topological spaces in exactly the same way as the smooth singular homology groups were defined in 2°, with only the difference that in the definition of a singular simplex op= (llp, g) the mapping g is assumed to be continuous, and the orientation of op is given by the order of the vertices of the simplex !lP.
A topological space Xis said to be a polyhedron if it admits a triangulation, i.e., it can be broken up into nondegenerate simplexes(4 ) o;;> (i = 1, 2, ... ,p = 0, 1, ... ) with disjoint interiors. The family K(X) = { o;;>} is called a simplicial
(3) See, for example [30], [114], [157], [163] or [175].
( 4 )A simplex aP =(tip, g) is nondegenerate if g: tip-+ Xis a homeomorphism.
10 PRELIMINARIES
complex of the polyhedron X. For a polyhedron we introduce the concept of a homology group of a simplicial complex, defined in the usual way, but only on the simplexes appearing in K( X). The homology groups of a simplicial complex of a polyhedron are isomorphic to the singular homology groups of the polyhedron and, consequently, are invariant with respect to triangulation. A differentiable manifold admits a smooth triangulation, so all three defini tions given above for the homology groups are equivalent for it. Instead of a simplicial complex it is sometimes more convenient to consider a cell complex: a partition of the polyhedron into cells op= (Mp, g), where MP is a polytope in RP, and g : MP ..... Xis a continuous mapping that is a homeomorphism interior to Mr
For a simplicial complex and a cell complex of a compact polyhedron X we have the Euler-Poincare formula
n n
L (-l)qaq = L (-I)qpq, (0.9) q=I q=I
where aq is the number of q-dimensional simplexes (cells of the complex K( X)), pq is the q-dimensional Betti number of X, and n = max{ q: aq =F O} is the dimension of X.
From the definition of homology groups it is easy to get the following propositions.
PROPOSITION 0.15. If Xis an arcwise connected space, then H0(X):::::: Z, where Z is the group of integers.
PROPOSITION 0.16. If X = U aeA Xa, where the Xa (a E A) are the arcwise connected components of X, then H/X) = ffiaeA H/Xa).
PROPOSITION 0.17. The homology groups for then-dimensional sphere Sn= {x E Rn+I :lxl= l} are
ifp = 0, n, if p =F 0, n.
It can be shown that a 2-dimensional orientable manifold S is homeomor phic to some 4p-gon o2 = a 1b1aj1b( 1 • • • aPbPa; 1b;1 in which the first and third and the second and fourth sides in each quadruple are identified pairwise, with the opposite orientation with respect to a circuit along the boundary of 0 2 (a sphere with p "handles"). The integer p is called the genus of the manifold (surface).
PRELIMINARIES 11
PROPOSITION 0.18. The homology groups of a 2-dimensional orientable mani fold S of genus p are
H0(S) '::!!. Hi{S) '::!!. Z, H (S) '::!!. Z ffi · · · ffiZ · I '-----v-----"'
2p
the I-dimensional cells a 1,b1, ... ,aP,bP form a basis for the I-dimensional homology of S, and are called "canonical cuts" of S.
Continuous mappings/, g: X ..... Y are said to be homotopic, f ~ g, if there exists a continuous mapping F( x, t) : X X [O, I] ..... Y, such that F( x, 0) = /( x) and F(x, I)= g(x).
A subset A of X is called a retract of X if there exists a continuous mapping /: X ..... A such that /IA = IA (the identity mapping). The set A is called a deformation retract of X if f 0 i ~ Ix: X ..... X, where i is the imbedding A C X.
THEOREM 0.19. If J~ g: X-> Y, then/*= g*: Hp(X) ..... Hp(Y).
COROLLARY 0.20. If A is a deformation retract of X, then H/A) '::!!. H/X).
Let A be a subspace of X. The factor group C/X, A)= CP(X)/C/A) is called the group of relative chains of the pair (X, A). The boundary homomor phism a: C/X) ..... CP_ 1(X) induces a homomorphism a: C/X, A) ..... CP_ 1(X, A) in the natural way. Consequently, the relative homology group
H/X, A) is defined. We define a homomorphism a*: Hp(X, A) ..... HP_ 1(A) that assigns to each class {cp} E H/X, A) the class {Clcp} E HP_ 1(A). Ob serve that cP E C/X) determines a relative cycle if acP EA.
A sequence of group homomorphisms
fk-1 fk ........ Ak-1 ..... Ak ->Ak+I ........
is said to be exact if the kernel of each homomorphism coincides with the image of the preceding homomorphism (Ker fk = Im fk- 1 ).
THEOREM 0.21. The sequence of homomorphisms
where i * and j * are the homomorphisms induced by the imbedding i : A ..... X and the factor mapping j : C/ X) ..... C/ X, A), is exact.
Let A be an abelian group. We denote the group of homomorphisms from A to C 1 (the group of complex-valued linear functions on A) by A* = {A: A - C 1}. A homomorphism A: A ..... B of abelian groups induces a dual homomor phism l\.*: B* ..... A*, defined as follows: if b* EB*, then l\*b*(a) = b*(l\(a))
for any a EA.
THEOREM 0.22. If a sequence of homomorphisms
of abelian groups is exact, then the sequence of dual homomorphisms
'Pk- I 'Pk ... -A!-1 - At-A!+1 - ...
is also exact.
5°. Main notation. The space of n complex variables is denoted by en, and its points are denoted by z, r. z0 , r 0 , etc. If z = (z1·· .. ,zn), then z = U1·· . . ,'in)· For z, r E en let (z, n= z1r1 + ... +znrn, 1z1= v(z, Z)' and llzll=max1,..;.,.nlz;I· We introduce notation for the ball B,=B,(z0 )= B(z0 , r) = {z: lz - z0 1< r}, the polydisk U = U(z0 , r) = {z: lz; - z?i< r, i = l, ... ,n }, and the generalized ball B/(z0 ) = { z: I z1 - z? l2k' + · · · +I zn - z~l2k• < r 2}, where k = (k1, ... ,kn). In general, the expression {z: · · ·} de notes a set in en, with other cases stipulated in special ways. The space of n
real variables is denoted by an, and its points are denoted by x, y, etc. If J and i are vectors with nonnegative integer coordinates (multi-indices),
then 111= i 1 +···+in; the inequality J.;;;;; i means that ik ..;;jk, k = 1, ... ,n where J = (i1, ... ,in) and i = (j1, ... ,jn). The expression k f£. J means that k-=/= i1, 1=1, ... ,n. Further,J!= i 1! ···in!,
zJ = z(1 • • • z~·,
a1J1+1i1 D;·!= . . . . '
' awj 1 ••• aw~·az{' ... az~n
dz =dz. /\ ... /\dz. J 11 1p'
where J = (i1, ... ,ip); and dz1J 1= dzj, /\ · · · /\dzj.-p· The notation [ ] will often be used in the same sense; for example, a 1, ... , [ j 1, ... ,j k ], ••• , an means that the elements aN ... ,ajk are omitted, and dzffl = dzf1 /\ • • • [k] · · · /\dz:·.
A neighborhood of a point a E en is denoted by U = U0 (sometimes by other letters).
