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I've been trying to solve the following equation for $\Theta$ and $r$: $z^{3}=icon(z)|z|$

where $con(z)$ is the complex conjugate of z.

The steps I've followed are:

Convert to exponential form: $ z = re^{i\Theta}, z^3 = r^3e^{3i\Theta}, con(z)=re^{-i\Theta}$

By substituting I get:

$$re^{i3\Theta}=ie^{-i\Theta}$$

Solving for $r$:

$$r=ie^{-4i\Theta}$$

How should I proceed from this point?

  • 1
    $icon(z)=\overline{-iz}$ ? – 2017-01-28

1 Answers 1

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Since $r\ge0$ is a real number, and $|ie^{-4i\theta}|=1$ for $\theta\in[0,2\pi)$, it follows that $r=1$. Thus,

$$1=ie^{-4i\theta}\implies e^{4i\theta}=i=e^{\frac\pi2i}\implies \theta=\frac\pi8,\frac\pi8+\dots$$