Are there complex solutions for $z^3=\bar z$ I'm asked to solve $z^3=\bar z$. I got $z=0, 1, -1$. Are there any complex solutions $a+bi$  to this though? 

Writing $z = re^{it}$, we have that
$$r^3 e^{3it} = re^{-it}$$
Taking absolute values, we find that $r^3 = r$, so that $r = 0$ or $r = \pm 1$. In the second two cases, we get that
$$e^{3it} = e^{-it} \implies e^{4it} = 1$$
It follows that $4t$ is an integer multiple of $2\pi$, so there are corresponding complex solutions.
