A Cauchy kernel for slice regular functions

Abstract

In this paper 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.AMS Classification: 30G35.

The Weierstrass factorization theorem for slice regular functions over the quaternions

“…Recently, Gentili and Struppa proposed a different approach, which led to a new notion of holomorphicity (called slice regularity) for quaternion-valued functions of a quaternionic variable [15,16]. Unlike Fueter's, this theory includes the polynomials and the power series of the quaternionic variable q of the type n≥0 q n a n , with coefficients a n ∈ H. Furthermore, the analogs (sometime peculiarly different) of many of the fundamental properties of holomorphic functions of one complex variable can be proven in this new setting, like the Cauchy and Pompeiu representation formulas and Cauchy inequalities, the maximum (and minimum) modulus principle, the identity principle, the open mapping theorem, the Morera theorem, the power and Laurent series expansion, the Runge approximation theorem, to cite only some of the most significant (see [4][5][6][7][12][13][14][15][16]24]). In fact, the theory of slice regular functions is already rather rich and well established on steady foundations, and appears to be of fundamental importance to construct a functional calculus in non commutative settings [8].…”

The Runge theorem for slice hyperholomorphic functions

“…This partial inverse of the previous corollary is based on the following result (see [12]), which is a consequence of the validity of the Identity Principle on slice domains; see [8]. In the case of slice regular functions over quaternions in the sense of [16], an analogous formula appeared in [2] (and see also [11] …”

“…slice regular and slice monogenic functions, were introduced in [8], [16], [17] and further studied in a series of papers; see e.g. [2], [9], [10], [11], [13]. In their recent work [18] the authors show that an alternative definition which works for functions with values in any real alternative algebra can be given in terms of what they call (following the terminology introduced by Cullen in [14]) stem functions.…”

A Bloch-Landau Theorem for Slice Regular Functions

“…This yields an extension theorem (see [1,4]) that in the special case of functions that are regular on B(0, R) can be obtained by means of their power series expansion.…”

“…Another useful result is the following (see [1,4]) Theorem 2.6 (Representation Formula). Let f be a regular function on B = B(0, R) and let J ∈ S. Then, for all x + yI ∈ B, the following equality holds…”