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On a definition of logarithm of quaternionic functions

Graziano Gentili, Jasna Prezelj, Fabio Vlacci

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 $*$-exponential $\exp_*$, was given: if $f$ is a slice--regular function, then $\exp_*(f)$ is a slice--regular function as well. The study of a $*$-logarithm $\log_*(f)$ 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 $\log_*(f)$ 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 $*$-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.

On a definition of logarithm of quaternionic functions

Abstract

For a slice--regular quaternionic function the classical exponential function is not slice--regular in general. An alternative definition of exponential function, the -exponential , was given: if is a slice--regular function, then is a slice--regular function as well. The study of a -logarithm of a slice--regular function becomes of great interest for basic reasons, and is performed in this paper. The main result shows that the existence of such a depends only on the structure of the zero set of the vectorial part of the slice--regular function , 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.

Paper Structure

This paper contains 18 sections, 16 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Basic properties of the exponential
  4. $*$-Exponential of a quaternionic function
  5. Properties of the function $\mu$
  6. Globally defined vectorial class
  7. Local definition of $\log_*$.
  8. Case 0: $g\in \mathcal{S}\mathcal{R}(\Omega)$ is a constant function
  9. Case 1: $g\in \mathcal{S}\mathcal{R}_\mathbb{R}(\Omega)$ is slice--preserving, i.e. $g_v \equiv 0$
  10. Case 2: $g\in \mathcal{S}\mathcal{R}(\Omega)$ with $g_v \not\equiv 0, \, g_v^s \equiv 0$
  11. Case 3: $g\in \mathcal{S}\mathcal{R}(\Omega)$ and $z_0\in \Omega$ such that $Z([g_v]) \cap \{\mathbb{S}_{z_0}\} = \varnothing$
  12. Case 4: $g\in \mathcal{S}\mathcal{R}(\Omega)$ and $z_0\in \Omega$ are such that $z_0 \in \mathbb{S} Z([g_v])$
  13. Global definition of $\log_*$ and proof of Theorem \ref{['general']}
  14. Proof of Theorem \ref{['general']}
  15. Proof of Theorem \ref{['general']} $(a)$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\Omega\subseteq \mathbb{H}$ be a basic domain and let $g \in \mathcal{S}\mathcal{R}_{\omega}({\Omega})$ be a nonvanishing function. Then it holds: where $g^s=g_0^2+g_v^s$ denotes the symmetrization of $g$.

Theorems & Definitions (52)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Example 2.7
  • Example 2.8
  • Definition 2.9
  • ...and 42 more