I'm trying to represent these complex numbers:
$$ \left\{ \begin{array}{c} z^3=i\overline{z}|z| \\ ||z-i|-i|<\sqrt 2 \\ \end{array} \right. $$
The solutions of the first equation are:
$z_0=0$
$z_1=e^{i\frac{\pi}{8}}$
$z_2=e^{i\frac{5\pi}{8}}$
$z_3=e^{i\frac{9\pi}{8}}$
$z_4=e^{i\frac{13\pi}{8}}$
$z_5=e^{i\frac{17\pi}{8}}$
I seem to get stuck when I have to find the solutions of the second equation.
I think that the result might be a circle without boundary.
For second part, let $|z-i|=p$, where $p \in \Bbb R^+$.
We are given that $$|p-i|<\sqrt 2 \implies \sqrt{p^2+1} <\sqrt 2 \implies p <1$$
So you need to solve for $$|z-i|<1$$
Which is simply the interior of a circle of unit radius and centre at $(0,i)$.