Question

Find all non-zero complex $z$ satisfying $\overline{z}={iz}^2$

Answer 1

Let $z=x+iy$

$\overline{z}=i{z}^2$

$x-iy=i(x^2-y^2+2xiy)$

$x-iy=i{x}^2-i{y}^2-2xy$

$x=-2xy-y=x^2-y^2$

$x(1+2y=0$

$x=0,y=-1/2$

If $x=0$

$y^2-y=0$

$y(y-1)=0$

$y=0,1$

If $y=-1/2$

$+\frac{1}{2}=x^2-\frac{1}{4}$

$x^2=\frac{3}{4}$

$x=\frac{\sqrt{3}}{2},\frac{-\sqrt{3}}{2}$

So, the values of $(x,y)$ are $(0,1),\left(\frac{-\sqrt{3}}{2}\frac{-1}{2}\right),\left(\frac{\sqrt{3}}{2}\frac{-1}{2}\right)$

$\therefore z=i,\frac{\sqrt{3}}{2}-\frac{i}{2},\frac{-\sqrt{3}}{2}-\frac{i}{2}$