$\ast$-exponential of slice-regular functions

Abstract

According to [5] we define the $\ast$-exponential of a slice-regular function, which can be seen as a generalization of the complex exponential to quaternions. Explicit formulas for ${\exp }_{\ast }(f)$ are provided, also in terms of suitable sine and cosine functions. We completely classify under which conditions the $\ast$-exponential of a function is either slice-preserving or ${\mathbb{C}}_J$-preserving for some $J\in \mathbb{S}$ and show that ${\exp }_{\ast }(f)$ is never-vanishing. Sharp necessary and sufficient conditions are given in order that ${\exp }_{\ast }(f+g)={\exp }_{\ast }(f)\ast {\exp }_{\ast }(g)$, 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 ${\exp }_{\ast }(f)$. A number of examples is given throughout the paper.

15 pages; to appear in Proceedings of the American Mathematical Society