Journal
CONCRETE OPERATORS
Volume 9, Issue 1, Pages 6-18Publisher
DE GRUYTER POLAND SP Z O O
DOI: 10.1515/conop-2022-0002
Keywords
Slice-regular functions; product of slice-regular functions; exponential; sine and cosine; logarithm
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This paper analyzes five transcendental operators acting on the space of slice regular functions and examines their relationships. The study finds that there is a certain relationship between cos* and cosh* in specific domains.
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 Omega is product and when it is slice. In particular we prove that when Omega is slice, then cos*(f) = cosh*(f * I) holds if and only if f is C-I preserving, while in the case Omega is product there is a much larger family of slice regular functions for which the above relation holds.
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