Power series and analyticity over the quaternions

TL;DR

This paper explores power series and different notions of analyticity in quaternionic functions, establishing their relationships and showing that regular quaternionic functions share properties with complex holomorphic functions.

Contribution

It introduces the concept of sigma-analyticity in quaternions and proves its equivalence to regularity, linking quaternionic analysis to classical complex analysis.

Findings

Sigma-analyticity is equivalent to regularity in quaternionic functions.

Quaternionic analyticity is mostly near the real axis.

Regular quaternionic functions exhibit properties similar to complex holomorphic functions.

Abstract

We study power series and analyticity in the quaternionic setting. We first consider a function f defined as the sum of a quaternionic power series centered at 0 in its domain of convergence (which is a ball B(0,R) centered at 0). At each point p of this ball, f admits expansions in terms of appropriately defined 'regular power series centered at p'. The expansion holds in a ball Sigma(p, R-|p|), defined with respect to a (non-Euclidean) distance sigma. We thus say that f is 'sigma-analytic' in B(0,R). Furthermore, we remark that Sigma(p, R-|p|) is not always an Euclidean neighborhood of p; when it is, we say that f is 'quaternionic analytic' at p. It turns out that f is quaternionic analytic only near the real axis. We then relate these notions of anayticity to the class of regular quaternionic functions introduced in Adv. Math. 216 (2007), 279-301, and recently extended. Indeed, sigma-analyticity proves equivalent to regularity, in the same way as complex analyticity is equivalent to holomorphy. Hence the theory of regular quaternionic functions, which is quickly developing, reveals a new feature that reminds the nice properties of holomorphic complex functions.