It is known [1-5, 7] that if a set 
of any of the four types is given: an extended complex plane 
, a polynomial image 
of a unit circle 
(
), an arbitrary unit subset of the set , and finally, a union of finite number of nondegenerate polynomial lines 
augmented with 
point, then for any point 
there is such a deleted neighborhood 
of its and a defined in it poly-analytic (p.a.) function 
, i.e. the function of the kind
, (1)
where 
, 
( 
) - are analytic in functions, that 
. Let us remind that the number is called [1] the order of poly-analytic property of the function , and if 
, then it is called the proximate order of its poly-analyticity; the functions () are called the analytic components of the poly-analytic (or as it is spoken about,
- analytic) function .
However, the earlier offered in [2,3] method of finding the corresponding p.a. function for the last, the fourth, case, when the point 
is called [2, 3] the isolated singular l-point of the function , and the predetermined set is sure to have the form
, (2)
where 
, 
,
; 
, unlike the rest of the cases, was very complicated and tedious.
In this article essentially more simple modes of construction of a p.a. function possessing a limit set of the kind (2) in its isolated l-point are offered, ( ) being arbitrary predetermined polynomials different from those identical to the constants.
As compelling for any 
congruence 
allows considering that 
or 
, and vice versa, then, for the sake of simplicity, some results further will be formulated namely for .
Theorem 1
Let 
, where ,
,
, 
,
and let 
, where the polynomials 
,
, and the numbers 
are all complex -th roots of 1 (unity), the polynomial 
has no complex unit module roots, then for the function
the congruence 
is correct.
The deduction of the theorem 1 is in [6].
Samples
1. As 
, then 
is a union of a pair of parallels;
is a union of three parallels
;
is a union of tree concurrent in the 0 point lines.
2. As 
, то 
;
is a union of three half-lines, all of them centering in 0 point and making angles of 1200.
3. As 
is a parabola of the second order, then every of the sets, 
and 
, is a union of two parabolas.
In the conclusion of the article let us show some simple upper estimate of the number of polynomial lines, making up 
, where 
is isolated singular l-point of the p.a. function .
Theorem 2
For any p.a. function of the proximate poly-analyticity order 
, and for its every isolated singular l-point the set of all the elements from 
can be represented in the form of a union of finite number of nontrivial polynomial lines, the quantity of which 
satisfies the following conditions:
а). 
;
б). 
(with 
and with 
this estimate is exact).
The deduction of the theorem 2 is in [6].
Literature
- Balk M.B. Polyanalytic functions. Mathematical research.- Berlin: Akad.- Verlag, 1991.- Vol. 63.
- Gomonov S.A. About the Structure of Limit Sets of Poly-analytic Functions in Isolated singularities // Mathematica Montisnigri.- Podgoritsa, 1995.- Vol. 5 (1995). - Annals of Chernogorsk University. - p. 27-64.
- Gomonov S.A. About the Application of Algebraic Functions to the Research of Limit Sets in Point of Poly-analytic Polynoms // Some Questions of the Theory of Poly-analytic Functions and Their Synthesis. - Smolensk: SSPI 1991. - p. 16-42.
- Gomonov S.A. Limit sets and Isolated Singularities of Poly-analytic Functions // Sorosov Educational Magazine. - M., 2000. - V.6, №1 (50). - p. 113-119.
- Gomonov S.A. Theorem of Sokhotsky-Weierstrass for Poly-analytic Functions // Papers of Mathematical Institute. - Minsk, 2004. - V.12, №1. - p. 44-48.
- Gomonov S.A. Some Methods of Construction of a Poly-analytic Function with a Predetermined Limit
- Set in Its Isolated Singular L-Point // Research on Complex Analysis´ Boundary Problems and Differential Equations / Smolensk State University. - Smolensk, 2005. - №6. - p. 20-27.
- Gomonov S.A. On the Sokhotsky-Weierstrass Theorem for Polyanalytic Functions // European Journal of National History. - London, 2006, №2. - p. 83-85.
- About Some Methods of Research Limit Sets of Bi-analytic Functions in Their Isolated Singularities // Research on Complex Analysis´ Boundary Problems and Differential Equations / Smolensk State University. - Smolensk, 2006. - №7. - p. 38-58.
The article is admitted to the International Scientific Conference "Fundamental Research", Dominican Republic, 2007, April 10-20; came to the editorial office on 22.12.06