On a definition of logarithm of quaternionic functions

Abstract

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 $\ast$-exponential ${\exp }_{\ast }$, was given: if $f$ is a slice--regular function, then ${\exp }_{\ast }(f)$ is a slice--regular function as well. The study of a $\ast$-logarithm ${\log }_{\ast }(f)$ 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 ${\log }_{\ast }(f)$ 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.

References in corpus (3)

• The C-property for slice regular functions and applications to the Bergman space
• Slice conformality and Riemann manifolds on quaternions and octonions
• $\ast$-Logarithm for Slice Regular Functions

Cited by in corpus (2)

• Quaternionic Cartan coverings and applications
• On a continuation of quaternionic and octonionic logarithm along curves and the winding number