Notions of Regularity for Functions of a Split-Quaternionic Variable

TL;DR

This paper compares two notions of regularity for split-quaternionic functions, showing one is limited while the other is more comprehensive, and provides examples and extensions in the latter framework.

Contribution

It clarifies and contrasts two existing notions of regularity for split-quaternionic functions, introducing a subclass and examples of Cauchy-Kowalewski extensions.

Findings

One notion covers a small class of functions.

The other notion provides a richer collection.

Examples of Cauchy-Kowalewski extensions are given.

Abstract

Notions of a "holomorphic" function theory for functions of a split-quaternionic variable have been of recent interest. We describe two found in the literature and show that one notion encompasses a small class of functions, while the other gives a richer collection. In the second instance, we describe a simple subclass of functions and give two examples of an analogue of the Cauchy-Kowalewski extension in this context.