A Note on the Hypercomplex Riemann-Cauchy Like Relations for Quaternions and Laplace Equations

TL;DR

This paper develops new Laplace-like equations for quaternions using Riemann-Cauchy hypercomplex relations, aiming to enhance the mathematical tools available for physical theories involving quaternionic functions.

Contribution

It introduces a novel set of quaternionic Laplace-like equations derived from Riemann-Cauchy relations, expanding the mathematical framework for quaternionic analysis.

Findings

New quaternionic Laplace-like equations formulated

Potential applications in classical field theories suggested

Mathematical relations extend previous hypercomplex function theory

Abstract

In this note it is worked out a new set of Laplace-Like equations for quaternions through Riemann-Cauchy hypercomplex relations otained earlier \cite{BorgesZeMarcio}. As in the theory of functions of a complex variable, it is expected that this new set of Laplace-Like equations might be applied to a large number of Physical problems, providing new insights in the Classical Theory Fields.