УДК 517.55
BISHOP'S FORMULA FOR A MATRIX POLYHEDRO
WITH A NON-PIECLE SMOOTH BOUNDARY
Annotation. This work defines matrix polyhedral region using a matrix ball.
In this matrix polyhedral region is obtained Bishop's formula for meromorphic
functions of a special form.
Keywords: Holomorphic function mappings, matrix polyhedral set, matrix
polyhedral domain, meromorphic function.
Introduction
In the theory of functions of many complex variables, integral formulas play
an important place in the theory of holomorphic and meromorphic functions of a
special form. At the same time, the tasks of obtaining new integral formulas using
local residues; expansions into series of holomorphic and meromorphic functions of
a special type using integral formulas are considered target scientific research. In the
complex analysis of many variables, integral formulas were studied in the works of
L.A. Aizenberg [1-3], G. Khudayberganov [6-9], A.K. Tsikh [10]. Tsikh proved the
Weyl and Bishop integral formulas in a special analytical polyhedron using local
residues of many variables. In the polyhedral region, A. Weil [5] studied integral
formulas with a holomorphic kernel. In the works of B.A. Shaimkulov, the matrix
analogue of the Cauchy–Weil integral formula and the Carliman formula were
studied ([11-15]).
Main part
Let 𝑍 = (𝑍1 , 𝑍2 , ⋯ , 𝑍𝑛 ) −e a vector composed of square matrices 𝑍𝑗 , 1  j  n
order 𝑚, considered over the field of complex numbers . It can be considered that
Z  element of space n [m m]  nm [6].
Matrix "scalar" product for Z ,W  n[m m] let's define it this way [6]:
Z ,W  Z1W1*   Z nWn* .
region
Bm,n  Z  n [m m]: I  Z , Z  0 ,
2
is called a matrix ball, where I  unit order matrix m .
The skeleton of this region is a variety of the form:
X m,n  Z : Z , Z  I  .
Obviously, the real dimension of the skeleton is equal to m 2  2n  1 .
At m  n  1 , B1,1  unit circle of , а X1,1  unit circle.
Let D  bounded convex complete circular domain with Shilov boundary S,
which is smooth (class С1) diversity.
Let's define a class H 2  D  as a class of all functions f , holomorphic in D,
for which
sup  f  r  d    ,
2
0 r 1 S
here r   r 1, , r n  а d   normed Lebesgue measure on a manifold S,
invariant under rotations.
Theorem 1[6]. For every function f  H 1  Bm,n  the formula is valid
f Z  
f (W )d (W )


mn
I    Z ,W
X m , n det
m

Z  Bm,n
,
(1)
where d W   normalized Lebesgue measure on the skeleton X m,n .

Let the mapping be given f  f1,
domain G  nm .
In the future, display f  f1,

, f nm2 : G 
nm2
, holomorphic in some
2

  f111  Z 

f Z   
 f 1 Z 
  m1
f11m  Z  

,
1
f mm
 Z  

, f nm2 : G 
 f11n  Z 

,
 f mn1  Z 

nm2
let's present it in the form:
f1nm  Z   

:G 
n
f mm
 Z   
n
 m  m .
Definition 1. A matrix polyhedral set defined by a holomorphic mapping
f :G 
n
 m  m , called a set


f 1  Bm,n   Z  G  r 2 I  f  Z  , f  Z   0, r  0 ,
if it is relatively compact i.e. G f 1  Bm,n  Р G .
Definition 2. Connected component of a matrix polyhedral set f 1  Bm,n 
called a matrix polyhedron (generalized matrix ball), let's denote it by  f , r .
Skeleton region  f , r is defined as follows:


 f ,r  Z G  f  Z , f  Z   r 2I , r  0 .
Let f (Z )  D  G — holomorphic domain mapping
D  CnZ  m  m  G  CWn  m  m  и H  Z  
 Z 
 meromorphic to a function.
 Z 
Definition 3 [10]. Display f  Z  has a finite type if for each W  G the
equation f  Z   W has in the area D the same finite number of roots, taking into
account their multiplicities ([8, p.38]).
Definition 4 [10]. Following the functions H (Z) regarding display f  Z 
called function
[Tr H ](W)   H (Z( ) (W)) W  G \ f (  0)

where the summation is carried out over all roots (taking into account multiplicities)
of the equation f (Z)  W .
In this work, using formula (1), we obtain Bishop’s integral formula in the
region  f , r for a special type of function
hZ 
, where h  Z   H 1   f , r  ,
J f Z 
J f  Z   Jacobian mapping f  Z  , having a finite type.
Let f  D  G — holomorphic mapping of a finite type of domain
D
n
Z
 m  m
в G
n
W
 m  m и W 0  G
— arbitrary point. Consider the area
Bm ,n ,r W 0  в G с центром в точке W 0


Bm,n,r W 0   W : r 2 I  W  W 0 ,W  W 0  0 Р G .
Theorem 2. Let H  Z   H1  D . Then for the trail [Tr H ](W ) in field
Bm ,n ,r W 0  the integral formula is correct
Tr H W   
Hd  f  Z  

mn
I    f  Z  ,W
Г f , r det

m

where Г f , r  Z  D : f  Z  , f  Z   r 2 I  m .

