# $*$-exponential of slice-regular functions

created by stoppato on 05 May 2017
modified by altavilla on 11 Mar 2019

[BibTeX]

published

Inserted: 5 may 2017
Last Updated: 11 mar 2019

Journal: Proceedings of the American Mathematical Society
Volume: 147
Pages: 1173--1188
Year: 2019
Doi: https://doi.org/10.1090/proc/14307

ArXiv: 1806.10446 PDF
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 $\mathbb{C}_J$-preserving for some $J\in\mathbb{S}$ and show that $\exp_*(f)$ is never-vanishing. Sharp necessary and sufficient conditions are given in order that $\exp_*(f+g)=\exp_*(f)*\exp_*(g)$, finding an exceptional and unexpected case in which equality holds even if $f$ and $g$ do not commute. We also discuss the existence of a square root of a slice-preserving regular function, characterizing slice-preserving functions (defined on the circularization of simply connected domains) which admit square roots. Square roots of this kind of functions are used to provide a further formula for $\exp_{*}(f)$. A number of examples is given throughout the paper.