Extension results for slice regular functions of a quaternionic variable

TL;DR

This paper introduces a new representation formula for slice regular quaternionic functions, enabling their extension from balls to larger axially symmetric domains, akin to holomorphic functions' domains of holomorphy.

Contribution

It presents a novel representation formula that allows extending slice regular functions to broader axially symmetric domains.

Findings

Representation formula for slice regular functions.

Extension of properties to axially symmetric domains.

Axially symmetric domains act as domains of holomorphy.

Abstract

In this paper we prove a new representation formula for slice regular functions, which shows that the value of a slice regular function $f$ at a point $q=x+yI$ can be recovered by the values of $f$ at the points $q+yJ$ and $q+yK$ for any choice of imaginary units $I, J, K.$ This result allows us to extend the known properties of slice regular functions defined on balls centered on the real axis to a much larger class of domains, called axially symmetric domains. We show, in particular, that axially symmetric domains play, for slice regular functions, the role played by domains of holomorphy for holomorphic functions.