Power series and analyticity over the quaternions

By Graziano Gentili, Caterina Stoppato. 2009

66 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a1b376e1df769b2bebcc45948deb453163d4316e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Entire slice regular functions (2015, background): “The theory of slice regular functions was delevoped in the papers [47, 104, 106, 107, 109, 110, 112, 113], in particular, the zeros”
A new series expansion for slice regular functions (2011, background): “In [8], we considered series of the form f(q) = ∑”