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Math Help - Find solutions of z bar = z^2

  1. #16
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    Re: Find solutions of z bar = z^2

    Quote Originally Posted by terrorsquid View Post
    Im trying find all solutions for: \bar{z}=z^2
    Work in polar form.

    \overline{z}=z^2,

    so:

    |\overline{z}|=|z|=|z^2|=|z|^2

    so for non-zeros solutions |z|=1, and z=e^{i\theta} for some \theta \in [0,\2\pi)

    Then:

    \overline{z}=e^{-i\theta}=z^2=e^{2i\theta}

    etc

    CB
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  2. #17
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    Re: Find solutions of z bar = z^2

    Can someone explain how |\bar{z}| gives you |z|^2 ? I can see that z\bar{z} =|z|^2 but how does |z| = |z|^2 ?
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  3. #18
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    Re: Find solutions of z bar = z^2

    Quote Originally Posted by terrorsquid View Post
    Can someone explain how |\bar{z}| gives you |z|^2 ? I can see that z\bar{z} =|z|^2 but how does |z| = |z|^2 ?

    You are looking for solutions to:

     \overline{z}=z^2

    now just take absolute values and use |z^2|=|z|^2.

    CB
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  4. #19
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    Re: Find solutions of z bar = z^2

    Oh, right. I think I misread an equation earlier. \bar{z} = |z| so the new equation is:

    |z| = |z|^2

    I read a comment as |z| = |z|^2 as an identity separate to my equation - nevermind :S

    Thanks.

    Out of curiosity how would you derive these solutions if I converted the original equation into:

    r(cos(\theta) - isin(\theta)) = [r(cos(\theta)+isin(\theta))]^2

    can you?
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  5. #20
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    Re: Find solutions of z bar = z^2

    Quote Originally Posted by terrorsquid View Post
    So applying this geometrically I would have a solution set of cos(0+\frac{2\pi k}{3}) where k = 0, 1, 2

    Which is the same as the z = 1, z=-\frac{1}{2}+i\frac{\sqrt{3}}{2} and z=-\frac{1}{2}-i\frac{\sqrt{3}}{2}

    But where does the z = 0 fit in?
    These are the solutions assuming z was NOT 0. Since you have already determined that z= 0 satisfies the original equation, it has four solutions.
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