How do I solve (a+bi+cj+dk)^(f+gi+hj+nk)?

Question

I purposely skipped using e as a factor in the title because e is Eulers Number.
I have found $$e^{(a+bi+cj+dk)}$$ and $$(a+bi+cj+dk)^n$$ but no way to combine them together. My current theory is adding on to the formula for imaginary numbers to get: $$n=e^{(w_2*\ln{⁡(\sqrt{w_1^2+x_1^2})}+x_2*\arctan⁡(\frac{x_1}{w_1})-y_2*\ln{⁡(\sqrt{y_1^2+z_1^2})}-z_2*\arctan{⁡(\frac{z_1}{y_1})})}$$ where (n,w,x,y,z) is (a,a,b,c,d),(b,b,a,d,c),(c,c,d,a,b),(d,d,c,b,a) Could someone please give me the true formula? It would be very well appreciated. Thank you so very much!

Answer 1

A quaternion $q$ with zero real part has negative square, so with e.g. Taylor series you can show $e^q=\cos|q|+q\operatorname{sinc}|q|$ ($\operatorname{sinc}$ is explained here), in analogy with $e^{i\theta}=\cos\theta+i\sin\theta$ for imaginary $i\theta\in\Bbb C$. Now you know how to evaluate $e^z$ for any quaternion $z$, we can use $z=|z|e^q$ with $\Re q=0$ to obtain $z^w=|z|^we^{qw}=e^{w(\ln|z|+q)}$. The hard part is determining which $q$ satisfy $\Re q=0,\,e^q=z/|z|$, which I leave you to ponder; but there will be plenty because, as with complex exponentiation, quaternionic exponentiation is multi-valued.

Answer 2

If you want to define $p^q$ for quaternions $p$ and $q$, you ought to first start at $w^z$ for complex numbers $w$ and $z$. Plugging something like $i^i$ or $\log i$ into a calculator tells you they can be defined, but it doesn't tell you how it's defined, so you stopped short of doing more research you could've been doing.

For complex numbers $w$ and $z$, the value of $w^z$ is defined to be $\exp(z\ln w)$, assuming you have a way to define $\ln w$. Famously, $\exp$ is not one-to-one on $\mathbb{C}$ (since $\exp 0=\exp2\pi i$ for instance), so it doesn't have an inverse which is holomorphic on all of $\mathbb{C}^\times=\mathbb{C}\setminus\{0\}$. That means we have to make some compromises in order to even define $\ln w$ at all. The way this is done in practice is with branch cuts. The "standard branch cut" is considered to be the nonpositive real axis $(-\infty,0]$, in which case $\ln$ is defined on $\mathbb{C}^\times$ but is only holomorphic on $\mathbb{C}\setminus(-\infty,0]$. If you graph $\ln w$ as $w$ crosses $(-\infty,0]$, you'll see a jump discontinuity where the jump is $\pm2\pi i$. Specifically, if $w=r\exp(i\theta)$ with $r>0$ and $-\pi<\theta\le\pi$ then $\ln w:=\ln r+i\theta$. Many argue, though, that the branch cut is arbitrary, and even that $\ln w$ isn't, or shouldn't, be defined.

This works for quaternions too. All quaternions have a polar form $p=r\exp(\theta\mathbf{u})$ for some unit vector $\mathbf{u}$, and we can arrange for $-\pi<\theta\le\pi$. There is some redundancy, since $\theta\mathbf{u}=(-\theta)(-\mathbf{u})$, but the resulting expression $\ln p:=\ln r+\theta\mathbf{u}$ is well-defined regardless. Yet, it still would be nice to have unique polar forms, so I do recommend restricting to convex angles $0\le\theta\le\pi$, in which case $\theta$ is unique, and $\mathbf{u}$ is unique unless $\theta=0$ or $\pi$ in which case it is arbitrary.

For real values $s$, you can define $q^s:=\exp(s\ln q)$ using the previous discussion to make sense of $\ln q$. But for quaternion bases you come to a decision point: should you define $p^q$ to be $\exp(q\ln p)$ or $\exp\big((\ln p)q\big)$. Or you could even pick things even between these two, like $\exp(q^s(\ln p)q^{1-s})$ for real values $s$.

And even if you do make an arbitrary choice to define $p^q$ ... why? Rotational dynamics don't demand this - in the broader context of Lie theory it makes sense to express everything with the natural exponential function. The function isn't holomorphic on any domain - indeed, the only quaternion holomorphic functions are affine functions $ax+b$ or $xa+b$ (depending on whether you're considering the left or right derivative). Note even power functions like $x^2$ are holomorphic, let alone power series or whatnot.

Maybe you just think $p^q$ is such a basic operation for real numbers that it makes sense to trek for a sensible interpretation of it for other number systems out of a sense of completeness. But it just turns out that it's mostly ugly and unnatural. (I would argue writing $p^t$ for real $t$ as shorthand is debatably a positive choice in certain contexts, but besides that, a negative choice.)