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 exponential function, the ∗-exponential exp∗ , was given in [AdF]: if f is a slice–regular function, then exp∗ (f) is a slice–regular function as well. The study of a ∗-logarithm log∗ (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 log∗ (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