EXPONENTIAL OF SLICE-REGULAR FUNCTIONS

As in [Entire slice regular functions, Springer, 20161 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 E 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.