On a definition of logarithm of quaternionic functions

[摘要] For a slice-regular quaternionic function fff, the classical exponential function exp⁡f\exp fexpf is not slice-regular in general. An alternative definition of an exponential function, the ∗*∗-exponential exp⁡∗\exp_*exp∗​, was given in the work by Altavilla and de Fabritiis (2019): if fff is a slice-regular function, then exp⁡∗f\exp_*fexp∗​f is a slice-regular function as well. The study of a ∗*∗-logarithm log⁡∗f\log_*flog∗​f of a slice-regular function fff 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\log_*flog∗​f depends only on the structure of the zero set of the vectorial part fvf_vfv​ of the slice-regular function f=f0+fvf=f_0+f_vf=f0​+fv​, 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.

[学科分类] 神经科学

[关键词] Regular functions over quaternions;quaternionic logarithm of slice-regular functions