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.