Let Mand N be subsets of en; then aM is the boundary of M,
p(z,M}= inf 1r-z1, p(M,N}= inf 1r-z1, fEM fEM
zEN
Mis the closure of M, int Mis the interior of M, andM <:£; N means that M is compact and MC int N. If w = (w1, .. . ,wn), and cp = (cp1, ... ,cpn), then
( W, dcp) = w1dcp 1 + · · · +wndcpn,
PRELIMINARIES 13
and
( W m<a+I>) = W ma1+I + ... +w cpa.+I 'T )T) n n '
where a= (a1, ••• ,a,,) and I= (1 1, ••• , 1). Further, 0 = (0, ... ,0) is the origin of coordinates, and dcp/cp = dcp 1/cp 1 /\ • • • /\dcp,,/cp,,. The Jacobian of a map ping f = f( z) = (/1, ••• ,f,,) is denoted by
a(f1.·· .,/,,) _ af a(z1·· .. ,z,,) oz'
(of /oz)~ is the value of this Jacobian at a point a, and grad(> = (4>;1, ••• , 4>;,,). We frequently write o/(z) in place of i/l(z, z) for a function. The symbol $
denotes a direct sum, ~ik is the Kronecker symbol, JJg;)J is the matrix with elements gij• detJJgijll is the determinant of this matrix, and oz= (dz1, ••• ,dz,,).
Let F be a closed subset of C" and 0 an open subset. Then c:m>(F) (m = 0, 1,2, ... ,oo) is the class of r-dimensional forms(5) whose coefficients can be extended to a neighborhood of Fas m times continuously differentiable functions. The subclass of c;~~(F) consisting of forms of type (p, q) is denoted by c;r::/(F). When m = 0, this index will be omitted. The projective limit of the classes c;'.';>(F), F ~ 0, is denoted by c;'.';>(O). For
we set
a=~' aJ'J.dzJ /\ di'J. E c;'.';>(O), J.~
(for m = 0 the differentiation is understood as differentiation in the space of generalized functions). Here the prime on the summation sign indicates summation over 1.;;;; i 1 < · · · < iP.;;;; n, 1 o;;;;j1 < · · · <jq.;;;; n, where J = (i 1, ••• ,iP) and i = (j1, ••• ,jq).
The class of functions holomorphic in a domain D (on a compact set K) is denoted by A(D) (by A(K)). Furthermore, Ac(D) = A(D) n C(D), and A"(D) is the class of mappings f= (/1, ••• ,f,,) that are holomorphic in D. Moreover, ReAc(D) is the class of functions that are real parts of functions in Ac(D). Let Ph(D) be the class of pluriharmonic functions in D, and R(D) the class of holomorphic functions in D with nonnegative real part. Finally. Phc(D) = Ph(D) n C(D), and Rc(D) = R(D) n C(D). If a measureµ is given on 'OD, then [ReAc(D)]; is the set of cp E C('dD) such that f;wh dµ = 0
( 5 )The word .. form" is used instead of "exterior differential form" here and frequently below.
14 PRELIMINARIES
for all/ E ReAc(D). The closure of Ac(D) in the metric of L~ .... is denoted by HJ .....
An index r (or z) on a domain of integration indicates that r (respectively, z) is the variable of integration; I aD I is the image of aD under the mapping z --+ (I z ii , ... , I z n I), and a r = { z : I z; I= I r;1 , i = I, ... , n}. Finally, A x B de notes the topological product of sets A and B. Some other notation was introduced in 1°-4°.
If a E Cp,q(O), then supp a= {z E 0: a(z) =fo O}. Suppose that the form a is defined in some set U c X, and that a function x is defined on the manifold X and has supp x a; U; then, as is common, we regard 1/1 = x a as defined on X by the equalities 1/1 = xa for x E U and 1/1 = 0 for x fl. U.
The symbol D marks the end of a proof.
CHAPTER I
§1. The Martinelli-Bochner integral representation
1°. THEOREM 1.1. Suppose that f E Ac( D ), where D is a bounded domain in en with piecewise smooth boundary aD. Then the Martinelli-Bochner formula holds:
1t(r)w(r-z,f-z)={f(z}, zE~ an 0, z fl D,
(1.1}
where
w(r _ z f _ z) = (n - I}! ~~=1(-l)k-t(fk - zk)dfckl /\ dr. (1.2} ' (2'7Ti}n 1r - z12n
PROOF. I. It is easy to establish by direct calculation that the Martinellil Bochner kernel w is closed with respect tor on the set cn\{z}. From this and Corollary 0.4 we get (1.1) for the case z fl i5. But if z ED, then, by the same Corollary 0.4, the integration over aD can be replaced by integration over the sphere aB,( z ), where the radius r is sufficiently small.
2. In a sufficiently small neighborhood of a point z E D the holomorphic function f<n can be expanded in a multiple power series about z; hence, to prove (1.1) it suffices to establish that
(n - I)! 1 (r - z}P ± (-l)k-t(fk - zk)dfckJ /\ dr (2'1Tir 2 r aB, k= I
= {~ if fl* 0, if fl= 0. (I.3)
15
16 I. INTEGRAL REPRESENTATIONS
3. Note that under the change of variables rk - zk = e;'k(rlc - zk) with all the tk real (k = I, ... ,n) the set aB,(z) is carried into itself, and the integrand form in (1.3) is multiplied by e;11P1 • • • eil•P•; therefore, (1.3) holds for the case p =I= 0.
4. If P = 0, then, by Stokes' formula, the left-hand side of (1.3) is equal to
n! J df /\ df (2'1Tir 2r s,
(\ ... /\dlrn - znl
= n!l d-r1 /\ • • • /\d-rn =I. D T1+ · ·· +T.<l,T;;;.O
2°. For n >I the Martinelli-Bochner kernel w(r - z, f- z) is not holomor phic in z but only harmonic; nevertheless, the following assertion holds.
THEOREM 1.2 (Aronov and Kytmanov). If D is a bounded domain with piecewise smooth boundary and the Martinelli-Bochner integral representation (1.1) is valid for z ED for a particular f E c<1>(D), then f E A(D).
To prove this theorem we need some lemmas.
LEMMA 1.3 (The Martinelli-Bochner formula for smooth functions). If f E c<1>(i5'), then
1 f{K)w(r - z, f - z) -f af(r) (\ w(r - z, f - z) av v
= {f(z), 01
z ED,
z fl. D. (1.4)
This lemma can be proved in the same way as Theorem 1.1, except that the integral of drU<nw<r - z, f - z)) = ar1<n (\ w(r - z, f - z), over D also appears, and it is not possible to expand f(n in a neighborhood of z in a Taylor series with respect to powers of r - z, but is instead necessary to approximate fin this closed neighborhood by a polynomial in r - z and f - z. The formula (1.4) enables us to easily compute the saltus of an integral of Martinelli-Bochner type in passing across the boundary of a domain D.
LEMMA 1.4. Let f E C(l>(aD), and define
F"'(z)=1 f(r)w(r-z,f-z), av
§I. THE MARTINELLI-BOCHNER REPRESENTATION 17
where D + = D and v-= C" \ i5. Then the functions F "' ( z) are continuous all the way up to the boundary of D"' , respectively, and
F+(z)- p-(z) =f(z), z E ()D. (1.6)
PROOF. Suppose that FE c< 1>(D) and FlaD = f. The function
cp(z) = J.aF(r) /\w{r-z,f-z) D
is continuous in C" (see Theorem 0.10). By (1.4),
p+ (z) = F(z) + cp(z), p-(z) = cp(z),
and this and the continuity of cp and F give us the continuity of F "' ( z) all the way up to the boundary of D"' , along with equality (1.6). D
We shall also make use of the following result of Keldysh and Lavrent'ev, which we present in the even-dimensional case we need.
