What does $i^i $ equal and why?

Question

I've been reading up on why the value of 0^0 is controversial (see Zero to the zero power - is $0^0=1$?) and I wondered: is it possible for $i^i$ to have a value? I plugged it into a TI-83 calculator and it returned 0.2078795764 (!) How is this possible and why is the result a real number not a complex number?

Update I realize how to use Euler's Identity of $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ and I understand the oldest answer's value of $\theta = \pi/2$ to solve for $i^i$, but doesn't this mean I can pick any odd multiple of $\pi/2$ (such as $3\pi/2, 5\pi/2$, etc.) as a value of $\theta$? Does that mean that $i^i$ is somehow "periodic" like the cosine curve is?

Answer 1

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

for $k=0$ you get $0.207879576350761908...$

for $k=1$ you get $0.000388203203926766...$

....

Or

$i=1\times i$, then $i^i=1^i\times i^i$: as tomasz commented, this post may be helpful: What is the value of $1^i$?

Answer 2

That's not the only answer in fact.

For any $\alpha\in\mathbb{C}$, we define, for any complex $z=a e^{i\theta }\neq 0$, the multifunction $$[z^{\alpha }]:=\{e^{\alpha (\log |z|+i\theta )} : \theta\in\{\arg (z)+2k\pi i: k\in\mathbb{Z}\}\},$$ taking $z^{\alpha }$ as $w=be^{i\phi }\in [z^{\alpha }]$ such that $\phi\in (-\pi , \pi ]$; we take the "principle value" in the set defined above. [Recall that $e^{2\pi }=1.$.]

Take $z=\alpha =i$.

[This definition is from "Introduction to Complex Analysis: second edition," by H. A. Priestley.]