On the Differentiability of Quaternion Functions

Abstract

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