LEMMA 1.5 (Keldysh and Lavrent'ev). The linear manifold generated by the fractions of the form
1
Ir - z !2"-2 '
where K is a compact set in C" of total Lebesgue measure zero and n > 1, is dense in the space C( K ).
PROOF. By the Stone-Weierstrass theorem, it can be assume that/ E c< 1l(K). We extend fas a smooth function to some closed ball jj with B ::J K, and we represent it by formula (1.4), where D = B. If we now replace the domain of integration Bin the second integral in (1.4) by B\U, where U is a neighbor hood of K of sufficiently small volume, and the integrals themselves by appropriate integral sums, then, using the fact that the coefficients of the form w(r - z, f - z) are the derivatives of the fundamental solution
g(r, z) = (1 - nr11r- z12 - 2 " (1.7)
of the Laplace equation and replacing the derivatives of g by difference quotients, we obtain linear combinations of the desired fractions that uni formly approximate f on K. D
PROOF OF THEOREM 1.2. Consider the integrals of Martinelli-Bochner type (1.5); then p+ =/in D. It now follows from (1.6) that F-lao = 0. We note that p- is harmonic in v- and has a zero of order 2n - I at infinity. By the uniqueness theorem for harmonic functions,
p-(z) =o. z Ev-. ( 1.8)
Since/is representable by formula (1.1), it is harmonic in D. Let us represent it by Green's formula for harmonic functions:
where
z ED,
z fl 15,
P (?) = (-l)n-1 (n - I)! ~ (:...l)k-1 at d? /\ d;:. f :\ (2 .)n ~ a? :\[k] :\
'1Tl k=I :\k
(1.9)
This "complex" Green's formula can be proved in the same way as its "real" analogue, the ordinary Green's formula (see also the proof of Lemma 1.3). It follows from (1.1) and (1.9) that
1 Pi(r> = 0 (I.IO) av1r- zfn-2
for z ED. Comparing (1.8) and (1.9), we get that (1.10) is also valid for points z En-. Lemma 1.5, applied to the compact set aD, now shows that v/r) lav = 0.
The function f is harmonic in D; therefore, dv/n = c!l.f dr /\dr = 0, i.~ the form v1 is closed in D, and so d(f(r) X v/n), extends continuously to D. Stokes' formula gives us that
(2itf. ~ I at 1 2 dv= J. d(t(r> · v1(r)) = 1 t(r> v1(r) = o.
Vk=I ark V av
From this and the continuity of the derivatives a11afk (k = I, ... ,n) it follows that these derivatives are equal to zero in D, i.e.,/is holomorphic in D. D
The following result is harder to prove, and we present it without proof.
'THEOREM 1.6 (Kytmanov and Aizenberg). If a function f E C(D) can be represented in D by the M artinel/i-Bochner integral ( 1.1 ), then it is holomorphic in D in each of the following cases:
a) an E c<2>; b) n = 2, and the boundary aD is connected and belongs to the class c<1>. §2. Multiplicity of a zero of a holomorphic mapping. lbe Rouche principle
Suppose that the mapping
w = f(z) (2.1)
is holomorphic in the domain D c en, where w = (w1, ••• ,wn) and/= ( / 1, ••• ,fn ). A point a E D is called a zero of the mapping (2.1) if/( a) = C. Let
§2. THE ROUCHE PRINCIPLE 19
E1 denote the set of zeros off. If the Jacobian of (2.1) satisfies (of /oz) la =fa 0 at an isolated point a of E1, then a is called a simple zero of the mapping.
We have the following statement, which we prove a little later.
PROPOSITION 2.1. If the closure of a neighborhood U0 of a zero a of the mapping (2.1) does not contain other zeros, then there exists an e > 0 such that for almost all r E B. the mapping
w = f(z} - r (2.2)
has only simple zeros in U0 , and their number depends neither on r nor on the choice of the neighborhood U0 •
The number of zeros of (2.2) in U0 indicated in this proposition is called the multiplicity of the zero a of the mapping (2.1 ), and denoted by µ 0 ( f ).( 1)
EXAMPLE. The point (0,0) is a zero of multiplicity 2 for the mapping w1 = z1,
W2 = z~ + zf. Indeed, if 1r1 is small and rf =fa ri. then the mapping W1 = z, - r,,w2 = z~ + zf - ri has the two simple zeros <r1, vr2 - r?) and (r,. - vr2 - r? in a neighborhood of this point.
The next result follows from the local invertibility of a holomorphic map ping at points where of /oz =fa 0.
PROPOSITION 2.2. The multiplicity of a simple zero is equal to 1.
We pass to the case of a zero that is not simple.
PROPOSITION 2.3. If a is an isolated zero of (2.1) and(of/oz) la= 0, then its multiplicity µ 0 (/) is greater than 1.
This statement justifies calling an isolated zero of (2.1) multiple if (a f /oz) la =O.
Just as the logarithmic residue expresses the number of zeros of a holomor phic function in the theory of functions of a single complex variable, its multidimensional analogues are connected with the number of zeros of a holomorphic mapping. The following multidimensional analogue of the classi cal logarithmic residue of Cauchy was constructed with the help of the Martinelli-Bochner integral representation (1.1).
THEOREM 2.4. Let f E An(D), where the domain D is bounded, and its boundary 'OD is piecewise smooth and does not contain zeros off. Then f has only
e) This is the so-called dynamic definition of multiplicity. See §22 about the connection of /LaU) with the coefficients of the Taylor expansion of (2.1) at the point a.
isolated zeros in D, and their number, with each zero counted as many times as its multiplicity,( 2 ) is expressed by the formula
N = N(f, D) =1 w(/(z), /(z) ), aD
(2.3)
where
w /, f - (2'1Tit lfi2n j~I (-1) ldfu1 /\ df.
THEOREM 2.5. (The Rouche principle). Suppose that the mapping (2.1) and the domain D satisfy the conditions of Theorem 2.4, and let g E An(D). If for each point z E aD there is at least one index j (j = l, ... , n) such that
lgiz)l<ll(z)I, (2.4)
w = /( z) + g( z)
(2.5)
REMARK. Condition (2.4) holds, for example, if lgl<l/I or llgll < 11/11 on aD. This condition can be relaxed. In fact, we can require only that/( z) + tg( z) -=!= 0 for z E aD and 0 .;;;;; t .;;;;; 1. It suffices that at each point z E aD there is at least one indexj such that Re g/z) <Re t(z) or Im g/z) < Im l(z).
We need some lemmas to prove these assertions.
LEMMA 2.6. If f E An(D), then/or any piecewise smooth cycle
y E Z2n-1(D\E1)
the integral f 1 w(f, J) is an integer.
PROOF. Since (2.1) maps the domain D\E1 into Cn\{0}, and the sphere aBr forms a basis for the (2n - 1)-dimensional homology, the image of the cycle y
under the mapping (2.1) is homologous to N · aBr, where N is an integer. Using Theorem 0.1, Corollary 0.5, and the Martinelli-Bochner formula (1.1), we get that
J w(f, J) = J w(t, f) = N 1 w(t, f) = N. D 1 [<Y> as,
( 2 ) Henceforth we adhere to this convention wherever the number of zeros is concerned.
