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]
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.
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.
For any function given in an unlimited set (i.e. ) and for any polynomial
where and .
For any function given in an unlimited set and for any polynomial , , the congruence is valid:
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).
For any function with and any the congruence is valid:
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 . Let us further suppose that .
For any complex-valued functions and with unlimited
with , then ; if , then
The following fact arises from Property 5 (generalizing of Property 2).
For any complex-valued functions , and defined in one and the same unlimited set of the plane , if
For any complex-valued functions , and defined in an unlimited set of the plane , if
The analogous property can be formulated for complex-valued functions of two real variables as well.
For any complex-valued functions , and of two real variables and with coinciding domains, for which the point is a cluster one, if
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)
are corresponding Laurent series expansion coefficients in the neighborhood of the point of the analytic components and of the b.a. function , i.e.
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 .
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 .
It is obvious that for any b.a. function given in some , if the corresponding to it function , then always.
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:
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 ) 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.
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 .
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 .
If , then either a straight line (augmented by ∞ point) or a singleton .
The following theorem of Sokhotsky-Weierstrass type was formulated in , but without its full substantiating [5, 6, 10].
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 (i.e. or is a straight line of the plane .
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:
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
(besides, on its proper inception ), we infer the following theorem [see 10]:
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 .
- Colligwood E., Lovater A. Theory of Cluster Sets. - M.: Mir/World, 1971. - p. 312.
- Balk M.B. Polyanalytic 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.- Vol. 63.
- Balk M.B., Polukhin A.A. Cluster Set of Single-Valued Analytic Function in Its Isolated Singularity. // Smolensk Mathematical Collection / SSPI. - Smolensk, 1970, V.3. - pp. 3-12.
- 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. Theorem of Sokhotsky-Weierstrass for Poly-Analytic Functions. // Papers of Mathematical Institute. - Minsk, 2004. - V.12, N1. - pp. 44-48.
- Gomonov S.A. On the Sohotski-Weierstrass theorem for polyanalytic functions // European Journal of Natural History.- London, 2006, N2.- p. 83-85.
- Gomonov S.A. About Cluster Sets of Multi-Valued Mappings of Topological Spaces. // Reports of AS USSR. - M., 1989. - V.306, N1. - pp. 20-24.
- Gomonov S.A. About Some Methods of Research of Cluster Sets of Bi-Analytic 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