Solution
= Solution
All <complex logarithms> of $i$ are $i(\pi/2+2\pi k)$ for integer $k$. By <complex exponentiation>,
$$
\boxed{i^i=\exp\!\left[-\frac\pi2-2\pi k\right],\qquad k\in\mathbb Z.}
$$
Every value is positive and real. If the power is defined using the <principal logarithm>, the single value is $\boxed{e^{-\pi/2}}$. Keeping the logarithm branch explicit distinguishes this value from the full multivalued set.