§2. THE ROUCHE PRINCIPLE 21
LEMMA 2. 7. Suppose that the mapping (2.1) and the domain D satisfy the conditions of Theorem 2.4. If, moreover, all the zeros of (2.1) in Dare simple, then for any function cp E Ac( D)
1 cpw(f, J) = ~ cp(a). aD aEE1
(2.6)
In particular (for cp = 1), (2.3) holds.
PROOF. The form cpw(f, f) is closed in D\E1; this can be verified directly as in the proof that the Martinelli-Bochner form (1.2) is closed. According to Corollary 0.5,
1 cpw(f, J) = ~ 1 cpw(f, J ), aD aEEI aua
(2.7)
where Ua is a sufficiently small neighborhood of a. We take Ua to be the connected component of the set {z: z ED, lf(z) I< e} containing a. Since ('df /'dz)~ =fa 0, f is biholomorphic in U,, for sufficiently small e; therefore, by Theorem 0.1 and the Martinelli-Bochner formula (1.1),
From this and (2.7) we get (2.6). 0
LEMMA 2.8. If the conditions of Theorem 2.5 hold, then
1 w(f,f)=1 w(f+g, f+g). aD aD
(2.8)
PROOF. According to (2.4), f(z) + tg(z) =fa 0 for z E 'dD and 0.;;;; t.;;;; 1. Consequently, the integral
J( t) = 1 w ( f + tg, f + tg ) aD
is a continuous function of t on [O, l], and, by Lemma 2.6, its values are integers. Therefore, J( t) is constant, and so J(O) = J(I ), i.e., (2.8) is valid. 0
PROOF OF PROPOSITION 2.1. Let e > 0 be such that e > minzEau., lf(z) 1. Then, by Lemma 2.8,
(2.9)
for r E Be. By Sard's Theorem 0.11, meas f({z: z ED, 'df/'dz = O}) = O; consequently, for almost all r EB. the mapping (2.2) has only simple zeros. According to Lemma 2.7, the integral on the right-hand side of (2.9) is equal to
N( I - r. Cla) for such r. The equality (2.9) implies that
N(f-f,Cla)=1 w(J,f}; aua
hence, N(f- f, lla) does not depend on the choice off EB •. If U~ is another neighborhood of a, then, by Corollary 0.5,
1 w(f, J} = 1 w(f, J }. D au~ aua
In passing we have proved that the multiplicity of an isolated zero of a holomorphic mapping (2.1) can be expressed by the formula
(2.10}
PROOF OF THEOREM 2.4. I. Assume first that 'df /'dz= 0. We show first of all that (2.1) can have only isolated zeros in D. Take e such that 0 < e =;;;;;; minaol/I. From Sard's theorem and Lemma 2.8 it follows that for almost all f E B. the system (2.2) has only simple zeros in D, and their number k ;;;.. 0 does not depend on r E B •.
If E1 =F 0, then for any point a< 1> E E1 we can choose a sequence of points r<m> .... 0, r<m> EB., such that for f = r<m> the mapping (2.2) has the simple zeros {zm·1, ••• ,zm,k}, k;;;.. l, and zm,l-> a< 1> as m-> oo. Indeed, since 'd//'dz = 0, there is a point zm,l at which 'd//'dz = 0 in any neighborhood of a(I>. Take r<m> = /(zm·1); then zm,l is a simple zero of (2.2) for r = r<m>. According to Sard's theorem, by varying r<m> slightly we can make the remaining zeros of this mapping also simple. It is clear that r<m) .... 0 as zm,l .... a(I>.
On the other hand, we show that for any choice of r<m> .... 0 the simple zeros {zm·•, ... ,zm·k} have a single finite set(3) {a(I>, .. .,a<k>} CD of limit points; consequently, E1 = { a(I>, ... ,a<k>}. Indeed, replacing the sequences {zCm,j)}:= 1,
j = l, ... ,k, by subsequences, we can make them converge to some points aU> E D,j = 1, ... ,k (because there are no zeros on 'dD). The set E1 consists of these points alone. Suppose not: suppose that there exists a point b<1> E E1 with b(I> =F aU>, j = l, ... ,k. Then just as before we get a sequence of points r<m> .... 0 such that for r = ~(m) the system (2.2) has only simple zeros {wm·•, ... ,wm·k}, and wm.j .... bU>,j = l, ... ,k, as m .... oo. We choose a 'PE Hc(D) such that 'P(a(I>) = · · · = 'P(a<k>) = 'P(h<2>) = · · · = 'P(b<k>) = 0, and
<3> Here some of the points a(I>, ... ,a<k> may coincide.
§2. THE ROUCHE PRINCIPLE
1 qiw(t-r<m>, /-r<m>) = ~ qi(zm·j); av j=l
k
1 qiw( /- ~<m>, /- ~(m)) = ~ qi{ wm·j). av j=l
Passing to the limit as m -+ oo, we get a contradiction: k
1 qiw(f, j) = ~ qi(aU>) = O; av j=l
k
Thus, E1 is finite.
23
Let us prove (2.3). We enclose each point a E E1 in a neighborhood Ua in such a way that fl,, c D and fl,, n flb = 0 for a =I= b, a, b E E1. Since the form w(/, f) is closed in D\E1, Corollary 0.5 and (2.10) give us that
1w(f,J)=~1 w(f,J)= ~ µAf)=N(!,D). av aEE1 auu aEE1
Moreover, if E1 =I= 0, then N( f, D) = k > 0. 2. Suppose now that a/ ;az = 0. In this case the integral on the right-hand
side of (2.3) is equal to zero, since df = (a f/az) dz appears in the form w(f, j). We show that N(/, D) is also equal to zero, i.e., f has no zeros in D. Suppose the opposite: /(a)= 0 for some point a ED. Since flav =I= 0, there is a holomorphic mapping g: D-+ en such that g(a) = o, lgl<l/I on aD, and au+ g)/az 21':: 0 (for example, we can take the linear mapping g(z) = X(z - a), where A= llAjkll is a matrix with sufficiently small elements such that ranklla.9(a)/azk + Xjkll = n).
1 w(f, J) = 1 w(f + g, f + g) > 1. av av
This is a contradiction. D The proof of Theorem 2.5 follows from Theorem 2.4 and Lemma 2.8. D PROOF OF PROPOSITION 2.3. By the definition of the multiplicity of an
isolated zero, ILaU>;;;;.. 0. Observe that af/az 21':: 0, for otherwise there would not be any isolated zeros at all (see part 2 of the proof of Theorem 2.4). There is a point z0 E Ua such that (af/az)b0 =I= 0 and l/(z0 )1< minau.1/1. It now follows from Proposition 2.2 and Theorem 2.5 that µ 0 (/);;;;.. I. If µa(/)= 1, then for any f E BE, where e is sufficiently small, (2.2) has exactly one zero in
U0 , i.e., (2.1) is one-to-one in a neighborhood of a. Then Osgood's theorem (see Theorem 4.15 below) gives us that this mapping is biholomorphic in a neighborhood of a, and (of/oa) ~=I= 0. 0
We explain the geometric meaning of formula (2.3). Consider the (n - 1) dimensional closed piecewise smooth surface (cycle) S = {x: x = x(t), t E r- 1} in Rn\{O}, where
in-I= {t:t=(t1>···•tn-l),O.;;;;tjE;;l,j= l, ... ,n-1},
x(t1, ••• ,0, ... ,tn_ 1) = x(t1, ••• , 1, ... ,tn_ 1),
j= l, ... ,n - 1.
