On a definition of logarithm of quaternionic functions
Abstract
For a sliceregular quaternionic function $f,$ the classical exponential function $\exp f$ is not sliceregular in general. An alternative definition of exponential function, the $*$exponential $\exp_*$, was given: if $f$ is a sliceregular function, then $\exp_*(f)$ is a sliceregular function as well. The study of a $*$logarithm $\log_*(f)$ of a sliceregular 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 sliceregular function $f=f_0+f_v$, besides the topology of its domain of definition. We also show that, locally, every sliceregular nonvanishing function has a $*$logarithm and, at the end, we present an example of a nonvanishing sliceregular function on a ball which does not admit a $*$logarithm on that ball.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.08595
 Bibcode:
 2021arXiv210808595G
 Keywords:

 Mathematics  Complex Variables;
 30G35;
 32A30;
 33B10