Singularities of slice regular functions

TL;DR

This paper extends the theory of singularities of slice regular quaternionic functions beyond balls centered at zero, introducing Laurent-type expansions and classifying singularities in broader domains.

Contribution

It generalizes the classification of singularities for slice regular functions to larger domains, constructing Laurent-type expansions and analyzing poles and essential singularities.

Findings

Constructed Laurent-type expansions at non-zero points

Classified singularities as removable, poles, or essential

Proved a Casorati-Weierstrass theorem for essential singularities

Abstract

Beginning in 2006, G. Gentili and D.C. Struppa developed a theory of regular quaternionic functions with properties that recall classical results in complex analysis. For instance, in each Euclidean ball centered at 0 the set of regular functions coincides with that of quaternionic power series converging in the same ball. In 2009 the author proposed a classification of singularities of regular functions as removable, essential or as poles and studied poles by constructing the ring of quotients. In that article, not only the statements, but also the proving techniques were confined to the special case of balls centered at 0. In a subsequent paper, F. Colombo, G. Gentili, I. Sabadini and D.C. Struppa (2009) identified a larger class of domains, on which the theory of regular functions is natural and not limited to quaternionic power series. The present article studies singularities in this new context, beginning with the construction of the ring of quotients and of Laurent-type expansions at points other than the origin. These expansions, which differ significantly from their complex analogs, allow a classification of singularities that is consistent with the one given in 2009. Poles are studied, as well as essential singularities, for which a version of the Casorati-Weierstrass Theorem is proven.