Differentiable functions in a three-dimensional associative noncommutative algebra

Open Access

Original research article

Author(s):

Kuzmenko Tetiana^a, Shpakivskyi Vitalii^b

^a Department of Fundamental Sciences, Zhytomyr Military Institute, Zhytomyr, Ukraine.
^b Department of Complex Analysis and Potential Theory, Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine.

Advances in the Theory of Nonlinear Analysis and its Applications 6 (1), 66-73.

Received: April 9, 2021

Accepted: November 21, 2021

Published: November 23, 2021

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.

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APA Style

Tetiana, K., & Vitalii, S. (2022). Differentiable functions in a three-dimensional associative noncommutative algebra. Advances in the Theory of Nonlinear Analysis and its Applications , 6 (1), 66-73.