The index of the surf ace S with respect to the origin is defined to be the number
( ) _ 1 1 1 ~ ( )j-1 10 S -r -1 In ~ -1 xjdxu1 n S X k=l
x 1(t) xn(t)
= _l !. l x) 1(t) · · · x~1 (t) dt, ~n r- 1 lx(t) r ................. .
xin-1 . . . x~n-1(t)
(2.12)
where ~n = nwnl2 /f(n/2 + 1) is the measure (volume) of the unit sphere in Rn. The integrand form in (2.12) expresses the measure of the projection of a surface element onto the unit sphere about zero (the solid angle under which this element is "seen" from the origin). The integral (2.3) can be represented in the form
f ( - ) -J ( -) _ ( n - 1) ! f 1 ~ j-1 - - }, w f,f - w t,t - (2 .)n l"l2n ~ (-1) tjdtu1Adt
CJD y 'ITl Y ;\ j= I
§3. THE LERAY-KOPPELMAN FORMULA 25
by Corollary 0.3. We c~n thus formulate the following statement.
THEOREM 2.9 (The argument principle). If the conditions of Theorem 2.4 hold, then the number of zeros of the mapping fin (2.1) in the domain D is equal to the index with respect to the origin of the image f(aD) of the boundary of D under f (the number of" circuits" of the surface f( aD) around 0).
This fact can also be obtained from topological considerations. Since the mapping (2.1) has no zeros on aD, the image/(aD) of the cycle aD under this mapping is in Z2n_ 1(Cn\ {O}). Consequently, /(aD) - N · aB,, where N is an integer indicating how many times the cycle /(aD) "goes around" zero. From (1.1), Theorem 0.1, and Corollary 0.5 it now follows that N = N(f, D). 0
§3. The fundamental integral fonnula of Leray and Koppelman
1°. THEOREM 3.1 (Yuzhakov and Roos). Suppose that Dis a bounded domain with piecewise smooth boundary, and that f is in An(D) and does not have zeros on aD. Then for any <p E Ac(D)
1 <pw(f, J) = ~ P.a(f) · <p(a). ao aeE1
(3.1)
PROOF. Choose e > 0 such that e < mina 0 I/I. For almost all ~EB, the mapping (2.2) has only simple zeros in D, and, by Lemma 2.7,
(3.2)
If we now let~ tend to zero, then (see part 1 of the proof of Theorem 2.4) the zeros of (2.2) tend to the zeros of (2.1 ), and, moreover, the number of simple zeros off- ~ tending to any particular zero off is equal to its multiplicity. If we pass to the limit in (3.2) as~--> 0, we get (3.1). 0
2°. Let us consider the following exterior differential form (important for what follows), which depends on a holomorphic mapping/, a continuous vector-valued function w<0l, and continuously differentiable vector-valued functions w(I>, ... , w<n- Il:
~( w<0>, w(I>, ... , w<n-1), f)
- (-1r<n-1)/2 (w<Ol,d/) /\d(w(ll,d/) /\ ... /\d(w<n-1),df)
{2'1Tir (w<0>,/) (w<I),/) (w<n-1),/) ·
{3.3) THEOREM 3.2 (Fundamental integral formula). Let D be a bounded domain
with piecewise smooth boundary, and suppose that the mapping f E An(D) does not have zeros on aD and that the vector-valued functions w<0l E C(aD) and
w<i> E c< 1>(an),j = 1, ... ,n - 1, satisfy the condition
( w<il(z ), f(z) )-:!= 0, z E an,}= 0, 1, ... ,n - 1. (3.4)
Then any function cp E Ac( n) satisfies the formula
1 cpO( w<0>, w(ll, ... ,w<n-I>, f )=~ µ.aC!) · cp(a). (3.5) aD aEE1
We indicate other ways of writing the kernel 0. Let
then (3.3) gives us
w<j) u<i>-----
- ( wU>, f)' j=0,1, ... ,n-1;
( )n(n-1)/2 0 = - ~ f ( u<0>, df) /\ d ( u(I>, df) /\ · · · /\d ( u<n-I>, df), {3.6)
2m
z E an,}= 0, l, ... ,n - 1. (3.7)
where u)t> = aujk>;az;,, and the summation is over all possible (n - 1)-tuples (i1, ... , in_ 1) of numbers taken from the set { 1, 2, ... ,n }.
We mention also that 0( j, ... ,j, f) = w( f, j ), i.e., (3.1) is a particular case of (3.5), and condition (3.4) means that lfi2 =I= 0 on an.
To prove Theorem 3.2 we need some simple properties of the kernel 0.
LEMMA 3.3. O(w0 , w1, ... ,wn-i, f) does not depend on the vector-valued
function w0 •
PROOF. Let 0 1 and 0 2 be two forms (3.6) depending on u<0>, u(l>, ... ,u<n- 1>
and v<0>, u(I>, ... ,u<n- 1>, respectively. We must show that 0 1 = 0 2 • Let s<0> = u<0> - v<0>. Then
(s0{z, Z), f(z)) = 0, z E an. (3.9)
§3. THE LERAY-KOPPELMAN FORMULA 27
Using the representation of 0 1 and 0 2 in the form (3.8), we get that the form 01 - 02 has the determinants
............ ' (3.10)
( auu~~~· z) ,/{z)) = o, z E aD,j = 1, .. . ,n - l,
and this and (3.9) imply that the determinants (3.10) vanish. 0
LEMMA 3.4. If the vector-valued functions w(I>, ... , w<n-l) and p(I>, ... ,p<n- 1>
are in the class c<2>(3D), then the difference
O(w<0>, w(l>, ... ,w<n-I), f) - O(p<0>, p(l>, ... ,p<n-1), f) {3.11)
is a a-exact form.
PROOF. The difference (3.11) can be represented as a sum of analogous differences of forms that differ only in one argument, so it suffices to prove that the difference
O{ w<O)' w(I) , ... 'w<n-1)' f) - O{ w<O)' p(I)' w<2)' ... 'w<n-1)' f) is a-exact. Indeed,
a{ {-1 r<n- l)/2 ( w(I>, df) /\ ( p(I>, df) /\ d ( w<2>, df)
(2'1Ti}n ( w<I), f) ( p(I>, f) ( w<2>, f)
( w<n-1), df)} /\ ... /\d~----~-
(2.,,;r (w(l>,f) (p(l>,f)
/\ ... _ ( w(I>, df) /\ d ( p(I>, df) /\ ... } ( w(I>, f) ( p(I>, f)
= o( p(I>, w(I>, w<2>, ... , w<n- 1>, f) - o( w(I>, p(I>, w<2>, ... , w<n- 1>, f)
= 0(w<0>, W(I), w<2>, ... ,w<n-l), f) - 0{w<0>, p(I>, w<2>, ... ,w<n-I>, f), where Lemma 3.3 was used in passing to the last equality. 0
PROOF OF THEOREM 3.2. If w(I>, ... , w<n-I) belong to the class c<2>(<W), then
the difference of the kernels in (3.1) and (3.5) is a a-exact form, by Lemma 3.4, i.e., this difference is orthogonal to holomorphic functions when integration is considered over aD (see Corollary 0.3). Therefore, (3.5) follows from (3.1). But if wU> t£. c<2>(aD), then we can approximate them on aD by smoother vector valued functions with preservation of the condition (3.4), write the formula (3.5) for the approximating functions, and pass to the limit. D
We mention some immediate corollaries of (3.5).
