On calculus with a quaternionic variable and its characteristic Cauchy-Riemann type equations

TL;DR

This paper investigates the conditions under which quaternionic functions possess a continuous derivative, characterized by specific partial differential equations, extending the understanding of quaternionic calculus and its relation to Cauchy-Riemann type equations.

Contribution

It provides necessary and sufficient PDE-based conditions for quaternionic functions to have a continuous derivative, advancing quaternionic analysis.

Findings

Derived PDE conditions for quaternionic differentiability

Extended Cauchy-Riemann equations to quaternionic functions

Established criteria in terms of variable coefficient PDEs

Abstract

We show sufficient and necessary conditions, in terms of some partial differential equations with variable coefficients, for a quaternionic function to admit a continuous derivative in a open set in the sense of C. Schwartz.