American Journal of Research in Humanities and Social Sciences 
ISSN (E): 2832-8019                Volume 16, | Sep., 2023                      

 

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3 

GENERALIZATION OF THE HUA LO-KEN FORMULA IN THE MATRIX 

POLYHEDRON 

Yusupbayeva Hilola Ergashbek daughter 

 

A B S T R A C T K E Y W O R D S 

Integral formulas generalizing the Cauchy integral formula in the theory 

of functions of one complex variable serve as an important constructive 

tool. Local residues of many complex variables and integral formulas in 

multivariate complex analysis serve as a basis in the problems of 

connecting the values of a function within a domain with the boundary 

of the domain or with a part of the boundary. Therefore, the study of 

integral formulas and residues of many variables in function theory 

plays an important role and is considered one of the topical trends in 

modern mathematics. 

 

Mathematics, HUA LO-

KEN, polyhedron, 

theorem, formula. 

 

Introduction 

Consider the space of ˆ n  n  —skew-symmetric matrices whose elements are complex numbers, 

and the classical domain of the third type 

 D {Z  ˆ n  n  I n   ZZ  0}, 

where   I n   is the identity matrix of order  n , Z - matrix, the complex conjugate matrix Z (Recall 

that the condition 

H  0 for the Hermitian matrix H means that H is positively defined, i.e. all eigenvalues are positive) 

([1],[2]). 

The boundary of the D3 domain is defined as follows ([2]): 

D3 {Z  ˆ n  n  det I n   ZZ   0, I n   ZZ  0}. 

srt in the D3 region is defined as follows [1]: 

Г Z  ˆ n  n : I n   ZZ  0 . 

It is known [4,p.96] that with the help of the Hua Lo-ken integral formula any holomorphic function 

in the form of an integral (for even n  h Z D3   С D3   you can imagine in  

 

where  

the order of the differentials and the constant 

h(Z) = cn  
  h( X )dX  

n−1 

, (1) 

dX =  
i=1, j=1 i j 

cn 



American Journal of Research in Humanities and Social Sciences 
Volume 16 Sep., 2023 

 

P a g e  | 78  www.americanjournal.org 
 

are chosen so that  

 
 Let be given a mapping  

 of 

some domain. 

We introduce the concept of a matrix polyhedron. A matrix polyhedral set defined by a holomorphic 

image   

is called a set 

f 1 D 3,r  {Z G  r2I 
n  

f Z  f Z   0,r  0}, 

 

The lemma is proved.Now we give an integral interpretation of the local deduction, which follows 

from the general integral representation for the local deduction obtained in [3] and will be applied to 

prove the theorem. 

Let the map f (Z ) be holomorphic in a closed neighborhood 

and 

have at point  

isolated zero. For the sprout  

according to the formula f (Z ), the effect of the local deduction at point A is determined by  

 

where  

is a small enough number. 

 

CONCLUSION 

In this paper, a matrix analogue of the Weyl formula in a polyhedron is found,defined using a classical 

domain of the third type (i.e.

 
 

 
cn  

 
 

 
dX 

n− 1 

 
= 1. 

f = 
 

f1, 

f : G → ˆ n  n,  

U A     

A 

ˆ n  n 

h UA →  

h(Z )  
i=1, j=1 

dzij 

res A f (h(Z )) = cn   
  i  j  

n−1 (4) 
 

  0 −  

 
f ,r 

) 



American Journal of Research in Humanities and Social Sciences 
Volume 16 Sep., 2023 

 

P a g e  | 79  www.americanjournal.org 
 

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1. Hua Lo-ken. Harmonic analysis of functions of many complex variables in classical domains. M., 

Ed. foreign lit., 1959. 

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