COROLLARY 3.5 (Leray and Koppelman). If w<0>, ... , w<n- I) satisfy the condi
tion
( wU>(z, a), z - a) =I= 0, zEaD,aED,j=0,1, ... ,n-1,
then every q> E Ac(D) satisfies the generalized Cauchy-Fantappie formula
(3.12)
COROLLARY 3.6 (Leray). If w E c<1>(aD) and (w(z, a), z - a)=I= 0 for z E aD and a ED, then every qJ E Ac(D) can be represented by the Cauchy-Fantap pie formula
where
( ) k-1 _ (n - l}! ~k=I -1 wkdw[kl /\dz
w(z-a,w)- n · {2'1Tir (w, z - a)
PROOF. From (3.6) we get that
( ) _ (n - l)! ~n { )k-1 ~ w, ... ,w,z-a - -1 ukdu[kJ/\dz,
{2'1Tir k=I
Further, taking (3.8) into account, we find
_ (n - l)! n g(w,w, ... ,w,z-a)- ~
(2'7Tit k=I [k] dilkJ /\dz
u~n
[k] · · · dz1k1 /\dz
( u, z - a)= l, z E aD, a ED. (3.15)
We give (3.13) a more abstract form. Fix a point a ED and consider the surface Ma= {(z, u): (u, z - a)= l, z E aD} in the space C 2n of complex variables (z, u).
COROLLARY 3.7 (Leray). Suppose that the vector-valued function u(z, a), z E aD, a ED, belongs to the class c<1>(aD) with respect to z and satisfies condition (3.15). Let a be the cycle on Ma described by the point (z, u) as z runs along aD. This cycle is in some class h E H2n_ 1(Ma). Then any cp E Ac(D) and any cycle fJ E h satisfy the Cauchy-Fantappie formula
(n - l)! 1 n k-1 cp(a) = . n cp(z) ~ (-1) ukdu[k) /\dz.
(2'7Tl) fJ k= I (3.16)
PROOF. Observe that (3.16) is (3.13) for fJ =a. To get (3.16) with any fJ Eh from (3.13) it suffices to see that the form cp(z) · w(z - a, u) is closed on Ma. Indeed, only 2n - 1 of the differentials dz1, ••• ,dzn, du 1, ••• ,dun are indepen dent on Ma, and the form under consideration has maximal dimension 2n - 1, so it is closed. D
We point out that the Cauchy-Fantappie formula (all the more so, the generalized Cauchy-Fantappie formula) is not a single integral representation for functions cp E Ac(D), but a whole collection of such representaiions, depending on the choice of the vector-valued function w. For various classes of domains D it is possible to choose w in such a way that the conditions of Corollary 3.6 hold and w is holomorphic with respect to a in D. We thus obtain integral representations with holomorphic kernels (see §§4 and 8-11).
The classical Cauchy formula
for holomorphic functions of a single complex variable has two remarkable properties which have been responsible for many of its applications:
I) It is universal, i.e., true for any domain D with a sufficiently nice boundary av, and the Cauchy kernel l/2'11'i(Z1 - a1) does not depend on the shape of D.
2) The Cauchy kernel is holomorphic in a 1 ED for fixed z1 E 3D.
There is no integral representation with two analogous properties for holo morphic functions of several complex variables; instead, there exist formulas that either are universal but have a nonholomorphic kernel (for example, the Martinelli-Bochner formula (1.1)) or have holomorphic kernels but are not universal (see §§4 and 8-11).
The following assertion can be obtained similarly to the way Corollary 3.6 was obtained.
COROLLARY 3.8. If the domain D and the mapping f satisfy the conditions of Theorem 3.2 and the vector-valued function w E c0 >(3D) is such that ( w, f) =F 0 on 3D, then for every <p E Ac(D)
1 <pw(/, w) = ~ µ. 0 (/)<p(a), (3.18) 3D aEE1
where
(n - l)! };k=1(-l)k-lwkdw[kJ /\ df w(!, w) = (2'1Tit ( w, f) n •
We remark in conclusion that (3.5), (3.12), and (3.18) can be given a more abstract form similarly to the way (3.13) was reduced to the form (3.16).
§4. The Cauchy formula. The logarithmic residue with respect to the skeleton*
1°. Let D; c C~ be a domain with piecewise smooth boundary, i = l, ... ,n. The domain D ~ D 1 X · · · XDn C en is called a polycylinder domain (in particular, a polydisk if all the D; are disks). The set f = 3D1 X · · · X 3Dn, provided with the natural orientation induced by the orientation of D, is called the skeleton of D.
THEOREM 4.1. Let D be a polycylinder domain and a any point of D, and suppose that <p E Ac(D). Then the Cauchy formula is valid:
1 ( <p(z) dz <p(a) = (2'11'ir Jr z - a . (4.1)
*Editor's note. "Skeleton" is a literal rendering of "ocrpoB", but "distinguished boundary" is common in the English literature.
§4.THECAUCHYFORMULA 31
Note that the skeleton r plays the same role in the Cauchy formula (4.1) as the whole boundary of the domain plays in the classical Cauchy formula for n = 1. The existence of the integral representation over the skeleton implies that the functions in Ac(D) satisfy the maximum modulus principle with respect to the skeleton (see Theorem 6.9).
The Cauchy formula (4.1) is an example of a formula with a holomorphic kernel that is not universal (it is true only for polycylinder domains).
PROOF. Formula (4.1) is easy to get by repeated application of the classical Cauchy formula for n = 1, but we give another proof by showing that ( 4.1) follows from the Cauchy-Fantappie formula (3.16) for a suitable choice of the cycle of integration /3. The same method is used later in deriving Weil's formula.
Let us represent aD in the form u~ Y;. where Y; = {z: z ED, Z; E aD;}· On each face Y; we consider the vector-valued function u<;> = (uii>, ••. ,u~;>), where
Ci> - ~ ( )-1 · k - 1 Ob . 1 h Ci> . f d" . (3 15) Uk - U;k Z; - a; 'l, - , ••• ,n. VIOUS y, t e u satls y con ltlOn . . Now consider the oriented surface /J; = {(z, uCi>), z E Y;} in c2n. The union U~ /J; is not a cycle, since each face Y; has its own vector-valued function u<i).
At the same time, aD is a cycle, and the set {(z, u), z E aD} also would be a cycle if u were continuous in z E aD. The set U~ /3; has "holes" located over the points where the faces Y; intersect, i.e., over the edges. We "stop up" these "holes" as follows: if the k faces Y;1, ••• ,'f;k meet at the edge Y;1 ... ;k' then we construct the set
We add all the surfaces /3;1 •• • ;k to U~ /3;; the resulting (oriented) surface f3 is a cycle. We apply (3.16) to /3. All the uCi) are holomorphic in z; therefore, dum /\ dz = 0, i, k = 1, ... , n, and, hence, the integral in (3.16) over each /3; is equal to zero. If k < n, then all the du[Jl /\ dz are 0 on /3;1 •• N because u depends holomorphically on z and on the k - 1 independent parameters A, i.e., there are n + k - 1 < 2n - 1 independent differentials in all. Therefore, the integral in (3.16) over /3;1 •• • ;k is also equal to zero. It remains to compute this integral over /3; .. ·i. On r = Y1 .. ·n the components Uk of the vector-valued
I n
k = l, ... ,n,
where we have used the elementary equality
1 ~ ( )k-1 1 >..1+ ... +A.=1k-:-I -1 XkdXlkl = (n - l)!.
