Mathematische Annalen

, Volume 352, Issue 1, pp 113–131

# Power series and analyticity over the quaternions

• Graziano Gentili
• Caterina Stoppato
Article

## Abstract

We study power series and analyticity in the quaternionic setting. We first consider a function f defined as the sum of a power series $${\sum\nolimits_{n \in \mathbb{N}} q^n a_n}$$ in its domain of convergence, which is a ball B(0, R) centered at 0. At each $${p \in B(0,R)}$$, f admits expansions in terms of appropriately defined power series centered at p, namely $${\sum\nolimits_{n \in \mathbb{N}} (q-p)^{*n} b_n}$$. The expansion holds in a ball Σ(p, R − |p|) defined with respect to a (non-Euclidean) distance σ. We thus say that f is σ-analytic in B(0, R). Furthermore, we remark that Σ(p, R − |p|) is not always a Euclidean neighborhood of p; when it is, we say that f is strongly analytic at p. It turns out that f is strongly analytic in a neighborhood of $${B(0,R) \cap \mathbb{R}}$$ that can be strictly contained in B(0, R). We then relate these notions of analyticity to the class of quaternionic functions introduced in Gentili and Struppa (Adv. Math. 216(1):279–301, 2007), and recently extended in Colombo et al. (Adv. Math. 222(5):1793–1808, 2009) under the name of slice regular functions. Indeed, σ-analyticity proves equivalent to slice regularity, in the same way as complex analyticity is equivalent to holomorphy. Hence the theory of slice regular quaternionic functions, which is quickly developing, reveals a new feature that reminds the nice properties of holomorphic complex functions.

## Mathematics Subject Classification (2000)

30G35 30B10 30G30

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