1. For every poly-analytic function (p.a. function) [1-4]
, (1)
set in a deleted neighborhood 
of its isolated nonessential singularity 
, the representation 
in is possible in the following form [4, 6]:
, (2)
where p.a. functions 
and 
are identically determined [4, 6] by expansion coefficient of analytic in functions 
in a Laurent series. Functions and allow to describe identically a cluster set [1, 2, 4, 6] 
of a poly-analytic function in a point , in particular, to establish if the 
point is an isolated l-point for ; [4, 6], without loss of reasoning generality, being always possible to pass on from studying the behaviour of an arbitrary p.a. function in a deleted neighborhood of its isolated singular l-point to studying of the behaviour of a poly-analytic polynomial 
, i.e.
, (2)
, in the neighbourhood of ∞point; besides
where 
,
,
.
In the clauses [3, 4] it was got the criterion of availability of the not identical to the constant p.a. polynomial (2) of finite limited values in 
point; the speciality of getting this criterion being that, that allows describing all the sequences generating all finite limited values in point of the p.a. polynomial; and it means it allows describing all the sequences generating all the finite limited values of an arbitrary p.a. function in its any isolated singular l-point.
2. To formulate the corresponding results, let us enter some auxiliary notions; besides let us suppose further that for any element 
any sequence of complex numbers convergent to 
is considered to be known.
Definition 1
About the sequence of complex numbers 
we shall speak that it is made up of the sequences
, (3)
if every component of any of the sequences of the set (3) is the component (and the only one component) of the sequence , and there are no other components in this sequence.
Thus, either any two components of different sequences from (3) or two any components of different numbers of the same sequence from (3) will turn out to be different components of the sequence 
; the values of the components of these sequences being not considered.
Now, taking into account that any sequence of complex numbers has a convergent subsequence (to the finite element from 
or to ) and that any sequence possessing the property 
where 
is some not identical to the constant analytic polynomial from 
, is limited, and also that this polynomial has a nonempty finite collection of roots in , it is possible to formulate the following lemma.
Lemma 1
For any not identical to the constant analytical polynomial the sequence of complex numbers possesses the property when and only when it is made up of an arbitrary part of the finite set of any sequences, every one of which converges to one of the roots of the polynomial .
It is relevant, in addition to the notion of the composed sequence (or even just as another way to describe resembling situations), to formulate the notion of the sequence convergent to a finite set and the corresponding property (lemma 2).
Definition 2
About the sequence of complex numbers we shall speak that it converges to a nonempty infinite set 
, if for any set 
where 
is any 
-neighbourhood of the point 
, there is such a number 
that for any natural 
, if 
, then 
; if the sequence converges to 
, but doesn´t converge to any of its eigenparts, then we shall speak about the sequence , that it converges to all the set .
The following lemma 2 is evident.
Lemma 2
Any sequence of complex numbers possesses the following property , where is some not identical to the constant analytical polynomial, when and only when it converges to the set (not obligatory to the whole one) of all complex roots of the polynomial .
The fact, that the notions of the composed sequence and the sequence convergent to a set, is directly relevant to establishing the characteristic property of sequences generating finite limited values of p.a. polynomials in point 
, will be evident after establishing the following fact (theorem 1), that, however, is necessary to precede with some agreements about the nomenclature.
3. Let 
be an arbitrary p.a. polynomial of the form (2) and not identical to the constant, and let
In addition to it let us consider that informally depends both on and on 
, i.e. 
, but 
and 
.
Let us mention that it won´t violate the reasoning generality, as for any not identical to the constant polynomial from the ring 
or from the ring , its cluster set in point is settled with one element.
Let it further be that 
.
Besides (for the nomenclature brevity sake) let us agree to add formal summands with trivial coefficients to any from the polynomials 
as far as possible, and let us consider from now that 
, but at least one of the coefficients 
is nonzero.
