i to the power of i and other complex exponentials

Question

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?

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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)

Answer 1

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 $i^i=e^{-(\frac{\pi}{2}+2k\pi)}$

Answer 2

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.