The $*$-exponential as a covering map

TL;DR

This paper uses complex analysis to develop the $*$-logarithm for quaternionic slice regular functions, analyzing monodromy, conditions for $*$-exponentials, and derivatives, advancing quaternionic function theory.

Contribution

It introduces a method to construct the $*$-logarithm and analyzes properties of the $*$-exponential in quaternionic analysis, providing new tools and results.

Findings

Computed the monodromy of the $*$-exponential

Established conditions for the product of two $*$-exponentials to be a $*$-exponential

Calculated the slice derivative of the $*$-exponential

Abstract

We employ tools from complex analysis to construct the $*$-logarithm of a quaternionic slice regular function. Our approach enables us to achieve three main objectives: we compute the monodromy associated with the $*$-exponential; we establish sufficient conditions for the $*$-product of two $*$-exponentials to also be a $*$-exponential; we calculate the slice derivative of the $*$-exponential of a regular function.