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arXiv · 2408.01523

A unified theory of regular functions of a hypercomplex variable

Riccardo Ghiloni·Caterina Stoppato
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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.

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BibTeXRIS

Riccardo Ghiloni, Caterina Stoppato. 2024-08-02. A unified theory of regular functions of a hypercomplex variable. https://doi.org/10.1016/j.bulsci.2026.103794

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