## Complex Powers

Exponents + Complex Numbers = Cool Stuff

I’ve been messing around with these sort of things and discovering all sorts of neat stuff. I haven’t got far enough in Needham to really understand all of it, but here are some things I have found.

The plots of $x^k$ where $k$ is a point on the complex unit circle are fascinating. Here’s an animation I made:

You can also watch a high-quality video on youtube.

At $k=i$ argument and magnitude of the output seem to be switched. They swirl into these circles and then swirl out such that the high magnitude and low magnitude ends are switched.

Another interesting thing is that $i^x$ and $e^{ix}$ are related. In fact: $i^{-2ix/\pi}=e^x$$i^{-2i/\pi}=e$

Update: The reason why this happens is fairly straight forward. Consider $x^{e^{\pi i n}} = (re^{i\theta})^{\cos(\pi n)+i\sin(\pi n)}=e^{\ln(r)\cos(\pi n)}*e^{\ln(r)i\sin(\pi n)}*e^{i\theta\cos(\pi n)}*e^{-\theta\sin(\pi n)}=e^{\ln(r)\cos(\pi n)-\theta\sin(\pi n)}*e^{i\theta\cos(\pi n)+\ln(r)i\sin(\pi n)}$ and thus magnitude and argument switch as $n$ increases.