Differentiable functions in a three-dimensional associative noncommutative algebra

Year 2022, Volume: 6 Issue: 1, 66 - 73, 31.03.2022

Tetiana Kuzmenko , Vitalii Shpakivskyi

https://doi.org/10.31197/atnaa.912344

Cited By: 2

Abstract

We consider a three-dimensional associative noncommutative algebra Ã2 over the field C, which contains the algebra of bicomplex numbers B(C) as a subalgebra. In this paper we consider functions of the form Φ(ζ)=f1(ξ1, ξ2,ξ3)I1+ f2(ξ1, ξ2,ξ3)I2+ f3(ξ1, ξ2,ξ3)ρ of the variable ζ= ξ1I1+ ξ2I2+ ξ3ρ, where ξ1, ξ2, ξ3 are independent complex variables and f1, f2, f3 are holomorphic functions of three complex variables. We construct in an explicit form all functions defined by equalities dΦ =dζ·Φ´(ζ) or dΦ = Φ´(ζ) ·dζ. The obtained descriptions we apply to representation of the mentioned class of functions by series. Also we established integral representations of these functions.

Keywords

noncommutative algebra, differentiable function, Cauchy-Riemann conditions, constructive description, power series, integral representation

Supporting Institution

Budget program "Support for the development of priority areas of research"

Project Number

KPKVK 6541230

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