*-Exponential of slice-regular functions

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 C J -preserving for some J ∈ 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.