(4.2)
A;~O
To verify (4.2) it suffices to observe that, by Stokes' formula, the left-hand side of ( 4.2) is equal to
COROLLARY 4.2. Under the conditions of Theorem 4.1,
a! 1 cp(z)dz D"cp(a) = (2'1Tit r{z - a)"+ 1 •
2°. The logarithmic residue formulas considered in §§2 and 3 for a holomor phic mapping involve integration over the whole boundary of a domain. We now investigate logarithmic residue formulas in which the integration is over an n-dimensional cycle. The logarithmic residue formulas in §§2 and 3 are essentially based on the Martinelli-Bochner formula (1.1), and the formulas of the present subsection are based on the Cauchy formula ( 4.1 ). The results here can, as a rule, be derived from the general formulas (3.5) or (3.17) in the same way as we just obtained (4.l) from (3.16), but we shall give simpler proofs based on the Cauchy formula ( 4.1 ). The form
df( z) = ( a f / / ) dz = a In/( z) dz f(z) 'dz az
appears here as an analogue of the logarithmic differential for a holomorphic mapping.
We need the concept of an analytic polyhedron. This is defined to be an open set of the form D =DP= {z: z E G, lf;(z) I< P;. i = 1, ... ,k}, where/; E A(G) (i = 1, ... ,n) and G is a domain in en, D ~ G. If k = n, then Dis called a
§4. THE CAUCHY FORMULA 33
special analytic polyhedron. The latter polyhedron play an important role in this subsection. We first mention the following facts about them.
LEMMA 4.3. ~very special analytic polyhedon V consists of finitely many connected components.
PROOF. Every (bounded) domain i> can be approximated by domains with piecewise smooth boundaries in such a way that the conditions of Theorem 2.5 hold on the boundaries of the approximating domains if they hold on the boundary of i>; therefore, the Rouche principle is applicable to i>. It follows from the definition of V that for each connected component V' c V there exists a point a EV' such that f(a) = t, lt;I< P;. i = l, ... ,n. Hence, the mappingf(z) - t has zeros in V'. On the other hand, by the Rouche principle (Theorem 2.5) the mappings f( z) and f( z) - r have the same number of zeros in V', so f(z) has zeros in V'. Thus, f(z) has zeros in each connected component V' of the special analytic polyhedron V. We conclude from Theorem 2.4 that the number of connected components of V is finite. D
LEMMA 4.4. The skeleton r = rp = {z: z ED, l/;(z) 1= P;. i = 1, ... ,n} of a special analytic polyhedron V is smooth for almost all p = (p 1, ••• ,pn), P; > 0, i = l, ... ,n.
A proof is obtained immediately by applying Sard's theorem (Theorem 0.11) to the mapping z -+ (l/i( z) 12 , ••• , If,,( z) 12 ). D
In all the results below concerning integration over the skeleton f P it is assume that f P is smooth. By Lemma 4.4, this situation can always be achieved by varying p by an arbitrarily small amount.
THEOREM 4.5 (Caccioppoli, Martinelli, and Sorani). Let V be a special analytic polyhedron given with the help of a vector-valued function f E An(D). Then any cp E Ac( V) satisfies the formula
l 1 df(z) _ (2 .)n cp(z) f(z) - ~ P.a(/)cp(a),
'1Tl r a EE/
(4.3)
where E1 = {z: z EV, f(z) = O}, and f is the skeleton of V.
PROOF. Since av = u;= ,{ z: z E ii, l~(z) 1= Pj}, f does not have zeros on av, and, by Theorem 2.4, it has only isolated zeros in V. We surround each zero a E E1 with a neighborhood Ua u;; V in such a way that ff,, n ff,, = 0 for a =fa b, a, b E E1. Let a> 0 be such that a< min{llf(z)ll, z ED\ u aEE1 Ua}· Then each connected component of the cycle y = {z: z EV, lf1(z)i= · · · =i f,,(z) i= 8} falls in one of the neighborhoods Ua, a E E1. Let Ya= y n Ua.
Note that f - y = aQ, where Q = {z: z ED, l_fj(z)I= 6t + p/1 - t), j = I, ... ,n, 0.;;;; t.;;;; l}, i.e., f - yin D1 = D\{z: ft(z) ... f,,(z) = O}. The form <p(df /f) is holomorphic in D1; therefore, by the Cauchy-Poincare theorem,
_1 _1 'Pd/= ~ _1 _J 'Pdf_ (4.4) (2'1Ti)" r f aEEI (2'1Ti)" Ya f
If a is a simple zero of /, then Ua can be chosen small enough that f is biholomorphic in it. Making the change of variables w = /( z) and applying the Cauchy formula ( 4.1 ), we get that
I J # 1 1 ~ -- <p- = -- 'P(/IU,) (w))~ = 'P(/lu;) (0)) = <p(a). (2wi)" Yu f (2'1Ti)" lw,j=6
i=l •...• n
(4.5)
Suppose now that a is a multiple zero of/. By Proposition 2.1, for almost all sufficiently small (in modulus) r the mapping /(z) - r has simple zeros a(l>, ... ,aU<> in Ua, where k =µ.if). For llfll < min{6,minau.ll/(z)ll- 6} the set Ya.r = {z: z E Ua, l/i(z) - f 1 I= · · · =lfn{z) - fnl= 6} forms a cycle in Ua, because Ya.r n aua = 0. The cycle Ya.r is obtained from Ya by the homo topy Ya.t~ = {z: z E Ua, l_fj(z) - tfjl= 6, j = 1, ... ,n}, 0.;;;; t.;;;; l, in the do main Ua\{z: II)= 1(_fj(z) - f) = O}. Applying the Cauchy-Poincare theorem once again along with formula ( 4.5) for the simple zeros, we have that
_1 _J <pd(!- r) = _1 _J <pd(!- r) = ± <p(aU>), (2wi)"y. f-f (2wi)"Y •. £ /-f J=I
and, passing to the limit as r-> 0,
1 J df (2'1Ti )" Y.<pf = P.a(/ )<p( a). (4.6)
Now (4.3) follows from (4.4)-(4.6). D
THEOREM 4.6. If f and D satisfy the conditions of Theorem 2.4, then (4.3) holds
for any <p E Ac(D), where f = {z: z E aD, l/2(z) I= · · · =lfn{z) 1= e}, and
e < min 0011/(z)ll.
PROOF. Since e < minaoll /II, it follows that Ir I c D1· Note that if D n E1 =F 0 , then I r I =F 0 . Indeed, by the Rouche principle (Theorem 2.5), the mapping ft( z) - e, ... ,In( z) - e has zeros in D. According to the second part of the proof of Theorem 2.4, this implies that the mapping 0, / 2( z) - e, ... ,f,,( z) - e has zeros on aD, i.e., Ir I is not empty. The cycle r is homologous to y = {z: z ED, l/i(z)I= e, ... ,lf,,{z)I= e} in D1 = D\{z:II}=i _fj(z) = O},
since r - y = aQ, where Q = {z: z E ii, lft(z)I;;;. e, l/2(z)I= ... =1fn(z)I= e}, Q c I!.J· By the Ca~chy-Poincare theorem and Theorem 4.5, applied to the domain D = {z: z CD, l.tj(z) I< e,j = l, ... ,n}, we get that
_l_f df=_l_f df= a D ( .)n }T cp J {2 .)n cp J ~ P.a(/)cp( ). 2m f WI Y aE~
COROLLARY 4.7. Under the conditions of Theorems 4.5 and 4.6, the number N of zeros off in D is expressed by the formula
N = N(f, D) =-1-nl df. {2wi) r f
The next theorem, which includes a variant of the Rouche principle and the logarithmic residue theorems, is useful for applications.
