$*$-exponential of slice-regular functions

Abstract

According to [5] we define the * -exponential of a slice-regular function, which can be seen as a generalization of the complex exponential to quaternions. Explicit formulas for exp * (f ) are provided, also in terms of suitable sine and cosine functions. We completely classify under which conditions the * -exponential of a function is either slice-preserving or CJ -preserving for some J ∈ S and show that exp * (f ) is never-vanishing. Sharp necessary and sufficient conditions are given in order that exp *

s-Regular functions which preserve a complex slice

“…Applying Corollary 3.2 in [3] and the above theorem to the case when Ω I0 = Ω ∩ C I0 is simply connected, i.e. : π 1 (Ω I0 ) = 0 for some, and then any, I 0 ∈ S, gives the following…”

“…Then we can choose a suitable spherical neighborhood U = B q0 (r) of the point q 0 where both f 0 and f v never vanish, which entails that also f s v is never-vanishing on U . Thus Corollary 3.2 in [3] gives the existence of ρ ∈ S R (U ) such that ρ 2 = f s v on U . Equality (7.1) thus implies that on U we have In particular, at any q ∈ U the point [f 0 (q) : ρ(q)] ∈ R \ {0} is a root of Q d and thus there exists ξ(q) ∈ Σ d such that [f 0 (q) : ρ(q)] = ξ(q).…”

“…If the first fundamental group of Ω I = Ω ∩ C I is trivial, then the set of slice regular functions whose d − th * -power belongs to S R (Ω) can be characterized even more precisely. Indeed we have Then the functions f s v has a square root f0 ξ ∈ S R (Ω); the hypothesis on the first fundamental group of Ω I together with Corollary 3.2 in [3], entail that this is equivalent to the fact that the zero set of f v does not contain non real isolated zeroes of odd multiplicities.…”

“…where exp * is the quaternionic * -exponential introduced in [8] and studied in [3]. The previous composition operator denoted by ⊛, is the one introduced in [20, Definition 4.1] (since E m has real coefficients the two definitions appearing in the cited paper coincide).…”

S-Polyregular Bargmann Spaces

“…Proof. The proof follows by making use of the fact that for C J -preserving functions ϕ and ψ, the L s -product satisfies ϕ L s ψ = ψ L s ϕ (see [8]). As basic example of computation with such L sp -product, we explicit the one of the following function…”