Geometric function theory over quaternionic slice domains

TL;DR

This paper explores the extension of quaternionic slice regular functions to non-symmetric slice domains, revealing new geometric phenomena and properties that differ from the symmetric case, including zero set behavior and function representations.

Contribution

It develops the geometric function theory for slice regular functions on non-symmetric slice domains, a significant advancement beyond previous symmetric domain studies.

Findings

Zero sets can differ drastically from symmetric cases

Includes differential, algebraic, and topological properties

Provides integral and series representations

Abstract

The theory of quaternionic slice regular functions was introduced in 2006 and successfully developed for about a decade over symmetric slice domains, which appeared to be the natural setting for their study. Some recent articles paved the way for a further development of the theory: namely, the study of slice regular functions on slice domains that are not necessarily symmetric. The present work is a panorama of geometric function theory in this new context, where new phenomena appear. For instance, the nature of the zero sets can be drastically different than in the symmetric case. The work includes differential, algebraic, topological properties, as well as integral and series representations, of slice regular functions over slice domains.