= Modulus of a complex self-power
{title2=$|z^z|=|z|^{\operatorname{Re}z}e^{-\operatorname{Im}z(\arg z+2\pi k)}$}
For $z\ne0$, every logarithmic value is $\log|z|+i(\arg z+2\pi k)$ with integer $k$. Multiplying by $z$ and taking the real part proves the displayed formula for the <modulus> of each value of <complex exponentiation>. For nonreal $z$, these values form a geometric progression with nonunit positive ratio $e^{-2\pi\operatorname{Im}z}$. For real nonzero $z$, changing the logarithm branch changes only the phase and leaves the modulus fixed. A fixed initial argument changes only the indexing of the set of values.
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