On a definition of logarithm of quaternionic functions
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929268849639424 |
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| author | Gentili, Graziano Prezelj, Jasna Vlacci, Fabio |
| author_facet | Gentili, Graziano Prezelj, Jasna Vlacci, Fabio |
| contents | 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: 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 $f_v$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_08595 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On a definition of logarithm of quaternionic functions Gentili, Graziano Prezelj, Jasna Vlacci, Fabio Complex Variables 30G35, 32A30, 33B10 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: 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 $f_v$ 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. |
| title | On a definition of logarithm of quaternionic functions |
| topic | Complex Variables 30G35, 32A30, 33B10 |
| url | https://arxiv.org/abs/2108.08595 |