Riemann slice-domains over quaternions II

TL;DR

This paper extends the theory of slice regular functions over quaternions to Riemann slice-domains, introducing new tools like holomorphic stem systems and tensor holomorphic functions to generalize key formulas and products.

Contribution

It generalizes the representation formula and $*$-product to Riemann slice-domains, connecting stem holomorphic, tensor holomorphic, and slice regular functions.

Findings

Extended the representation formula to Riemann slice-domains

Constructed power series expansions for slice regular functions

Established relations between different types of holomorphic functions

Abstract

We generalize the representation formula from slice-domains of regularity to general Riemann slice-domains. This result allows us to extend the $*$-product of slice regular functions on axially symmetric domains to certain Riemann slice-domains by introducing holomorphic stem systems and tensor holomorphic functions. In particular, we construct a power series expansions of slice regular functions on certain Riemann slice-domains, which implies a relation between stem holomorphic, tensor holomorphic and slice regular functions.