Scientific journal
European Journal of Natural History
ISSN 2073-4972

CLUSTER SETS OF BIANALYTIC FUNCTIONS IN THEIR ISOLATED SINGULARITIES

Gomonov S.A.
Brief

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:

  1.  ;
  2.  ;
  3.  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 (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:

,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

  1. Colligwood E., Lovater A. Theory of Cluster Sets. - M.: Mir/World, 1971. - p. 312.
  2. Balk M.B. Polyanalytic Functions and Their Synthesis. // INT. Modern Mathematical Problems. Fundamental Directions. - M., 1991. - V.85. - pp. 187-254.
  3. Balk M.B. Polyanalytic functions. Mathematical research.- Berlin: Akad.- Verlag, 1991.- Vol. 63.
  4. 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.
  5. 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.
  6. 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.
  7. Gomonov S.A. Theorem of Sokhotsky-Weierstrass for Poly-Analytic Functions. // Papers of Mathematical Institute. - Minsk, 2004. - V.12, N1. - pp. 44-48.
  8. Gomonov S.A. On the Sohotski-Weierstrass theorem for polyanalytic functions // European Journal of Natural History.- London, 2006, N2.- p. 83-85.
  9. 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.
  10. 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