$*$-exponential of slice-regular functions

Abstract:

As in [Entire slice regular functions, Springer, 2016] 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 function are used to provide a further formula for $\exp _{*}(f)$. A number of examples are given throughout the paper.