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The Behavior of Cauchy-Type Integral Near the Boundary of the Semicylindrical Domain
A. Gaziev1, M. Yakhshiboev2
1Faculty of Mechanical-Mathematics of Samarkand State University, Samarkand, Uzbekistan
2Samarkand Branch of Tashkent University of Informational Technology, Samarkand, Uzbekistan
Email address
(M. Yakhshiboev) Citation
A. Gaziev, M. Yakhshiboev. The Behavior of Cauchy-Type Integral Near the Boundary of the Semicylindrical Domain. International Journal of Mathematical Analysis and Applications. Vol. 3, No. 1, 2016, pp. 1-16.
Abstract
The purpose of this work is the elucidation of the behavior of Cauchy-type integrals near the boundary semicylindrical domain to 
- characteristics celebrated jordanovic of closed curves (in the case when θ(δ)~δ, this class of curves is much wider class of piecewise-smooth class of curves for which the chord length relation to the pulling together arch are limited (K-curves), and also in it existence of cusps is allowed). The main characteristics for functions 
- the mixed and private modules of continuity which was proven continuously extendibility of 
multiple Cauchy-type integral to the border of the semicylindrical domain and the limit values of the types of Sokhoskiy’s formulas.
Keywords
Closed Jordan Rectifiable Curve (c.j.r.c.), Semicylindrical Domain, Cauchy-Type Integrals, Private and Mixed Continuity Modules, Characteristic Curve Core, Continuous Extendibility, Sokhoskiy’s Formulas
1. Introduction
Let 
be a closed Jordan rectifiable curve (c.j.r.c.) with length 
and diameter 
in the complex planes of variables 
. A bounded domain 
with the bound we call an internal, the padding of 
we call an external and denote by 
.
The contours 
defines in whole complex space of 
variables 
various semicylindrical domains which are obtained by all possible combinations of characters in the topological multiplication
.
Among them: one is of the type of 
(
), which we denote by 
are domains of the type of 
(
) which we denote by
; similarly, 
are domains of the type of
(
)
which we denote by 
) and etc..
Borders of all these semicylindrical domains have the common part, namely, 
which is called the core.
If the function 
is defined in 
and for an arbitrary 
there exists
we say 
is continuously extended up to the bound of . Analogically, define continuously extendibility up to core of domains 
etc.. The corresponding limit values of the function is denoted by 
etc., respectively.
If is continuously extended up to core from the domain 
then we say that the function is continuously extendible up to core. We say that a function is continuously extended to the given boundary point of semicylindrical domain, if the function tends to a given boundary point along any path, while remaining at all times in this semicylindrical domain. The corresponding limits we call boundary values in this domain, and we denote them as well as the boundary values on the core
, with the replacement 
core of 
corresponding boundary point. It is easily seen that if the function is continuously extended to the core from every semicylindrical domain which boundaries have a common core of , then it will continuously be extended to any boundary point of each of these semicylindrical domains.
Let us consider - multiple integral of Cauchy-type
(1)
where 
- is the space of continuous functions on .
In the paper [1] was studied the behavior of integral (1) for smooth contours and functions of Holder's class and in papers [2], [3] and [4] (for n=2) under some assumptions on the curves
and the function 
the continuity up to core of the integral was investigated. In [5] and [6] (for n=2) the investigation of the integral (1) was extended to a case of summable density. The papers [7], [8], [9] and [12] are focused on study the integral near a bicylindrical fields and [13-16] contain results the behavior of integral Martinelli-Bochner, which (1) turns into a Cauchy-type integral for n = 1.
In the current work the behavior of -multiple integral (1) on the border of semicylindrical domain in terms of the continuity modulus and 
) characteristic curve 
is studied under the most common assumptions concerning function 
and curves (it was first given in [7], and then generalized in [10], [15-17].) The paper is organized as follows: in the next section are presented some results and notations which will be used in the formulation of the main theorems. In Section 3 we give our main results and their proofs.
2. Preliminaries
For the brevity of the writing we introduce the following notations as in 
,
,
.
=(
),
etc..
etc..
,
,
,
, etc..
etc..
Then be these notations (1) takes form
Let us denote by 
the following
It is easy to verify that holds the identity
holds. Using this and (1) we have
(2)
where 
The integrals on the right hand side in (2) we consequently denote by
which will be further used.
Let
be a closed rectiable Jordan curve (c.r.j.c.)
be the length of the curve and be an equation of the curve in arc coordinates 
. Let us denote
The function 
is chosen as the main characteristics of the curve 
Monotonically increasing function 
defined by
is called a generalized inverse with respect to 
. The concept of generalized inverse function is introduced and studied in [8]. To investigate the behavior of integral (1) on the boundary of semicylindrical domain appears the following main characteristics to function 
1). mixed continuity module (for the case 
was given in [11])
where
2). private continuity modules
where
where
,
Let us denote by 
a multiple nonnegative monoton increasing function 
on 
such that 
and 
monoton decreases. By
denote a set of functions 
defined on (
and lying in 
on each argument, i.e., 
by 
at fixed 
. It is clear that
1). Let 
be a nonincreasing function on 
Then the following
holds for arbitrary 
2). Let be a nonincreasing function on 
and 
satisfy the conditions
, 
, 
then
3). Let be nonincreasing function on 
. Then
.
