i to the power of i and other complex exponentials After stumbling accross $i^i$, I have been become quite obsessed with complex numbers and especially complex exponentials. 
This even increased after realising that $i^i = e^{-\pi(2k + \frac{1}{2})} $ with $ k \in \mathbb{Z}$ - which means that $i^i$ has infinite solutions. 
I do understand how to get to these  formulas, so I'm not searching for a proof for $i^i$ being the above stated.
Yet, I didn't find any hint for a visual "proof", meaningfull representation of these formulas, real world use or an application in another field of mathematics, which I would like to know about.
Do you know about any of those/ have any hints towards them? 

This is what I found/ know about so far (mainly videos explaining $i^i$ and $\sqrt[i]{i}$ and general complex stuff ):


*

*$i^i$ and co: The youtube channel of blackpenredpen

*Moivre, multiplikation and roots of complex numbers (from university)

*I do know 3blue1brown, but I did not watch all of his videos about complex numbers yet (so a hint that a certain video maybe helpfull for me, may help me)

 A: Step 1: $i^i=e^{iln(i)}$.  $ln(i)$ is multi-valued.  Next $i=e^{i(\frac{\pi}{2}+2k\pi)}$, so $ln(i)=i(\frac{\pi}{2}+2k\pi)$   Therefore $i^i=e^{-(\frac{\pi}{2}+2k\pi)}$
A: There's nothing deep.
We define that for any real $\theta$ that $e^{\theta i } = \cos \theta + i\sin \theta$.
And after that we have no choice.
......
For any non-zero complex number $z = Re(z) + iIm(z)$ we can define $r_z = |z| = \sqrt{Re^2(z)+Im^2(z)}$ and $\theta_z = \arctan \frac {Im(z)}{Re(z)}$ so that $z = r_z e^{\theta_z i}$.
And from that point on our hands are tied.
.......
For any non-zero complex numbers, $w$ and $z$ then the value $w^z$ MUST be equal to:
$(r_w e^{\theta_w i})^z =$
$r_w^z\cdot e^{z\theta_w i} = $
$e^{z\ln r_w + z\theta_w i} = $
$e^{[Re(z)\ln r_w - Im(z)\theta_w] + [Im(z)\ln r_w + Re(z)\theta_w]i}=$
$[e^{[Re(z)\ln r_w - Im(z)\theta_w]}][ e^{i [Im(z)\ln r_w + Re(z)\theta_w]}] = $
$R*(\cos W + i \sin W)$ where
$R$ is the positive real number $e^{[Re(z)\ln r_w - Im(z)\theta_w]}$ and $W$ is the real number $Im(z)\ln r_w + Re(z)\theta_w$.
That has to be how complex numbers to complex number powers must fall out.
We had no choice.
