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A unified theory of regular functions of a hypercomplex variable

Riccardo Ghiloni, Caterina Stoppato

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.

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 -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 -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 -regular functions over an alternative but nonassociative -algebra, such as the real algebra of octonions.
Paper Structure (26 sections, 62 theorems, 231 equations)

This paper contains 26 sections, 62 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Hypercomplex subspaces and monogenic functions
  3. Alternative real $*$-algebras
  4. Hypercomplex subspaces
  5. Monogenic functions on hypercomplex subspaces
  6. Properties of monogenic functions on hypercomplex subspaces
  7. Monogenic polynomial maps on a hypercomplex subspace
  8. Integral representation of functions on a hypercomplex subspace
  9. Properties of the reproducing kernel and harmonicity
  10. Series expansions of monogenic functions on a hypercomplex subspace
  11. Regularity in hypercomplex subspaces
  12. $T$-fans
  13. $T$-regularity
  14. Integral representation
  15. Polynomial regular functions
  16. ...and 11 more sections

Key Result

Lemma 2.8

Fix $m\geq1$. If $v_1,\ldots,v_m\in{\mathbb{S}}_A$ are linearly independent, then $(1,v_1,\ldots,v_m)$ can be completed to a fitted basis of $A$.

Theorems & Definitions (184)

  • Example 2.4: Dual quaternions
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Definition 2.9
  • Remark 2.10
  • Example 2.11: Division algebras
  • Example 2.12: $C\ell(0,3)$
  • Example 2.13: Dual quaternions
  • ...and 174 more