A Cauchy kernel for slice regular functions

By Fabrizio Colombo, Graziano Gentili, Irene Sabadini. 2009

79 citations recorded by the source corpus.

In this article, we show how to construct a regular, non-commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/cd1b506c1a941105921fca6033f4f401326105a5

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

Entire slice regular functions (2015, methodology): “The integral formulas in this chapter come from [47] and [55], and the Riemann mapping theorem was originally proved in [100].”
Regular Moebius transformations of the space of quaternions (2008, background): “The properties of Cullen-regular functions proven in [19] and in the subsequent papers [7,9,14–17,20,22,26] recall holomorphic functions of one complex variable rather than Fueter’s theory.”
On Two Approaches to the Bergman Theory for Slice Regular Functions (2013, methodology): “Nevertheless, the paper [9] tells us that this family generates the set of slice regular functions.”
Power series and analyticity over the quaternions (2009, background): “Hence the theory of regular quaternionic functions, which is quickly developing, reveals a new feature that reminds the nice properties of holomorphic complex functions.”
Extension results for slice regular functions of a quaternionic variable (2009, methodology): “…of a regular function at a point q belonging to an axially symmetric s-domain is related to the values that the function assumes at two conjugate points on a same plane LJ and belonging to the 2-sphere associated to q (see Theorem 2.26 in [1], as well as [5]): Theorem 3.1 (Representation Formula).”
Slice regular functions on real alternative algebras (2010, methodology): “We define a Cauchy kernel, which in the quaternionic case was already introduced in [3], and for slice monogenic functions in [5, 6].”
A new series expansion for slice regular functions (2011, methodology): “The previous lemma was the basis for the proof of a more complete Cauchy formula in [2].”
On Some Properties of the Quaternionic Functional Calculus (2009, methodology): “and similarly we can calculate the limit when z → μ. Note that the derivative f � (μ) coincides with ∂ ∂x f( μ)since f is an s-regular function (see [ 7 ]).”