$i^i\in\mathbb R$

Question

I knew that $^2i=i^i\in\mathbb R$ so I tried to find another number $n\in\mathbb N$ like that: $^ni\in\mathbb R$, so, there is no such number all the way up to a million, can we prove that two is the only number?

Answer 1

Let $r\in(0,1)$ and $θ ∈ (0,\pi/2)$. Then $$ i^{r\, e^{i\theta}} = e^{ir\pi e^{i\theta}/2} = e^{-rπ\cos(\theta)/2} e^{irπ\cos(\theta)/2} =: r_1 e^{i\theta_1} $$ with $r_1\in(0,1)$ and $θ_1 ∈ (0,\pi/2)$. So by induction, starting $^3i$, you get that $^ni$ will always have an angle $\theta\in (0,\pi/2)$ and so different from $0$.