On a definition of logarithm of quaternionic functions

Gentili, Graziano (author), Prezelj, Jasna (author), Vlacci, Fabio (author)

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

• Language: English
• Keywords: regular functions over quaternions, quaternionic logarithm of slice-regular functions
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2023
• Year: 2023
• Number of pages: Str. 1099-1128
• Numbering: Vol. 17, no. 3
• PID: 20.500.12556/RUL-155646
• UDC: 517.5
• ISSN on article: 1661-6952
• DOI: 10.4171/JNCG/514
• COBISS.SI-ID: 162763779
• Publication date in RUL: 10.04.2024

GENTILI, Graziano, PREZELJ, Jasna and VLACCI, Fabio, 2023, On a definition of logarithm of quaternionic functions. Journal of noncommutative geometry [online]. 2023. Vol. 17, no. 3, p. 1099–1128. [Accessed 26 April 2025]. DOI 10.4171/JNCG/514. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=155646

Record is a part of a journal

• Title: Journal of noncommutative geometry
• Shortened title: J. noncommut. geom.
• Publisher: EMS Publishing House, ETH-Zentrum FLI C4
• ISSN: 1661-6952
• COBISS.SI-ID: 17114713

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: regularne funkcije nad kvaternioni, logaritem, rezinsko regularne funkcije