On a definition of logarithm of quaternionic functions

For a slice-regular quaternionic function f , the classical exponential function exp f is not slice-regular in general. An alternative definition of an exponential function, the *-exponential exps, was given in the work by Altavilla and de Fabritiis (2019): if f is a slice-regular function, then exps f is a slice-regular function as well. The study of a *-logarithm logs f of a slice-regular function f becomes of great interest for basic reasons, and is performed in this paper. The main result shows that the existence of such a logs f depends only on the structure of the zero set of the vectorial part fv of the slice-regular function f = f0 + fv, besides the topology of its domain of definition. We also show that, locally, every slice-regular nonvanishing function has a *-logarithm and, at the end, we present an example of a nonvanishing slice-regular function on a ball which does not admit a *-logarithm on that ball.