A new series expansion for slice regular functions

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.

20 pages, 1 figure

References in corpus (1)

• Analytic cliffordian functions

Cited by in corpus (26)

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• Regular subspaces of a quaternionic Hilbert space from quaternionic Hermite polynomials and associated coherent states
• Regular vs. classical Möbius transformations of the quaternionic unit ball
• Approximation by polynomials on quaternionic compact sets
• Twistor interpretation of slice regular functions
• Geometric function theory over quaternionic slice domains
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• On the real differential of a slice regular function
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• A new approach to slice analysis via slice topology
• Kato S-spectrum in the quaternionic setting
• Some notions of subharmonicity over the quaternions
• A direct approach to quaternionic manifolds
• Riemann slice-domains over quaternions I
• The Mittag-Leffler Theorem for regular functions of a quaternionic variable
• Spherical coefficients of slice regular functions
• Julia theory for slice regular functions
• Boundary Julia theory for slice regular functions
• Noncommutative Cauchy integral formula
• On geometric aspects of quaternionic and octonionic slice regular functions