Let us denote now a p.a. polynomial composed of all monomials of the p.a. polynomial (2) of the form 
by the symbol 
, i.e.
.
It is evident that
,
where 
is a not identical to the constant analytical polynomial from 
, then the following statement makes sense.
Theorem 1
Every sequence of complex numbers generating a finite point of a cluster set in point of the not identical to the constant p.a. polynomial possesses the following property: the sequence 
converges to the set of all roots of a unit module of the analytic polynomial 
.
Deduction
Let be a p.a. polynomial differing the not identical to the constant one, and let a sequence of complex numbers possesses properties:
, and (
) converges to the finite limit, but then 
, but 
, that proves the theorem.
Complementation
If a polynomial 
constructed not for the identical to the constant p.a. polynomial has no roots of a unit module, then 
.
Theorem 1 allows concluding that every sequence of complex numbers generating a finite limited value of the not identical to the constant p.a. polynomial in point is made up of a part of a finite set of sequences (their own for every ) 
of which every one converges to , simultaneously "ranging" in the direction set by a straight line from the finite collection (for every a straight line from this collection is its own and it is possible to consider that it passes through the point 
). The "structure" of every of the sequences was studied in [3] and [4], that allows formulating the following theorem; the situation for b.a. functions is fully considered in [3, 7].
Theorem 2
For any p.a. function of poly-analyticity order 
and its any isolated singular l-point 
there is a finite collection of expressions of the following form:
, (4)
where complex numbers 
as well as the coefficients of the polynomials 
from the ring 
, are fully determined by the coefficients of analytic components of the p.a. function 
corresponding to the p.a. function and the point , besides 
; this collection of expressions (4) possessing the following properties:
1) For any finite limited value in the point of this p.a. function any generating this point sequence is made up of a part of the collection of sequences 
with 
, or
with 
, (5)
being obtained by substitution into the expressions (4) of some convergent real sequences 
and 
; converging to some finite, and - to infinite limits;
2) Vice versa: for any convergent real-valued sequences 
and 
, where converges to an arbitrary finite, and - to infinite limits, every of the sequences (5) generates a finite limited value of the p.a. function in its isolated singular l-point 
(with or accordingly).
References
- Balk M.B. Poly-Analytic Functions and their Synthesis // INT. Modern Mathematical Problems. Fundamental Directions. - M., 1991. - V. 85. - pp. 187-254.
- Balk M.B. Polyanalytic Functions. Mathematical Research. - Berlin: Akad. - Verlag, - 1991. - V. 63.
- Gomonov S.A. About Application of Algebraic Functions to the Research of Cluster Sets in Point of Poly-Analytical Polynomials. // Some Questions of Theory of Poly-Analytic Functions and Their Synthesis. - Smolensk SSPI, 1991. - pp.16-42.
- Gomonov S.A. About Structure of Cluster Sets of Poly-Analytical Functions in Isolated Singularities. // Mathematica Montisnigri. - Podgoritsa, 1996. - V. 5(95). - Annals of Chernogoria University. - pp. 27-64.
- Gomonov S.A. About Characteristic Properties of Sequences Generating Finite Elements of Cluster Sets in Point of Poly-Analytical Polynomials. // Research on Boundary Problems of Complex Analysis and Differential Equations: Interuniversity Collection of Scientific Papers. / SSPU. - Smolensk, 1999. - pp. 34-39.
- Gomonov S.A. On the Sohotski-Weierstrass theorem for polyanalytic functions // European Journal of Natural History.- London, 2006, N2.- pp. 83-85.
- Gomonov S.A. About Some Methods of Research of Cluster Sets of Bi-Analytical Functions in Their Isolated Singularities. // Research on Boundary Problems of Complex Analysis and Differential Equations. / Smolensk State University. - Smolensk, 2006, N7, pp. 38-58.
The article is admitted to the International Scientific Conference «Actual Problems of Science and Education», Cuba, 2007. March 20-30; came to the editorial office on 22.12.06