hi,
'Find all the solutions of z bar= z^3 in cartesian form ?', thanks in advance for any help
Let $\displaystyle z = r\:e^{i\theta}$
$\displaystyle r\:e^{-i\theta} = r^3\:e^{3i\theta}$
leads to $\displaystyle r = r^3$ and $\displaystyle 3\theta = -\theta +2k\pi$
$\displaystyle r = r^3$ is solved by $\displaystyle r \r^2-1)=0$ which leads to $\displaystyle r=0$ or $\displaystyle r=1$
$\displaystyle 3\theta = -\theta +2k\pi$ leads to $\displaystyle \theta = k\:\frac{\pi}{2}$
The solutions are therefore $\displaystyle z=0, z=1, z=i, z=-1, z=-i$