Functional calculus for dual quaternions

Abstract

We give a formula for $f(\eta )$, where $f:\mathbb{C}\to \mathbb{C}$ is a continuously differentiable function satisfying $f(\bar{z})=\overline{f(z)}$, and $\eta$ is a dual quaternion. Note this formula is straightforward or well known if $\eta$ is merely a dual number or a quaternion. If one is willing to prove the result only when $f$ is a polynomial, then the methods of this paper are elementary.

The author recently came across a new approach by `MathFish' at https://oceanofmath.blog/?p=391. It is described at the end of this document, and is not available in the Version of Record

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