,
(2)
Proof. For simplicity, we prove the theorem for the case W 0  0 .
In view of the proposal [10, p. 33] for almost everyone W  B m ,n ,r W 0  roots of a
system of equations f  Z   W  0 simple; let's designate them Z (1) (W ), , Z(  ) (W )
. Let be, U  D family of disjoint neighborhoods of points Z ( ) (W ) and

Г  Г f  Z W ,  Z  D : f  Z   W , f  Z   W   2 I 
m
  cycle in U

.
Then, by the definition of a trace and the Cauchy-Szeguy formula (1), we have
H d  f  Z  

Tr H W    
 1
Г  
Z W  ,

det mn I    f  Z  ,W
m

.
Кроме того, в области регулярности подынтегральной формы (2) сумма

Г


1
homologous to the cycle Г f ,r . Therefore, applying the Stokes formula, we
obtain
H d  f  Z  




1 Г

Z   W  ,
det
mn
I
 m
 f  Z  ,W



Г f ,r
H d  f  Z  
det
mn
I
 m
 f  Z  ,W

.
The theorem has been proven.
Now, as a consequence of the proven theorem, we present an integral representation
for the trace of a meromorphic function of a special form.
Corollary 1. Let h( Z )  H1  D  , J f — Jacobian mapping f  D  G , finite type.
Then for the trace of a meromorphic function H  h / J f в B n,m,r W 0  the integral
formula is correct
[Tr h / J f ]W  

 f ,r
hZ d  Z 
det
mn
I
 m
 f  Z  ,W

.
(3)
Proof. d  normalized Lebesgue measure on  f , r , that's why [4, p. 153]
d  f  Z    J f d  Z  .
From the formula (2) for W  Bm,n,r W 0  we get


Tr h / J f  W    h / J f Z   W  




Г  
Z
W ,
h / J f d  f  Z  
det
mn
I
 m
 f  Z  ,W



Г f ,r
det
mn
I
h d  Z 
 m
 f  Z  ,W

.
The investigation has been proven.
The last corollary allows us to obtain an analogue of Bishop’s formula in the
matrix
polyhedron


 f , r  Z  D  r 2 I    f  Z  , f  Z   0, r  0
m
for
a
meromorphic function h  J f .
Theorem 3. If the function h  Z   H 1   f ,r  , Z  f r and at this point the
h
Jf
Jacobian J f  Z   0 , then for a meromorphic function
the integral
representation is valid
hZ 
h  X    Z , X  d  X 
 
.
J f  Z  Г f ,r   Z , Z  det mn I  f  Z  , f  X 


Proof. From the definition of trace it follows that the integral in the formula
(3) at W  f  X  equal to the sum of the function values
h
at points Z  X and
Jf
the meaning of this function in the prototypes X   f  X      2  points
W  f  X  . Let us introduce the weight function   Z , X   0 , having the
following properties: for any fixed Z from


 f ,r  Z  D  r 2 I    f  Z  , f  Z   0, r  0 Р D ,
m
function (Z X ) equal to zero at all points Z  X   , except Z  X . Such a function

exists. Indeed, suppose W 0  non-critical display value f , и g  Z  — linear


function i.e. g X  W 0   various. Then we can define the function


 Z , X    g  Z   g X 

 2

 


 g Z   g X 
2
    g  Z   g  X   

(4)
where we assume the numbering of preimages X ( )  X ( )  Z  such that
X (1)  Z   Z . Thus, the product in (4) is a polynomial in g  Z  , with coefficients
holomorphically dependent on X . So, we have
 1
  Z , X    ck  X  g k  Z  
k 1


where ck  X  holomorphic functions in  f r . By construction  Z , X    Z   0 ,
for points X ( )  Z   Z .
Using the corollary and the constructed weight function   Z , X  , we get
Bishop's formulas in  f r .
Really,



h Z

 X     Z , Z    X  


J f  Z    X 

 
 

h Z  Z , Z  h Z  X   Z , Z


2
J f Z 
J f Z   X 

2


2
 X 


h Z   Z , Z 
h  X    Z , X  d  X 
.
 
m
mn
J f Z 
I    f  Z , f  X 
Г f ,r det
The theorem has been proven.


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