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Math Help - 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 z^3 = -4 \bar z

    If someone is able to explain that to me, thanks.
    Last edited by topsquark; June 3rd 2008 at 04:46 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 z^3 = -4 \bar z
    Hello Wedric

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

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

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

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

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

    \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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