Imaginary^Imaginary

Algebra Level 4

$i^i = \big(e^{ i \pi / 2} \big)^i = e^{- \pi / 2}$ 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.

k \ln k = n , n \in \mathbb{N}

k \ln k = 2n, n \in \mathbb{N}

k \ln k = 2 n \pi, n \in \mathbb{N}

k \ln k = n \pi, n \in \mathbb{N}