\[ i^i = \big(e^{ i \pi / 2} \big)^i = e^{- \pi / 2} \] It is (initially) surprising that an imaginary number raised to an imaginary number results in a real number.

Characterize all positive \(k\) such that

\[ (ik)^{ik} \in \mathbb{R}. \]

Note: \( 0 ^ 0 \) is undefined.

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