Riemann slice-domains over quaternions I

TL;DR

This paper introduces Riemann slice-domains over quaternions, rectifies classical extension formulas for slice regular functions, and develops a new topological framework to better understand quaternionic analysis.

Contribution

It constructs a new topology on quaternions, rectifies extension formulas, and establishes a representation formula for slice regular functions over Riemann slice-domains.

Findings

Counterexample to a classical extension theorem

Development of a new topology on quaternions

Representation formula for slice regular functions

Abstract

We construct a counterexample to a well-known extension theorem for slice regular functions, which motivates us to develop a theory of Riemann slice-domains by introducing a new topology on quaternions. By some paths describing axial symmetry in Riemann slice-domains, we rectify the classical extension formula in the theory of slice regular functions and prove a representation formula over slice-domains of regularity. This proof involves an intertwining relation between imaginary units of quaternions and a fixed matrix corresponding to a complex structure.