On a quaternionic Picard theorem

TL;DR

This paper explores the extension of Picard's theorem to quaternionic slice regular functions, analyzing how many values such functions can omit, thus generalizing classical complex analysis results to quaternionic analysis.

Contribution

It establishes bounds on the number of values a non-constant quaternionic slice regular function can omit, extending Picard's theorem to quaternionic functions.

Findings

Quaternionic Picard theorem bounds the omitted values for slice regular functions.

Non-constant slice regular functions can omit at most finitely many values.

Results generalize classical Picard theorem to quaternionic setting.

Abstract

The classical theorem of Picard states that a non-constant holomorphic function $f:\mathbb{C}\to\mathbb{C}$ can avoid at most one value. We investigate how many values a non-constant slice regular function of a quaternionic variable $f:\mathbb{H}\to\mathbb{H}$ may avoid.