How to raise a number to a quaternion power

Question

Now, I know that it's (relatively) easy to calculate, say, $r^{a+bi}$ (using the fact that, for $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2\ln(z_1)}$ and $\ln(z_1$) can just be found using: $\ln(a+bi)=\ln[\sqrt{a^2+b^2}]+i \cdot \arctan(\frac{b}{a})$ ).

Anyway, how would I go about calculating, say, $i^j, k^i$ etc., or, more generally, $(a_1+b_1i+c_1j+d_1k)^{a_2+b_2i+c_2j+d_2k}$ (I know that exponentiating a complex number (to another non-real complex number) produces a non-unique result, so I assume the same would apply further up the hypercomplex ladder; if that's the case, I'm only concerned with the 'principal' value)?

I obviously don't want a general formula or anything like that; just some intuition and a method by which I could calculate such a thing.

And, finally (because I really like to push my luck), can this method for quaternions be extended to higher number systems (i.e. $\mathbb{O, S},$ etc.) to give an analogous result?

Thanks

Answer 1

There is no point in trying to generalise the base $x$ of exponentiation $x^y$ to be a quaternion, since already for $x$ a complex number (other than a positive real) and non-integral rational $y$ there is no unique natural meaning to give to $x^y$. (For instance $z^{2/3}$ could be interpreted as asking for the square of a cube root of$~z$, or for the cube root of $z^2$, and in both cases there are not one but (the same) three candidates; an attempt to force a single outcome for instance by fixing a preferred cube root for every complex number would make the two interpretations differ for certain$~z$.) Anyway, if anything $x^y$ is going to be equivalent to $\exp(\ln(x)y)$ or $\exp(y\ln(x))$ (giving you some choice in case of non-commutatvity) for some meaning of $\ln x$. So the whole effect of using a strange $x$ is to multiply the exponent by a constant; one is better off just writing that multiplication explicitly and sticking to the exponential function $\exp$.

There is no problem at all to extend $\exp$ to a function $\Bbb H\to\Bbb H$, by the usual power series. In fact every non-real quaternion spans a real subalgebra isomorphic to$~\Bbb C$, which will be $\exp$-stable, and restricted to it $\exp$ will behave just as the complex exponential function. Of course one can only expect $\exp(x+y)=\exp(x)\exp(y)$ to hold if $\exp(x)$ and $\exp(y)$ commute, which essentially is the case when $x$ and $y$ lie in the same subalgebra isomorphic to$~\Bbb C$ (and hence commute).

Answer 2

I do see a point in defining $x$ to the power of $y$ for general $x$ and $y$. It is the following rationale.

The famous Mandelbrot-Set for computer-graphics has an iteration that can nicely be generalized with meaningful results.

Originally a Julia-Set is generated by a non-divergence criterion on some complex number $z_0$ with respect to a complex parameter $c$. A series

$$z_{k+1} = z_k z_k + c$$

is calculated as divergent or non divergent for $z_0$ given. Whenever $c$ is replaced by the identity-mapping on the Eulerian plane, i.e.

$$c(z_0) = z_0$$,

matters simplify and the famous Mandelbrot-Thing appears.

The complex multiplication has a useful square mapping. Whenever a higher exponent than $2$, e.g. $3$, $4$, $5$, or, what you want, is applied, we get a meaningful object of studying a general Mandelbrot-Thing by calculating the divergence of

$$z_{k+1} = z_k \cdot z_k \cdot z_k \cdot\dots\cdot z_k + z_0$$

for $z_0$ around $0$ complex.

This meaningful object has got a non-trivial scale-appearance and an astonishing way, how symmetries resemble this natural exponent, used. This proposed natural exponent increased to great numbers seems somewhat to create an increasingly circle-like fractal in the complex plane, with inner and outer circular limit and with a narrow meander of a fractal curve in between.

The quarternions are the last thing useful for studying this fractal-jazz. Some saying from Euler, I remember cited, however, says, the easiest way to a real problem would make use of complex models. The Zeta-Function discussions for a famous Riemannian millenium-problem might benefit from proper terms for some way to circumvent all the particularities of pure complex models by actually defining everything it takes to work with $x$ to the power of $y$ for general $x$ and $y$.

I will comment to the first answer, if I have 50 reputation. For the time being this text must be part of the answer to the original question about quarternions. So, in brief, the way how to raise a number to a quarternion power, is in what it shall mean to everyone.