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
