Complex Roots

  • Thread starter Diode
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  • #1
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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 =]
 

Answers and Replies

  • #2
211
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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.
 
  • #3
8
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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?
 
  • #4
211
0
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]
 
  • #5
286
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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?
[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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