Domains of existence of slice regular functions in one quaternionic variable

TL;DR

This paper characterizes the domains where slice regular functions in quaternions exist, showing they are exactly the slice-regularly convex, 2-path-symmetric, slice-open sets, extending classical complex analysis results.

Contribution

It provides a complete characterization of domains of slice regularity in quaternions, including a Cartan-Thullen type theorem and the concept of 2-path-symmetry.

Findings

All 2-path-symmetric slice-open sets are domains of slice regularity.

A slice-open set is a domain of existence for some slice regular function iff it is slice-regularly convex.

The paper proves an interpolation theorem of independent interest.

Abstract

Recently, we introduced domains of slice regularity in the space $\mathbb{H}$ of quaternions and also proved that domains of slice regularity satisfy a symmetry with respect to paths, called $2$-path-symmetry. In this paper, we give a full characterization by showing that all $2$-path-symmetric slice-open sets are domains of slice regularity. In fact, we will prove a counterpart of the Cartan-Thullen theorem for slice regular functions, namely that a slice-open set is a domain of existence for some slice regular function if and only if it is a domain of slice regularity, if and only if it is slice-regularly convex, if and only if it is $2$-path-symmetric. As a tool, we also prove an interpolation theorem of independent interest.