*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

**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 $\mathbb{C}_J$-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.

**Tags:**
SIR2014-AnHyC

, SIR-NEWHOLITE

, FIRB2012-DGGFT