A representation formula for slice regular functions over slice-cones in several variables

Abstract

The aim of this paper is to extend the so called slice analysis to a general case in which the codomain is a real vector space of even dimension, i.e. is of the form ${\mathbb{R}}^{2n}$. We define a cone ${\mathcal{W}}_{\mathcal{C}}^d$ in $[End({\mathbb{R}}^{2n}){]}^d$ and we extend the slice-topology ${\tau }_s$ to this cone. Slice regular functions can be defined on open sets in $\left({\tau }_s,{\mathcal{W}}_{\mathcal{C}}^d\right)$ and a number of results can be proved in this framework, among which a representation formula. This theory can be applied to some real algebras, called left slice complex structure algebras. These algebras include quaternions, octonions, Clifford algebras and real alternative $\ast$-algebras but also left-alternative algebras and sedenions, thus providing brand new settings in slice analysis.

29 pages

References in corpus (1)

• A new approach to slice analysis via slice topology

Cited by in corpus (2)

• A new approach to slice analysis via slice topology
• Almansi-type decomposition for slice regular functions of several quaternionic variables