Modulus of a complex self-power ($|z^z|=|z{|}^{\mathrm{Re}z}{e}^{-\mathrm{Im}z(\arg z+2\pi k)}$)

For $z\ne 0$, 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 $e^{-2\pi \mathrm{Im}z}$. 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.