General methods of finding an explicit cluster set of any bi-analytic function in its arbitrary isolated singularity.
1. Let it be required to find a cluster set 
[1-6] of an arbitrary bi-analytic function (b.a. function) [2-6]
(1)
in its isolated singular point 
, where the functions 
and 
are arbitrary analytic functions in a deleted neighborhood 
of the point (so called analytic components of the b.a. function 
).
The first step in finding should be most reasonable to make determining the fact if the point is an isolated singularity at least for one of the functions - analytic components of the function , as the following result [2-4] makes sense.
Lemma 1
If the point is an essential isolated singularity for at least one of the functions 
, then is total (i.e. coincides with 
, where in a b.a. function (1) given in a deleted neighborhood of the point 
.
2. Lemma 1 allows considering only those cases, when the isolated singularity of the b.a. function is not an essential singularity for and ; besides, in this case we are always successful in converting finding to finding a cluster set in the point 
of a bi-analytic first-degree polynomial (i.e. a function from the ring 
) relative 
or a function of the form 
[5, 6, 10]. The following rather simple properties will help such changing.
Property 1
For any function 
given in an unlimited set 
(i.e.
) and for any polynomial 
,
in particular
,
where 
and 
.
Property 2
For any function given in an unlimited set and for any polynomial 
,
, the congruence is valid:
.
Property 3
For any defined and continuous in 
function , any point and any function 
of complex variable 
the congruence is valid:
,
in particular, if is an arbitrary polynomial 
from , then 
,
where is a function defined in a deleted neighborhood of the point (a natural agreement that 
being accepted).
Property 4
For any function with 
and any 
the congruence is valid:
.
Points
Property 4 evidently always allows passing on from the research of a cluster set of an arbitrary b.a. function in its arbitrary isolated singularity to the research of a cluster set of a b.a. function in its isolated singularity 
or 
; besides, a transition to any of these two variants is always possible [10]. Let us further suppose that .
Property 5
For any complex-valued functions 
and 
with unlimited
, if
with , then 
; if , then
.
The following fact arises from Property 5 (generalizing of Property 2).
Property 6
For any complex-valued functions , and 
defined in one and the same unlimited set of the plane , if
,
then 
.
Property 7
For any complex-valued functions , and defined in an unlimited set of the plane , if
,
then
The analogous property can be formulated for complex-valued functions of two real variables as well.
Property 8
For any complex-valued functions 
, 
and 
of two real variables 
and 
with coinciding domains, for which the point 
is a cluster one, if
,
then
3. Let be a b.a. function of the form (1) given in a deleted neighborhood of the point ; point not being an essential singularity for analytic components of this b.a. function, then the following representation in some
is valid for it:
F(z)=F∞(z) + Fω(z)+ F0(z) (2)
where
; (3)
;
,
where 
and
are corresponding Laurent series expansion coefficients in the neighborhood of the point of the analytic components and of the b.a. function , i.e.
, where
;
, where
.
As it is evident that 
, then 
and it is natural to consider two possible situations for F∞(z):
a) 
and then 
, where 
is a unit circle of the plane C,
. In this case ∞ point for is called an isolated singular О-point [6-8]. But if 
and, moreover,
, then 
and ∞ point is called a removable isolated point of the function [6-8].
b) 
. In this case the properties of this b.a. polynomial 
will be determining in the structure
; the influence of the function 
on will be equal to a parallel translation of the set 
[10].
Lemma 2
Let a bi-analytic in a deleted neighborhood function be represented in this domain in the form (2) (besides for the corresponding function 
, then for availability of finite elements in the cluster set it is necessary that:
-
; -
; - but if
, and 
, then 
.
Points
It is obvious that for any b.a. function given in some , if the corresponding to it function 
, then 
always.
Complementation
Let for a b.a. function the corresponding to it function 
satisfies the conditions
(1)-(2) of Lemma 2, then, if contains finite elements, then any sequence 
generating any of them satisfies the following condition:
.
Points
The facts which are analogous to Lemma2 and the complementation to it, are obviously valid for the isolated singularity as well; besides it is easy to see that the substitution of by the corresponding constant and the substitution of z in 
by the function 
with multiplying function 
by the function of the form 
(where 
is selected properly [10]) allows passing on from to 
, where 
is a b.a. function defined in a deleted neighborhood , and moreover is a b.a. polynomial from
.
4. The following two lemmas will give us an opportunity to substantiate the theorem of Sokhotsky-Weierstrass type for bi-analytic functions fully. Besides it is worth underlying that the above points allow further formulating of many facts without loss of reasoning generality only for the case of isolated singular ∞ point of b.a. polynomials.
Lemma 3
For any bi-analytic polynomial 
from 
there is a certain augmented with ∞ point straight line of the plane 
, in which there are all points from 
.
Lemma 4
Let it for a bi-analytic polynomial
,
where 
, be known that 
is any its limited in ∞ point value, which is generated by a sequence , then every sequence of the form
,
generates a finite point
,
of the cluster set .
Complementation
If 
, then either a straight line (augmented by ∞ point) or a singleton 
.
The following theorem of Sokhotsky-Weierstrass type was formulated in [4], but without its full substantiating [5, 6, 10].
Theorem 1
A cluster set in any point of any b.a. function (1)
, given in a deleted neighborhood of this point a either consists of one point of the plane , or represents a circle of the plane , or is total, i.e. coincides with . Vice versa, for any point and any set 
which is either a circle, or a straight line (augmented with ∞ point), or a one-element set (i.e.
with 
), or is the whole plane , there is a defined in a deleted neighborhood of the point a bi-analytic function , for which
.
5. Let us obtain the criterion which allows distinguishing if the point
is a pole for the b.a. function given in a
or
.
To do it let us pass on to the search of a cluster set in 
point corresponding to the function and, in general terms, an arbitrary bi-analytic polynomial of the form (3), as it is precisely this situation which a general case with 
is converted to; considering, of course, that the conditions 
and 
are hold.
1) Having applied Property 1 with 
to , we´ll infer:
;
supposing further that 
(we´ll mean any of its two values by the radical), and then applying Property 3 we infer:
,where 
.
Applying Properties 2 and 5 to 
we infer:
,
where 
is an integral part of a rational function
.
If 
, then we get immediately that
,
and all the sequences generating all finite points from are of the following form:
,
where 
and 
are arbitrary real-valued sequences possessing the property:
, and 
.
2) Denoting 
and passing on to real variables x and y, we infer
,
because (see Lemma 2) the coefficient 
can be thought as a real number (if not - 
).
3) Having substituted the variables 
, we´ll infer
and having applied not more than 
times Property 8 [5, 10], we´ll "delete" all the monomials containing 
symbol (in the corresponding true degree), excluding 
, and we´ll get that
.
4) Dividing the real and imaginary parts of the polynomial
, where
(besides, on its proper inception 
), we infer the following theorem [see 10]:
Theorem 2
Let be a nontrivial polynomial of the form (3). Then 
has finite elements when and only when , in addition
;
besides, all the sequences generating all finite points from , are given by the following formula:
,
where 
, 
are arbitrary real-valued sequences, and also converges to the finite limit 
, and - to .
References
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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