THEOREM 4.8 (Yuzhakov). Let f, cp, D, and r be the same as in Theorem 4.5, and suppose that the mapping g E An(D) satisfies on r the inequality
lg/z)l<l.tj{z)I, j=l, ... ,n. (4.7)
Then: a) the mappings f and f + g have the same number of zeros in D; and b)
_1_1cpd(f+g)= ~ P.a(/+g)cp(a). {4.8) {2wir f f + g aEE/+g
LEMMA 4.9. Under the conditions of Theorem 4.8 there exists a number 80 > 0 such that for 0 < 8 < 80 the cycle f is homologous to the cycle y = { z: z E D, 1.tj(z) + g/z) 1= 8, j = l, ... ,n} in Df+g = D\ { z: II.f= 1[.tj(z) + g1(z)] = O}.
PROOF. Let 'j = maxze£1g/z)I, j = l, ... ,n. By the maximum modulus principle (see Theorem 6.9), it follows from (4.6) that 'j < p1 (j = l, ... ,n ). Consider 80 = min {p I - r I• ••• 'Pn - rn} and the family of cycles rp = { z : z E
i5, l.tj(z) + g/z)I= 8,j = l, ... ,p, l/k(z)I= Pk• k = p + l, ... ,n}, 0 < 8 < 80 ,
p = 0, l, ... ,n. Obviously, f 0 = f and fn = y. Further, fP C Df+g• since/+ g =I= 0 on rP by the choice of 8. Moreover, f = f 0 - _!:1 - • • • - fn =yin Df+g· Indeed, fp - fp+t = aQP, where QP = {z: z ED, l.tj(z) + giz) I= 8, j = l, ... ,p - 1,1.f;,(z) + gp(z)I;;;. 8,1.f,,(z)I.;;;; Pp,lfk(z)I =Pk• k = p + 1, ... ,n} C
Df+g• p = 1, ... ,n. D PROOF OF THEOREM 4.8. By the maximum modulus principle for a special
analytic polyhedron (see Theorem 6.9), (4.7) gives us that I giz) I< p1 in i5 for j = l, ... ,n. Thus, the inequality lg1(z)l<l.tj(z)I, j = l, ... ,n, holds on the face ll1 = {z: z ED, l.tj(z)I= P)· But since aD = U~ ll1, we get assertion a)
from Theorem 2.5. Further, by Lemma 4.9 and Theorem 4.5, applied to the mapping f + g and the cycle y,
I 1 d(/ + g) _ 1 1 d(/ + g) (2wi)n rep I+ g - (2wir Yep I+ g
~ P.a(f+g)ep(a). D aEE/+g
EXAMPLE. Let m be the multiplicity of the zero of the mapping w1 = z1 - zi, w2 = Zf - z~ at the point (0, 1). Since the inequalities lz212 <lz1 I, and lz112 <I Z213, hold on r = {z: lz11= e7, lz21= e4 }, 0 < E < 1, it follows from Theorem 4.8 that
_ 1 J.d(z1-zi}Ad(zr-zi} m - (2wi)2 r (z1 - zi}(zr - zi}
= _l_J. (-3zi + 4z1z2)dz1 /\ dz 2
(2wi)2 r (z1 - zi}(zr - zi}
= _I_J (-3zi + 4z~)dz2 = 3 2wi izil=e4 z~ - z~ ·
THEOREM 4.10 (Caccioppoli, Martinelli, and Sorani). Suppose that f E An(D), <p E A(D), and that the cycle r satisfies the weak homology relation
(4.9)
D1 = D\(z: /i(z) · · · fn(z) = O}, where Yu= {z: lf1(z) 1= · · · =lfn(z) I= e} n Va; here Va is a neighborhood of a zero a E E1 and e is sufficiently small. Then
I !. df _ "1 ( 2 .)n <p-f - ~ naP.a(/)<p(a).
'ITI r aeE1
The proof follows at once from Corollary 0.5 and (4.6). D In connection with this theorem the problem arises of a topological description of cycles of the
form(4.9). Let Fj = {z: z E D,fj(z) = O},j= l, ... ,n and F= Fi U ··· UFn. Then D1 = D\F. The cycle Ya is said to be locally separating for the analytic sets Fi, . .. , Fn at the point a of their intersection. The subgroup of Hn(D\F) generated by the locally separating cycles is called the (globally) separating subgroup, and accordingly a cycle of the form (4.9) is called a separating cycle.
PROPOSITION 4.11. (Martinelli and Sorani). A separating cycle satisfies the condition
r-o inD\F[jJ• j= l, ... ,n, (4.10)
where F[j) = F1 U · · · [j] · · · UFn.
PROOF. Ya= oQa, where Qa = {z: Z E Va, lfi(z) 1= · · · [}] · · · =1/n(z) 1= E, l.lj(z) I..;; e}, Q0
C D\F[jJ· D The converse assertion is, in general, false.
ExAMPLE. D = C2\{z: IZ11.;;; 1, 1z21.;;; 1 }. f1 = Z1, fi = Z2, r = {z: IZ11=1z21= 2}. Then the cycle r is not homologous to 0 in D\ F, and it satisfies condition (4.10), but is not separating, since £1 = 0.
On the other hand, we have
THEOREM 4.12 (Tsikh). If Dis a domain of holomorphy, then a c:vcle f E Z,.( D\ F) is separating if it satisfies the conditions (4.9).
Above we considered logarithmic residue formulas in C" in which the integration was over cycles of dimension nor 2n - I. It is also possible to indicate formulas for cycles of intermediate dimension.
THEOREM 4.13 (Yuzhakov and Kuprikov). Under the conditions of Theorem 3.1
where
(-1) 00 - 1/.- • • ·/.- dl /\ dif X ~ _ a: a, ~[ao ... ·:pl _
11 p •
ao<··· <a, (f.f1 + · · · +fn-p-ifn-p-1 + fn-p/.1-p · · · · ·f,./.,)
with q = 4<n - p)(n - p - 1) and the summation over all ordered subsets (a0 , •.• ,ap) of {I .... ,n} such that 1 .;;; a 0 .;;; n - p, n - p .;;; a 1 < · · · < aP .;;; n;
f2n-p-I = {z: z E ClD,lfn-p-i(z)i= ··· =if.,(z)i= e}.
with E sufficient~v small.
Theorem 3.1 and Theorem 4.6 are particular cases of this theorem (for p = 0 and p = n - I respectively).
COROLLARY 4.14 (Martinelli, Sommer, and Sorani). Let qi E Ac( D), where D is a bounded domain with piecewise smooth boundary. Thenfora ED
(4.11)
This formula generalizes the Martinelli-Bochner formula (LI) and the Cauchy formula (4.1) to the case of integration over cycles of intermediate dimension, but (4.11) can itself be obtained from the Cauchy-Fantappie formula (3.16) by a suitable choice of the cycle /J, similarly to the way we obtained the Cauchy formula in subsection I 0 of this section.
In conclusion we p

Documents

Integral Representations and Residues in Multidimensional Complex Analysis