Ввeдение в методы теории функций пространственного компексного переменного

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Coordinate axes of materialized reality

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Mathematical science is actively developed since the invention of a Cartesian coordinate system $<$X,Y,Z$>$. Up to now, the Cartesian coordinate system forms the world view of space matter, its structure and processes occurring in it.


Scientific and technical progress continuously adjusts the coordinate system. Investigation of the structure of matter came to the foreground scene and Cartesian coordinates can be regarded as a first attempt in this direction.

Geometrically Cartesian coordinates are represented by three mutually perpendicular lines outgoing from the zero-dimensional point as a point of intersection of these lines. The geometry of such algebra does not correspond to the three-dimensional space. This deficiency was not corrected in the algebra of vector analysis.

As a result, in science there is no unambiguous link between the representation of numbers and its representation in the space, as this has been achieved in the algebra of complex analysis Cauchy. Planar algebra of complex analysis provides integrated cohesion between the coordinates on the plane (x, y).


Coordinate representation by the imaginary unit gives a numerical expression of a point on the plane


Point location is determined by the distance from the origin as magnitude $r$, and angle $\varphi $.

i -- is number $i=\sqrt{-1} $ (is called imaginary unit).

A more detailed study of square root extraction operation of -1 and Euler's formula gives a representation of number in three-dimensional space


Algebra of spatial number MATH complies with the operations of real numbers, it eliminates noncommutativity of the multiplication, which is used in many applications.

Complex space MATHcontains subspace of zero divisors; their presence is equivalent to the presence of root extraction operation from 0 in the analysis of real numbers.

Zero divisors are defined by next equation:


Multipliers do are zero divisors, they are numbers non-equal zero, but their product is zero.


Zero devisors are numbers having magnitude equal to square root of zero and isolated argument.

Great Lorenz developing SRT implicitly introduced these numbers.


When V=C we have MATH

So, complex space contains a subspace. In other words, the fact of the complex structure of the material world defined. The theory of relativity and its mathematical description introduced structure of the material world, but has not researched. Since the definition of the Cartesian space it is the second attempt to study the structure of the material world.

SRT was developed at the moment when there is no research of micro-particles space. As a result research of the micro-particles structure and material space microcosm ware postponed and the physics developed space geometrization.

The research of the space geometry end ups with definition of space operators of micro-particles.

For example space operator of proton is next:


This is proton space. Direction coefficients are calculated according to the proton structure as combinations of quarks.

Technical sciences determined that the matter in all its forms exists in the variation of constants: G - gravitational constant, and C - the speed of light, Planck's constant.

A set of these constants give next parameters.

MATH - Planck mass MATHg.

Limiting dimension


The mass exceeds the mass of micro-particles $10^{20} $times.

Limiting dimension geometrically defines the limit of epsilon tunnel in a medium filled with the Planck mass.

These two parameters limit the matter. In addition to the closure of the material medium, they make the calculation, by the Newton equation, between the two limiting masses at the limiting distance.


Reduction gives the well-known formula.

The mass of micro-particle is determined by a similar formula


We define the mass of micro-particle as a result of interaction between the two limiting Planck masses via exchange of mass $S_{g} c^{2} $


As a first approximation, we have cohesion between medium, micro-particle mass and exchange mass between the interacting quantum Planck masses.


As a result, we obtain all the necessary parameters for numerical calculation of the energy of space.

Coming back to the space of a proton, we transform the formula of the geometrization of space in the energy space.


To determine the four unknown directions, we set up a system of spaces in geometrized space, introducing an additional spaces MATH

We have a system of Geometrized spaces of stable micro-particles: a proton, lambda hyperon, pion and electron.

Solution of the system gives energy directions:


Finally, we have the formula of energy of the material space (based on example of proton energy).


Energetic material space mathematically and physically is a four dimensional. Coordinate axes X, Y, Z are isolated in the material space directions, which represent the direction of the light beam.

Light rays geometrically in the material world represent $\varepsilon $ - epsilon tunnels, and exchange mass is going through them with speed V = C, the exchange mass binds the central core of the system with the external space <X,Y,Z>.

In the center of the coordinate axes we have sphere having real radius equalMATH, from which coordinate axes are outgoing.


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