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

TL;DR

This paper extends the theory of monogenic functions to several hypercomplex variables over real alternative *-algebras, establishing fundamental properties like integral formulas and extension theorems.

Contribution

It introduces the study of monogenic functions over real alternative *-algebras in multiple variables, unifying and generalizing classical hypercomplex analysis theories.

Findings

Established Bochner-Martinelli formula for these functions

Proved Plemelj-Sokhotski formula in this setting

Extended Hartogs' theorem to hypercomplex variables

Abstract

The notion of monogenic (or regular) functions, which is a correspondence of holomorphic functions, has been studied extensively in hypercomplex analysis, including quaternionic, octonionic, and Clifford analysis. Recently, the concept of monogenic functions over real alternative $\ast$-algebras has been introduced to unify several classical monogenic functions theories. In this paper, we initiate the study of monogenic functions of several hypercomplex variables over real alternative $\ast$-algebras, which naturally extends the theory of several complex variables to a very general setting. In this new setting, we develop some fundamental properties, such as Bochner-Martinelli formula, Plemelj-Sokhotski formula, and Hartogs extension theorem.