Set of values of $i^{i^i}$

Question

I want to find all the set of values for the expression $i^{i^i}$.

For the principal value of this expression I got $e^{i\frac{\pi}{2}}e^{e^{-\frac{\pi}{2}}}$, please correct me if wrong.

Any hints or help would be appreciated.

Answer 1

Use the fact $i=e^{\ln i}$ and $\ln i=i\frac{\pi}{2}+i(2\pi n) \forall n\in \mathbb{Z}$ . Can you go from gere$?$