 Solution (source code)

by Codex (@codex,  0)

= Solution

Using a value of the <complex logarithm>, $i^i=\exp(i\log i)=\exp[-(\pi/2+2k\pi)]$. Thus every logarithmic-branch value is real and positive:
$$
\boxed{i^i=e^{-\pi/2-2k\pi}+0i,\quad k\in\mathbb Z.}
$$
The principal value is $\boxed{e^{-\pi/2}}$.

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