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Math Help - sqrt of a complex number

  1. #1
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    sqrt of a complex number

    hello

    let's say I have

    z^2 = exp (1.5 * pi *i)

    how do I find the values of z which satisfy this equation

    thanks
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by parallel View Post
    hello

    let's say I have

    z^2 = exp (1.5 * pi *i)

    how do I find the values of z which satisfy this equation

    thanks
    Well, \left ( e^a \right ) ^b = e^{ab}, even when a and/or b is complex. Does this help?

    -Dan
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  3. #3
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    Quote Originally Posted by topsquark View Post
    Well, \left ( e^a \right ) ^b = e^{ab}, even when a and/or b is complex. Does this help?

    -Dan
    That is not correct when a,b\in \mathbb{C}.
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  4. #4
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    Quote Originally Posted by parallel View Post
    hello

    let's say I have

    z^2 = exp (1.5 * pi *i)

    how do I find the values of z which satisfy this equation

    thanks
    If z\not = 0 then \sqrt{z} = \sqrt{|z|} e^{i\arg(z)/2}.

    Now given e^{1.5 \pi i}. And \arg e^{1.5 \pi i} = -\frac{\pi}{2}. Thus, its square root is given by \sqrt{\left| e^{1.5\pi i} \right|} e^{-i\pi/4} = e^{-i\pi/4} = \frac{\sqrt{2}}{2}-i\frac{\sqrt{2}}{2}. The other root is the negative of this root.
    Last edited by ThePerfectHacker; September 20th 2007 at 08:15 AM.
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  5. #5
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    Quote Originally Posted by ThePerfectHacker View Post
    If z\not = 0 then \sqrt{z} = \sqrt{|z|} e^{i\arg(z)/2}.

    Now given e^{1.5 \pi i}. And \arg e^{1.5 \pi i} = -\frac{\pi}{2}. Thus, its square root is given by \sqrt{\left| e^{1.5\pi i} \right|} e^{-i\pi/4} = e^{-i\pi/4} = \frac{\sqrt{2}}{2}-i\frac{\sqrt{2}}{2}. The other root is the negative of this root.
    thank you very much for your help
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