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Complex Roots

  1. Aug 26, 2009 #1
    Hello all,

    I've got a bit of a problem trying to find and plot solutions to this equation:

    [tex]z^{4}=1-i\sqrt{3}[/tex]

    I'm ok to plot things on an Argand diagram and I know there will be 4 to find, but my sources only explain explicitly how to find the nth roots of unity. Any help would be greatly appreciated =]
     
  2. jcsd
  3. Aug 26, 2009 #2
    express the right hand side in the polar form: |r| e^(x + 2*n*pi)

    then you can take roots on both sides explicity and solve from there.
     
  4. Aug 26, 2009 #3
    Would I be right in doing this this?

    [tex]z^{4}=2e^{in\frac{\pi }{3}}[/tex]

    [tex]z=2e^{\frac{in\frac{\pi }{3}}{4}}[/tex]

    with different roots being multiples of n, up to 5?
     
  5. Aug 26, 2009 #4
    If you draw it out on an Argand idagram, the argument is [tex] - tan^{-1} \sqrt{3} = -\frac{\pi}{3}[/tex]

    so [tex]z^4 = 2e^{(-\frac{\pi}{3} + 2n\pi)}[/tex]
     
  6. Aug 26, 2009 #5
    [tex]1-i\sqrt{3} = 2 e^{i(5\pi/3)}[/tex]

    [tex]z^n=re^{i\theta} \Rightarrow z = \sqrt[n]{r}\; e^{i(\theta +2k\pi)/n} \text{ for }k=0,1,2, \dots, n-1[/tex]

    --Elucidus
     
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