Constructive description of monogenic functions in a finite-dimensional commutative associative algebra

TL;DR

This paper provides a constructive method to describe monogenic functions in a finite-dimensional commutative algebra using holomorphic functions, showing they possess derivatives of all orders.

Contribution

It introduces a new constructive description of monogenic functions in such algebras via holomorphic functions, extending understanding of their differentiability properties.

Findings

Monogenic functions are expressible through holomorphic functions of complex variables.

These functions have derivatives of all orders.

The description applies to functions in any finite-dimensional commutative algebra with idempotents.

Abstract

Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,e_3$ be elements of $\mathbb{A}_n^m$ 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 $xe_1+ye_2+ze_3$ where $x,y,z$ are real, and obtain a constructive description of all mentioned functions by means of holomorphic functions of complex variables. It follows from this description that monogenic functions have Gateaux derivatives of all orders.