A unified notion of regularity in one hypercomplex variable

TL;DR

This paper introduces a comprehensive framework for regularity of functions valued in hypercomplex algebras, unifying existing theories and developing new ones, especially over quaternions.

Contribution

It proposes a general notion of regularity in hypercomplex analysis that encompasses and extends existing concepts like monogenic, slice-monogenic, and Fueter-regular functions.

Findings

Unifies various notions of regularity in hypercomplex analysis.

Develops a new theory of regularity over quaternions.

Shows that existing concepts are special cases of the new framework.

Abstract

We define a very general notion of regularity for functions taking values in an alternative real $*$-algebra. Over Clifford numbers, this notion subsumes the well-established notions of monogenic function and slice-monogenic function. Over quaternions, in addition to subsuming the notions of Fueter-regular function and of slice-regular function, it gives rise to an entirely new theory, which we develop in some detail.