Synergetics. Etudes 70.
Is Dedicated to Centennial of Birthday of
Outstanding Scientist, Professor
Basin Abram M.
Basina G. I., Basin M. A.
SIC ”Synergetics” of Saint-Petersburg
Association of Scientist and Scholars.
Etude 5.
Algebraic and Spiral Complex Numbers.
April 24. 2010.
Here we
shall introduce one of possible modifications of complex numbers, using of
which allows, if it is necessary, to consider power functions of complex
variable with complex degrees of power as functions, having a single meaning,
and consequently to apply to there investigation all apparatus of contemporary
analysis.
Every
complex number has as minimum two possible representations: algebraic 
and exponential -
. To each
algebraic representation of complex number corresponds enumerable set of
exponential representations, in which value 
differs on 
. Shall we
suppose, following B. Riemann[1] and G. Weil[2], that there is any screw spiral
structure, crossing complex plane on positive axis 
with aspiring to zero step 
[3-5]. To every point of such spiral we shall
correspond spiral complex number, which may be described with the formula 
. Value 
characterizes
distance from the point of spiral surface to the axis, perpendicular to the
plane and passing through the point 
. At such
geometric interpretation all points, corresponding to the spiral complex numbers, having meanings of , differing
on lay on the
one straight line, which projection on plane
is one
algebraic complex number , where 
. To every spiral complex number corresponds
one algebraic complex number. But to every algebraic complex number correspond
enumerable set of spiral complex numbers. Shall we introduce in consideration
another complex plane, which corresponds to the field of algebraic complex
numbers, which will get, if we shall formal take the operation of logarithm
from every spiral number 
. Function 
and inverse to it 
are one-for-one functions, reflecting each on
other area of determination of spiral numbers and complex plane. Each horizontal
streak of the area with height 
corresponds
to one blade of spiral
.
Shall we consider in the area totality of
one-for-one complex linear reflections 
. As
coefficients of such reflections we shall take the fields of algebraic complex
numbers
. These
reflections are the one-for one reflections in all points 
, except 
.
Shall we
introduce another one-for-one reflection of complex plane 
on the area
of spiral complex numbers 
. Then we
shall receive one for one reflection
/ Constant 
also lays in the area of determination of
spiral numbers. So the multi-ciphered power function with complex numbers of
degree at the consideration of it in the area of determination of spiral
numbers is one-for-one reflection. So we can to do with it all operations how
with power functions, determining in the area of real varieties. Introduction
of spiral complex numbers creates the conditions for the development of power
geometry with algebraic complex power numbers. In any cases, when 
is integer we may work with projections of
spiral numbers on complex planes 
.
Being in frames of algebraic complex
numbers we have a risk to lose possible elements of volume of points, on which
reflects any power function given algebraic complex number. Arise
multi-ciphered and infinity-ciphered functions and with them concept of variety
of choice of any branch of multi-ciphered reflection.
It is the
problem of relation of built mathematically construction with real objects,
which are describing with this mathematically construction. We not always “see” the spiral complex
number, often we observe its projection on the complex plane 
and describing with usual power functions dynamics of system
“seems” us as bifurcation dynamics.
While built
by us one-for-one power functions are the exponents of the algebra of linear
reflections on the complex numbers, then in the area of determination of spiral
numbers naturally may be introduced multiplication of spiral complex numbers as
the spiral analog of summarizing of powers of exponents of corresponding algebraic
complex numbers. Analogous statement may be done about multiplication of power
functions of any spiral complex variable 
. Shall we
have 
-two power functions of spiral variable
.
Then
function 
also will be one-for-one power function from.
, где 
.
With
analogous matter we can prove, that exponentiation of power function in complex
degree 
generates
new degree function with power, equal
, and coefficient
representing spiral number, which is equal to the coefficient of primary spiral
number exponented in the degree .
It is more
difficult to introduce the summarizing
of spiral complex numbers, and the summarizing of power monomers. This
problem may be called the problem of Lagrange
[6]. We shall show only the ways of decision of this problem. Shall we
have two spiral numbers or two power functions of spiral variable 
. We must to
determine the sum of this spiral numbers or functions.
In the area
of spiral numbers to give determination of the sum if difficult. But we may to
project spiral numbers on the plane of algebraic complex numbers and to compute
the sum of two algebraic complex numbers. If we shall consider that sum of
algebraic complex numbers is the projection of spiral complex number, which is
the sum of spiral complex numbers, then our determinasion allows with the
accuracy of infinity values of meanings of 
,
districting one for one on 
, such sum
to determine. More detail this problem will be considered in th next Etude.
References.
1.Riemann B.
Theorie der Abelschen Funktionen. Borhardt’s Journ. für reine und
angewandte Math. 54.). 1857. Werke. Leipzig 1876.S.81-135.
2. Weyl H. Die Idee der Riemanischen Fläche. Leipzig-Berlin. 1913 (1-ste Aufl.). 1923 (2-te Aufl.). Stuttgart.1953 (3-te Aufl.)
3.Шабат Б. В.Введение в комплексный анализ. Ч.1. Функции одного переменного. М. 1976
4.Стоилов С. Теория функций комплексного переменного. В 2 томах. Основные понятия и принципы. М.:1962.
5. Басин М. А. Спиральные числа. Степенные особенности. Волны. Вихри. Грибовидные структуры. Транспортно-информационные системы. Международная междисциплинарная научно-практическая конференция: «Современные проблемы науки и образования». Керчь 27.06-4.07.2001. Ч.1 Харьков. 2001. С.12-13.
6.Арнольд В.И., Авец А. Эргодические проблемы классической механики. Ижевск 1999