On the Differentiability of Quaternion Functions

TL;DR

This paper introduces a notion of H-derivative for quaternion functions, extending classical complex analysis concepts to quaternions, and demonstrates its application to elementary functions and logarithms.

Contribution

It defines the H-derivative for quaternion functions and establishes calculation rules, advancing the theory of quaternion analysis.

Findings

Elementary quaternion functions have H-derivatives.

Quaternion logarithm function possesses an H-derivative.

Rules for calculating H-derivatives are established.

Abstract

Motivated by the general problem of extending the classical theory of holomorphic functions of a complex variable to the case of quater- nion functions, we give a notion of an H-derivative for functions of one quaternion variable. We show that the elementary quaternion func- tions introduced by Hamilton as well as the quaternion logarithm function possess such a derivative. We conclude by establishing rules for calculating H-derivatives.