Functional calculus for dual quaternions

TL;DR

This paper develops a functional calculus for dual quaternions, providing formulas for applying complex functions to dual quaternion variables, extending known results for dual numbers and quaternions.

Contribution

It introduces a new formula for applying complex functions to dual quaternions, generalizing existing calculus for dual numbers and quaternions.

Findings

Derived a formula for $f( ext{dual quaternion})$ for differentiable functions

Extended known calculus from dual numbers and quaternions to dual quaternions

Provided elementary proofs for polynomial cases

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.