Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 7 / d

For $z=r{e}^{i\theta }\ne 0$, fix one argument $\theta$. Every value of the complex logarithm is $\log r+i(\theta +2\pi k)$, $k\in \mathbb{Z}$. By complex exponentiation, $z^z=\exp [z(\log r+i(\theta +2\pi k))]$. Taking the real part of its exponent gives all possible moduli:Equivalently these are $r^{\mathrm{Re}z}{e}^{-\mathrm{Im}z(\theta +2\pi k)}$. If $z$ is real and nonzero the modulus is the single value $r^{\mathrm{Re}z}$; otherwise the values form a geometric progression with ratio $e^{-2\pi \mathrm{Im}z}$. Changing the initially chosen argument just reindexes the same set. The logarithmic definition excludes $z=0$. This is the modulus of a complex self-power.