A new series expansion for slice regular functions

By Caterina Stoppato. 2011

48 citations recorded by the source corpus.

Abstract A promising theory of quaternion-valued functions of one quaternionic variable, now called slice regular functions, has been introduced by Gentili and Struppa in 2006. The basic examples of slice regular functions are the power series of type ∑ n ∈ N q n a n on their balls of convergence B ( 0 , R ) = { q ∈ H : | q | R } . Conversely, if f is a slice regular function on a domain Ω ⊆ H then it admits at each point q 0 ∈ Ω an expansion of type f ( q ) = ∑ n ∈ N ( q − q 0 ) ∗ n a n where ( q − q 0 ) ∗ n denotes the n th power of q − q 0 with respect to an appropriately defined multiplication ∗ . However, the information provided by such an expansion is somewhat limited by a fact: if q 0 does not lie on the real axis then the set of convergence of the series in the previous equation needs not be a Euclidean neighborhood of q 0 . 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.

Subfields: Applied Mathematics

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In Applied Mathematics

Citations in the corpus

Twistor transforms of quaternionic functions and orthogonal complex structures (2012, methodology): “We first recall two theorems proven in [27].”
Entire slice regular functions (2015, background): “were treated in [103, 105, 111] while further properties can be found in [33, 44, 85, 92, 93, 102, 140, 141, 150, 151, 152, 154].”