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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