Monogenic Functions in Commutative Algebras

Abstract

Let An be an arbitrary n-dimensional commutative associative algebra over the field of complex numbers. Let e1 = 1, e2, e3 be elements of An which are linearly independent over the field of real numbers. We consider monogenic (i.e., continuous and differentiable in the sense of Gateaux) functions of the variable xe1 + ye2 + ze3, where x, y, z are real, and obtain a constructive description of all mentioned monogenic functions by means of holomorphic functions of complex variables. It follows from this description that monogenic functions have Gateaux derivatives of all orders. The relations between monogenic functions and partial differential equations are investigated.