Differentiable functions in a three-dimensional associative
noncommutative algebra

We consider a three-dimensional associative noncommutative algebra
Ã_{2} 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)=f_{1}(ξ_{1},ξ_{2},ξ_{3})I_{1}+f_{2}(ξ_{1},ξ_{2},ξ_{3})I_{2}+f_{3}(ξ_{1},ξ_{2},ξ_{3})ρ
of the variable
ζ=ξ_{1}I_{1}+ξ_{2}I_{2}+ξ_{3}ρ,
where ξ_{1}, ξ_{2}, ξ_{3} are
independent complex variables and f_{1},
f_{2}, f_{3} 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.

23 Nov 2021