Abstract
Abstract The aim of this paper is to carry out an analysis of five trascendental operators acting on the space of slice regular functions, namely *-exponential, *-sine and *-cosine and their hyperbolic analogues. The first three of them were introduced by Colombo, Sabadini and Struppa and some features of *-exponential were investigated in a previous paper by Altavilla and the author. We show how exp*(f ), sin*(f ), cos*(f ), sinh*(f ) and cosh*(f ) can be written in terms of the real and the vector part of the function f and we examine the relation between cos* and cosh* when the domain Ω is product and when it is slice. In particular we prove that when Ω is slice, then cos*(f ) = cosh*(f * I) holds if and only if f is ℂ I preserving, while in the case Ω is product there is a much larger family of slice regular functions for which the above relation holds.
Highlights
The aim of this paper is to carry out an analysis of ve trascendental operators acting on the space of slice regular functions, namely *-exponential, *-sine and *-cosine and their hyperbolic analogues
In particular we prove that when Ω is slice, cos*(f ) = cosh*(f * I) holds if and only if f is CI preserving, while in the case Ω is product there is a much larger family of slice regular functions for which the above relation holds
In this paper we investigate the behaviour of some operators de ned on the space of slice regular functions
Summary
In this paper we investigate the behaviour of some operators de ned on the space of slice regular functions. In Proposition 2.19 we are able to express the value of these operators on a function f in terms of its real and vector parts, by means of the functions μ and ν which were introduced in [2] These relations allow us to show that *-sine and *-cosine still satisfy an analogue of the Pytagorean Trigonometric Identity, see Proposition 2.20. When the domain Ω is slice, the function J cannot be de ned anymore, so that an analogue of the relation in the complex case is given by cos*(f ) = cosh*(f *I), where I ∈ S is an imaginary unit. We deal with the case when the domain Ω is product, here the statement of Theorem 4.5 gives a complete list of all slice regular functions f which satisfy the equality cos*(f ) = cosh*(f * I).
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