The value of i raised to the power i

Compute $i^{\,i}$, where $i=\sqrt{-1}$.

$e^{-\pi/2}\approx0.208$ (principal value), a real number. From Euler, $i=e^{i\pi/2}$, so $i^i=e^{i\cdot i\pi/2}=e^{-\pi/2}$. Other branches $i=e^{i(\pi/2+2\pi k)}$ give $e^{-\pi/2-2\pi k}$.