A unified theory of regular functions of a hypercomplex variable

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 $\ast$-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 $\ast$-algebra, such as the real algebra of octonions.

76 pages, to appear in the Bulletin des Sciences Mathématiques

References in corpus (10)

• Slice regular functions on real alternative algebras
• Poles of regular quaternionic functions
• The algebra of slice functions
• The zero sets of slice regular functions and the open mapping theorem
• Slice regular functions in several variables
• Volume Cauchy formulas for slice functions on real associative *-algebras
• Some properties for quaternionic slice-regular functions on domains without real points
• Cauchy-Riemann operators and local slice analysis over real alternative algebras
• Generalized partial-slice monogenic functions
• A unified notion of regularity in one hypercomplex variable

Cited by in corpus (1)

• Monogenic functions over real alternative *-algebras: the several hypercomplex variables case