Trying to get more intuition on how exponentiation with complex numbers works, I began entering various functions with complex exponents into WolframAlpha. Both $e^{ix}$ and $i^x$ wasn't all that much of surprise, then I entered $x^i$ and...
What the heck am I seeing? Supposedly in $e^{\theta i \pi}$ the $e$ is just a scale factor, chosen for convenience, so that argument would remain 1. Meanwhile, as scale factor the base doesn't behave all that nice either...
Could someone explain what's going on here?
By Euler's formula,
$$x^i=e^{i\log x}=\cos(\log x)+i\sin(\log x).$$
The argument is $\log x$, reduced to some $2\pi$ range.
There is no big mystery.