A unified theory of regular functions of a hypercomplex variable

TL;DR

This paper introduces a comprehensive unified framework for regular functions across various hypercomplex algebras, integrating multiple existing theories and extending to nonassociative structures like octonions.

Contribution

It develops the theory of T-regular functions, unifying and generalizing existing classes of hypercomplex regular functions, with foundational results and integral formulas.

Findings

Includes integral formulas and series expansions for T-regular functions.

Proves an Identity Principle and a Maximum Modulus Principle.

Extends the theory to nonassociative algebras like octonions.

Abstract

This work proposes a unified theory of regularity in one hypercomplex variable: the theory of $T$-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises Fueter-regular functions, slice-regular functions and a recently-discovered function class. In the special case of Clifford-valued functions of one paravector variable, it encompasses monogenic functions, slice-monogenic functions, generalized partial-slice monogenic functions, and a variety of function classes not yet considered in literature. For $T$-regular functions over an associative $*$-algebra, this work provides integral formulas, series expansions, an Identity Principle, a Maximum Modulus Principle and a Representation Formula. It also proves some foundational results about $T$-regular functions over an alternative but nonassociative $*$-algebra, such as the real algebra of octonions.