Let us emphasize
….
.
3. Main Results
Theorem 3.1. Let be a closed rectiable Jordan curve and 
Then for arbitrary 
the following estimate holds
(3)
,
(4)
(5)
(6)
where
Proof. Let us denote by 
an arbitrary point of the border 
Then the identity takes the place
(7)
the validity of which is easily shown by direct calculations of items. It is easy to see also that
(8)
Let us consider the difference
(9)
Using (8) and (7) in (9) we obtain
(10)
Let us first denote items on the right hand side of equality in (10) by 
, respectively, and we estimate each of them separately.
Before proceed to assess 
remind the is an identity
=
.
Let us consider
(11)
The difference 
standing under the integral in 
, we replace with a right member of identity (11). Then we have
. (12)
Let 
The integral (12) is represent table a type of the sum of two integrals of 
and 
, taken, on 
and 
respectively, where
Let us denote by
the integrals taken piecemeal cores
respectively. As for every
(13)
we have
where
Consequently applying 1), 2), 3) of Lemma 2.1 and choosing 
we obtain
.
Thus,
(14)
Now we estimate 
For this aim we first estimate integral 
:
Now we estimate the integral 
as follows
.
From this and (13) we obtain the following
𝗑
𝗑
for every 
Taking into account that 
and 
, from the last estimate we have
𝗑
. (15)
Taking similar transforms which were used in the estimate of (15
we have for the integral 
the following:
(16)
(17)
, (18)
(19)
Summarizing the obtained estimates in (15
, we get an estimate for
Now we estimate 
As 
applying items 1) and 2) of Lemma 2.1 we have
Owing to a lack of growth 
Therefore,
(20)
Similarly estimating the rest integrals we have
(21)
(22)
(23)
For get an estimate for the integral 
we apply Theorem 2 in [18]. Summarizing all estimates for integrals 
and 
and estimates (14), (
and (
-(23
we finally obtain the required estimate (
.
To estimate the following difference
we use the following identity
the validity of which follows from (7). Then 
is represented in a form of a difference (10) and is estimated also as estimates for 
and the distinction consists only among integrals. Therefore, we obtain
Similarly, we have
By continuing this process we show estimates for the differences
, 
have the forms:
.
These prove the theorem.
From the theorem immediately follow the following equalities
Theorem 3.2. If 
then for every 
the following estimate
holds.
The proof follows from Theorem 3.1.
Theorem 3.3. Let 
(k=
) - c.j.r.c., 
Then function 
continuously extendable on a core Δ from each of 
semicylindrical domains for which the core is common.
By Theorems 3.1 and 3.2 and taking into account (2), we get that the function 
is continuously extended to the cores Δ and for the limiting values of the function equitable Sokhotskii’s formulas:
,
+
, (24)
,
.
In particular, for the case 
, the Sokhotskii’s formulas (24) take the forms
(25)
and for the case 
, (26)
(27)
(28)
(29)
Above we have shown the behavior of Cauchy-type integrals on the core 
of the border. Now we explain the behavior of the integrals on the whole boundary of the semicylindrical domain. To reduce the entries in detail, consider the case n=3 for
.
The boundary of this domain consists of the sets: 
; 
; 
; 
;
;
and 
.
The integral (1) for n=3 we write as follows
(30)
(31)
Let us consider integral (30) as integral of Cauchy-type of a complex variable with a core
depending on the parameters 
and applying to them Sokhotskii's formulas (
of the variable 
and we obtain
(32)
At p=1,2,3 the formula (32) gives boundary values of integral (1) in points of boundary sets: ; ; .
Considering integrals (31) as integral of Cauchy-type of two complex variables of with the core
,
depending on the parameter and applying to them Sokhotskii's formulas (, (
and (
on the corresponding variables and obtain
(33)
(34)
(35)
Formulas (33), (34) and (35) give values of integral (1) at points of sets ; ; 
. All shifts integrals at a conclusion of formulas (32)-(35) are admissible as only shifts of special integrals with routine were applied. On a core Δ boundary values 
are defined by Sokhotskii's formula (24).
Thus the following theorem is proven.
Theorem 3.4. Let 
r.j.c.c., 
Then the function defined as (1) is continuously extended to the entire of border of the semicylindrical domain and the limiting values of the function Ф(z) are formulas such as Sokhotskii’s formulas for the case of the core (24).
References
