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Thread: Find all complex solutions

  1. April 26th 2009, 08:47 AM #1
    posix_memalign
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    Find all complex solutions

    How can I find all complex solutions of the following?

    z^4 = -1 + sqrt(3*i)
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  2. April 26th 2009, 11:45 AM #2
    running-gag
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    Quote Originally Posted by posix_memalign View Post
    How can I find all complex solutions of the following?

    z^4 = -1 + sqrt(3*i)
    Hi

    Isn't it z^4 = -1 + \sqrt{3}\:i

    One way is to go to exponential form
    z^4 = 2\left(-\frac12 + \frac{\sqrt{3}}{2}\:i\right) = 2\:e^{i\frac{2\pi}{3}}

    Then z = r\:e^{i\theta}\implies z^4 = r^4\:e^{4i\theta}

    r^4 = 2 and 4\theta = \frac{2\pi}{3} + 2k\pi

    r = 2^{\frac14} and \theta = \frac{\pi}{6} + k\frac{\pi}{2}
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