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Thread: Complex numbers

  1. #1
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    Complex numbers

    There's another question that I'm -completely- stuck on, I can't even think of how to start solving this one:

    Find all complex numbers z for which $\displaystyle z^3 = -4 \bar z$

    If someone is able to explain that to me, thanks.
    Last edited by topsquark; Jun 3rd 2008 at 03:46 AM.
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  2. #2
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    Quote Originally Posted by Wedric View Post
    Find all complex numbers z for which $\displaystyle z^3 = -4 \bar z$
    Hello Wedric

    first write z in "mod/arg" form. $\displaystyle z = Re^{i \theta }$ so $\displaystyle z^3 = R^3e^{i 3\theta }$ and $\displaystyle \bar z = Re^{-i \theta}$

    $\displaystyle z^3 = -4 \bar z \ \ \Rightarrow \ \ R^3e^{i 3\theta } = -4Re^{-i \theta} $

    $\displaystyle R^2e^{i 4\theta } = -4$

    $\displaystyle R^2 \cos 4\theta + R^2 i \sin 4 \theta = -4$

    Require $\displaystyle \sin 4 \theta = 0$
    and $\displaystyle \cos 4\theta = -1$
    $\displaystyle R=2$
    $\displaystyle 4 \theta = 2 \pi n +\pi $
    $\displaystyle \theta = \frac{ \pi }{4} (1+ 2n) \ \ \forall n \in \mathbb{Z}$

    $\displaystyle \therefore z = 2e^{i\frac{ \pi }{4} (1+ 2n)}$

    Bobak
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  3. #3
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    Thanks for the reply, I'll mull over that for a while.
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