Poles of regular quaternionic functions

TL;DR

This paper classifies singularities of Cullen-regular quaternionic functions, introduces semiregular functions as quotients, and analyzes the distribution of poles in the quaternionic setting.

Contribution

It provides a comprehensive classification of singularities and introduces semiregular functions as quaternionic analogues of meromorphic functions.

Findings

Cullen-regular functions have Laurent series expansions.

Singularities are classified as removable, essential, or poles.

Semiregular functions are quotients of Cullen-regular functions.

Abstract

This paper studies the singularities of Cullen-regular functions of one quaternionic variable. The quaternionic Laurent series prove to be Cullen-regular. The singularities of Cullen-regular functions are thus classified as removable, essential or poles. The quaternionic analogues of meromorphic complex functions, called semiregular functions, turn out to be quotients of Cullen-regular functions with respect to an appropriate division operation. This allows a detailed study of the poles and their distribution.