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Thread: equation involving complex numbers

  1. #1
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    equation involving complex numbers

    hi,
    'Find all the solutions of z bar
    = z^3 in cartesian form ?', thanks in advance for any help
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  2. #2
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    Quote Originally Posted by speckmagoo View Post
    hi,
    'Find all the solutions of z bar
    = z^3 in cartesian form ?', thanks in advance for any help
    Hi

    You can solve by setting $\displaystyle z=x+iy$ and expand $\displaystyle z^3 = (x+iy)^3$
    or more simply by setting $\displaystyle z = r\:e^{i\theta}$ and solve $\displaystyle r\:e^{-i\theta} = r^3\:e^{3i\theta}$
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  3. #3
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    hi thanks for ur assistance!!, id really appreciate if you could explain a little further as im not quite understanding how to go about using the approach u suggested, cheers
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  4. #4
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    Let $\displaystyle z = r\:e^{i\theta}$

    $\displaystyle r\:e^{-i\theta} = r^3\:e^{3i\theta}$

    leads to $\displaystyle r = r^3$ and $\displaystyle 3\theta = -\theta +2k\pi$

    $\displaystyle r = r^3$ is solved by $\displaystyle r \r^2-1)=0$ which leads to $\displaystyle r=0$ or $\displaystyle r=1$

    $\displaystyle 3\theta = -\theta +2k\pi$ leads to $\displaystyle \theta = k\:\frac{\pi}{2}$

    The solutions are therefore $\displaystyle z=0, z=1, z=i, z=-1, z=-i$
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  5. #5
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    Talking

    hi,
    thanks for ur assistance again, it now makes more sense, thanku
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