Extension results for slice regular functions of a quaternionic variable

Abstract

In this paper we prove a new Representation Formula for slice regular functions, which shows that the value of a slice regular function f at a point q = x + yI can be recovered by the values of f at the points q + yJ and q + yK for any choice of imaginary units I, J, K. This result allows us to extend the known properties of slice regular functions defined on balls centered on the real axis to a much larger class of domains, called axially symmetric domains. We show, in particular, that axially symmetric domains play, for slice regular functions, the role played by domains of holomorphy for holomorphic functions.

The Runge theorem for slice hyperholomorphic functions

“…We will say that a singularity q = q 0 of a function f is a pole if it belongs to an isolated 2-sphere of singularities and it is such that lim q→q 0 |f (q)| = +∞. If one considers the case of slice regular functions in the sense of [16] which are defined on axially symmetric slice domains, an analogous result is proven in [11].…”

“…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

“…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…”

On the Transformation Formula of the Slice Bergman Kernels in the Quaternion Variables

“…Each regular function on a slice domain has the unique regular extension on the axially symmetric slice domain (see Theorem 4.1 in [2]). Thus axially symmetric slice domains play, for slice regular functions, the similar role as the domains of holomorphy for holomorphic functions on C n .…”