On a definition of logarithm of quaternionic functions

Abstract

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 $\ast$-exponential ${\exp }_{\ast }$, was given in the work by Altavilla and de Fabritiis (2019): if $f$ is a slice-regular function, then ${\exp }_{\ast }f$ is a slice-regular function as well. The study of a $\ast$-logarithm ${\log }_{\ast }f$ 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 ${\log }_{\ast }f$ depends only on the structure of the zero set of the vectorial part $f_v$ of the slice-regular function $f=f_0+f_v$, besides the topology of its domain of definition. We also show that, locally, every slice-regular nonvanishing function has a $\ast$-logarithm and, at the end, we present an example of a nonvanishing slice-regular function on a ball which does not admit a $\ast$-logarithm on that ball.