# On a definition of logarithm of quaternionic functions

### Graziano Gentili

Università di Firenze, Italy### Jasna Prezelj

Univerza v Ljubljani; Univerza na Primorskem, Slovenia### Fabio Vlacci

Università di Trieste, Italy

## Abstract

For a slice-regular quaternionic function $f$, the classical exponential function $expf$ is not slice-regular in general. An alternative definition of an exponential function, the $∗$-exponential $exp_{∗}$, was given in the work by Altavilla and de Fabritiis (2019): if $f$ is a slice-regular function, then $exp_{∗}f$ is a slice-regular function as well. The study of a $∗$-logarithm $g_{∗}f$ 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 $g_{∗}f$ 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 $∗$-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.

## Cite this article

Graziano Gentili, Jasna Prezelj, Fabio Vlacci, On a definition of logarithm of quaternionic functions. J. Noncommut. Geom. 17 (2023), no. 3, pp. 1099–1128

DOI 10.4171/JNCG/514