$*$-exponential of slice-regular functions (1806.10446v1)

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.