# $∗$-logarithm for slice regular functions

### Amedeo Altavilla

Università degli Studi di Bari “Aldo Moro”, Italy### Chiara de Fabritiis

Università Politecnica delle Marche, Ancona, Italy

## Abstract

In this paper, we study the (possible) solutions of the equation $exp_{∗}(f)=g$, where $g$ is a slice regular never vanishing function on a circular domain of the quaternions $H$ and $exp_{∗}$ is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function $f$ which satisfies $exp_{∗}(f)=g$ is called a $∗$-logarithm of $g$. We provide necessary and sufficient conditions, expressed in terms of the zero set of the “vector” part $g_{v}$ of $g$, for the existence of a $∗$-logarithm of $g$, under a natural topological condition on the domain $Ω$. By this way, we prove an existence result if $g_{v}$ has no non-real isolated zeroes; we are also able to give a comprehensive approach to deal with more general cases. We are thus able to obtain an existence result when the non-real isolated zeroes of $g_{v}$ are finite, the domain is either the unit ball, or $H$, or $D$ (the solid torus obtained by circularization in $H$ of the disc contained in $C$ and centered in $2−1 $ with radius $1$), and a further condition on the “real part” $g_{0}$ of $g$ is satisfied (see Theorem 6.19 for a precise statement). We also find some unexpected uniqueness results, again related to the zero set of $g_{v}$, in sharp contrast with the complex case. A number of examples are given throughout the paper in order to show the sharpness of the required conditions.

## Cite this article

Amedeo Altavilla, Chiara de Fabritiis, $∗$-logarithm for slice regular functions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 34 (2023), no. 2, pp. 491–529

DOI 10.4171/RLM/1016