Scientific journal
European Journal of Natural History
ISSN 2073-4972

ABOUT SOME METHODS OF CONSTRUCTION POLYANALYTIC FUNCTIONS WITH A PREDETERMINED LIMIT SET IN THEIR ISOLATED SINGULAR L-POINT

Gomonov S.A.
In this paper we consider a new method of quick construction of polyanalytic functions with a predetermined cluster set in isolated singular B-points (radiant points) of these functions.

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

  1. Balk M.B. Polyanalytic functions. Mathematical research.- Berlin: Akad.- Verlag, 1991.- Vol. 63.
  2. 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.
  3. 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.
  4. Gomonov S.A. Limit sets and Isolated Singularities of Poly-analytic Functions // Sorosov Educational Magazine. - M., 2000. - V.6, №1 (50). - p. 113-119.
  5. Gomonov S.A. Theorem of Sokhotsky-Weierstrass for Poly-analytic Functions // Papers of Mathematical Institute. - Minsk, 2004. - V.12, №1. - p. 44-48.
  6. Gomonov S.A. Some Methods of Construction of a Poly-analytic Function with a Predetermined Limit
  7. 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.
  8. Gomonov S.A. On the Sokhotsky-Weierstrass Theorem for Polyanalytic Functions // European Journal of National History. - London, 2006, №2. - p. 83-85.
  9. 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