Question

Solve the equation $z^2=\overline{z}$

Answer 1

$z^2=z$

$(x+iy{)}^2=(x+iy)$

$x^2-y^2+2ixy=x+iy$

$x^2-y^2=x$ $x=\frac{1}{2}$

$\left(\frac{1}{4}\right)-y^2=\frac{1}{2}$

$-y^2=\frac{1}{2}+\frac{1}{4}=\frac{2+1}{4}=\frac{3}{4}\Rightarrow y=\frac{+\sqrt{3}i}{2}$

$z=\frac{1}{2}-\frac{i\sqrt{3}}{2}$