Past exam of the mathematical methods I course of the University of Oxford / 2015 / 1 / Solution

Use polar coordinates $z=r{e}^{i\theta }$, where $r=|z|\ge 0$. The equation sets $r=\theta$, giving an Archimedean spiral. The origin is excluded because the complex argument of zero is undefined. With the principal complex argument in $(-\pi ,\pi ]$, the locus isIt approaches the excluded origin along the positive real axis, passes through $i\pi /2$, and ends at $-\pi$. The choice of argument matters: the convention $[0,2\pi )$ instead gives $0<\theta <2\pi$, while allowing all values of the complex argument gives every turn with $\theta >0$.