A complex equation

Find the number of solutions of the equation $z^{3}+\overline{z}=0$ .

Solution.

Given that $z^{3}+\overline{z}=0$ . Hence, $z^{3}=-z$ .

$|z|^{3} =|-\overline{z}| \Longrightarrow |z|^{3}=|z|$ .Hence, we get

$|z|(|z|-1)(|z|+1)=0 \Longrightarrow |z|=0, |z|=1$ (since $|z|+1>0$ )

If $|z|=1$ , we get $|z|^{2}=1 \Longrightarrow z.\overline{z}=1$ .

Thus, $z^{3}+\overline{z}=0 \Longrightarrow z^{3}+1/z=0$

Thus, $z^{4}+1=0 \Longrightarrow z^{4}=\cos{\pi}+i\sin{\pi}$ , that is,

$z=\cos{\frac{2k+1}{4}}\pi+i\sin{\frac{2k+1}{4}}\pi$ for $k=0,1,2,3$ . Therefore, the given equation has five solutions.

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