Differentiable functions in a three-dimensional associative noncommutative algebra

Authors : Tetiana KUZMENKO, Vitalii SHPAKİVSKYİ

Pages : 66-73

Doi:10.31197/atnaa.912344

View : 12 | Download : 4

Publication Date : 2022-03-31

Article Type : Research Paper

Abstract :We consider a three-dimensional associative noncommutative algebra Ã2 over the field C, which contains the algebra of bicomplex numbers Binsert ignore into journalissuearticles values(C); as a subalgebra. In this paper we consider functions of the form Φinsert ignore into journalissuearticles values(ζ);=f1insert ignore into journalissuearticles values(ξ1, ξ2,ξ3);I1+ f2insert ignore into journalissuearticles values(ξ1, ξ2,ξ3);I2+ f3insert ignore into journalissuearticles values(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´insert ignore into journalissuearticles values(ζ); or dΦ = Φ´insert ignore into journalissuearticles values(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