A new series expansion for slice regular functions

TL;DR

This paper introduces a novel series expansion for slice regular quaternionic functions that overcomes previous convergence limitations, enabling polynomial series expansions valid in open subsets and aiding in zero multiplicity and derivative computations.

Contribution

It presents a new type of polynomial series expansion for slice regular functions that is valid in open sets regardless of the center's position.

Findings

New polynomial series expansion valid in open subsets

Applications to zero multiplicity computation

Applications to partial derivative analysis

Abstract

A promising theory of quaternion-valued functions of one quaternionic variable, now called slice regular functions, has been introduced in 2006. The basic examples of slice regular functions are power series centered at 0 on their balls of convergence. Conversely, if f is a slice regular function then it admits at each point of its domain an expansion into power series, where the powers are taken with respect to an appropriately defined multiplication *. However, the information provided by such an expansion is somewhat limited by a fact: if the center p of the series does not lie on the real axis then the set of convergence needs not be a Euclidean neighborhood of p. We are now able to construct a new type of expansion that is not affected by this phenomenon: an expansion into series of polynomials valid in open subsets of the domain. Along with this construction, we present applications to the computation of the multiplicities of zeros and of partial derivatives.