$*$-exponential of slice-regular functions

TL;DR

This paper introduces the $*$-exponential for slice-regular quaternionic functions, providing explicit formulas, classifications of preservation properties, conditions for exponential additivity, and square root characterizations, with numerous examples.

Contribution

It generalizes the exponential function to quaternionic slice-regular functions, offering explicit formulas, preservation classifications, and new insights into exponential properties and square roots.

Findings

$oxed{ ext{Explicit formulas for } ext{exp}_*(f)}$

$oxed{ ext{Classification of preservation properties}}$

$oxed{ ext{Conditions for } ext{exp}_*(f+g)= ext{exp}_*(f)* ext{exp}_*(g)}$

Abstract

According to [5] we define the $*$-exponential of a slice-regular function, which can be seen as a generalization of the complex exponential to quaternions. Explicit formulas for $\exp_*(f)$ are provided, also in terms of suitable sine and cosine functions. We completely classify under which conditions the $*$-exponential of a function is either slice-preserving or $\mathbb{C}_J$-preserving for some $J\in\mathbb{S}$ and show that $\exp_*(f)$ is never-vanishing. Sharp necessary and sufficient conditions are given in order that $\exp_*(f+g)=\exp_*(f)*\exp_*(g)$, finding an exceptional and unexpected case in which equality holds even if $f$ and $g$ do not commute. We also discuss the existence of a square root of a slice-preserving regular function, characterizing slice-preserving functions (defined on the circularization of simply connected domains) which admit square roots. Square roots of this kind of functions are used to provide a further formula for $\exp_{*}(f)$. A number of examples is given throughout the paper.