Scientific journal
European Journal of Natural History
ISSN 2073-4972


Gomonov S. A.
The classification of isolated singular points [5-7] of poly-analytic functions [3-5, 10], appearing due to the theorem of Sokhotsky- Weierstrass type [1, 2-5, 10], does not exclude the possibility of its being worked out in detail for some particular kinds of these functions, for example, for those of them, the structure of limit sets in l-points and o-points of which can be specified. In particular, such a specification can be offered for tri-analytic functions [8, 9], as a theorem, which is a nontrivial specifying and adding to a corresponding theorem for poly-analytic functions, is possible to be established for them.


A limit set in an arbitrary point  of any tri-analytic function

,          (1)

prescribed in some deleted neighborhood of this point, consists of a point (finite or infinite), or a line of the plane , or grouping of two lines, or a half-line, or a parabola of the second order, or a circle, or  a Pascal snail of any of the four kinds with the isolated point elimination, or finally, is total, i.e. coincides with the whole plane ; here and further the lines, half-lines and parabolas of the plane  being considered as completed with  point.

Vice versa, for a point  and a set  , which is a line or a grouping of two arbitrary lines (parallel or concurrent), or a half-line, or a parabola of the second order, or a circle, or a Pascal snail of any of the four kinds with the isolated point elimination, or finally, for  with c as an arbitrary point of the plane , a tri-analytic function  with analytical  components - rational functions from the field , for which , exists.


If the point  is an essentially singularity for at least one analytic function , it is the only case when  is total, i.e. coincides with  [3-4, 7].

Now let the point  be not an essential singularity for none of the analytic components  of the tri-analytic function , but then a representation (3) of [10] takes place, if , or an analog representation for the function  in some deleted neighborhood of the null point. Now the structure of the limit set  will be fully defined by the corresponding functions  и  [5, 10] (the function  can be ignored, as ).

If , and , then ; and if  and  is not identical to the constant, then  where  is a unit circle and  is a first or second degree polynomial [5, 9, 10]  from the ring . In the first case   is, obviously, a circle; and in the second one - a Pascal snail of any of the four kinds [11] or again a circle.

The given conclusion for the second case results from an easily fixed auxiliary fact.


The image of the circle  with mapping given by a second degree polynomial  will be:

with  - a circle of radius  and centre ;

with  - a Pascal snail (with its isolated point elimination), the snail having a juncture if ; the snail being a cardioid  with a cuspidal point if ; the snail having the only point in which right and left tangents coincide (and the snail will have two flex points) if если ; the snail having the only one tangent in its every point , but not having a cuspidal point if .


Taking and , let us assume the polynomial ,  as follows


Now let us write a nonvanishing number   in an exponential form  with , , and finally, reduce the polynomial to the form


Now, obviously, it is enough to clear up what is the image of the unit circle  with mapping given by the following auxiliary polynomial: , with ; but if  with , then ,

that means [11]:  is a Pascal snail (with the isolated point elimination) with the parameters  and , which is biased to the vector . The lemma is proved.

Now let the function  be not identical to the constant. Then  or  is a join of a finite number of nontrivial polynomial lines augmented with point. The last situation for tri-analytic functions needs to be specified (1). To do it we have to change the function with the corresponding constant [5, p. 58], then apply the described in [5] (theorem 3.1) poly-analytic property deflation mode of the function .  The made transformations allow to make sure that the above join can be a line, or a grouping of two lines, or a half-line, or a parabola of the second order (with adding to every of these sets point).

As a supplement to the deduction of the first part of the theorem it is worth taking into account the first part of the theorem 3.2 from [5], and also the possibility to transform the case  to the case  (and vice versa) by means of a simple change of the variables - changing symbol  into the expression  and multiplying the given function by the fraction with the further considering of this factor´s possibility to converge to one of not more than two complex numbers ([5], p. 57).

The deduction of the second part of the theorem becomes easy when denoting concrete samples of realization of all the numerated in the theorem cases, the structuring of the corresponding samples being obvious or being contained in [5,6].


It is obvious that the given theorem allows to offer a more delicate classification of isolated singularities of tri-analytic functions compared to the classification of these points for poly-analytic general functions, and namely, to excel, alongside with the essential singularity, removable (on continuity) singularity and a pole, four kinds of l- and o-points more.


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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