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$\def\Arg{\operatorname{Arg}}$I have here a complex equation:

$$z^2-\Arg(z)=z\overline z, \qquad z\in \mathbb C$$

where $\Arg(z)$ is the argument of $z$, and $\overline z$ is complex conjugate of $z$.

How do we get the solutions of this equation? I started replacing $z=x+iy$ but I can't continue. Any suggestions please?

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  • $\begingroup$ Try using polar form ($z = re^{i\theta}$). $\endgroup$ – Javier Nov 7 '12 at 12:38
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$$z=re^{it}\Longrightarrow z^2-\arg z=r^2e^{2it}-t\,\,,\,\,z\overline z=|z|^2=r^2\Longrightarrow$$

$$r^2e^{2it}-t=r^2\Longleftrightarrow r^2\cos 2t-t+r^2i\sin 2t=r^2$$

Now compare real and imaginary parts:

$$r^2\cos 2t-t=r^2$$

$$r^2\sin 2t=0$$

Can you take it from here?

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    $\begingroup$ Yes, thank you! $\endgroup$ – Mark Nov 7 '12 at 13:38

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