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

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.

• I believe the principal value would actually be $e^{i\frac{\pi}{2}e^{-\pi/2}}=0.947+0.321i$. – Jam Nov 23 '19 at 18:11

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$$?$$
• Well, but I get 2 parameters $n\in\mathbb{Z}$. One for each of the powers. I guess I have to define 2 parameters $n,n’$ and define the set of values in terms of that, right? – Phil Mett Nov 23 '19 at 17:55