exponent with a Quaternion relating to eulers equation

Question

Considering exponentiation of the Quaternion unit vectors

For the first unit vectors we have

$e^{1*Pi}=e^{Pi}$

and

$e^{I*Pi}=-1$

What are the properties of $e^{Jx}$ and $e^{Kx}$?

Any reading materiial on the subject would also be appreciated

Answer 1

No, it's not true that $e^{Ix} = -1$ (if the unit quaternions are $I$,$J$,$K$). See Wikipedia. The subalgebra generated by $I$, $J$ or $K$ is isometrically isomorphic to the complex numbers, with $I$, $J$ or $K$ corresponding to the complex number $i$, so

$$ \eqalign{e^{Ix} &= \cos(x) + I \sin(x)\cr e^{Jx} &= \cos(x) + J \sin(x)\cr e^{Kx} &= \cos(x) + K \sin(x)\cr}$$