Transcendental operators acting on slice regular functions

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.