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Homework Help: Patterns found in complex numbers

  1. Mar 28, 2012 #1
    Patterns found in complex numbers URGENT!!!!

    • use de moivre's theorem to obtain solutions to z^n = i for n=3, 4, 5
    • generalise and prove your results for z^n = 1+bi, where |a+bi|=1
    • what happens when |a+bi|≠1?

    Relevant equations[/b]
    r = √a^2 + b^2
    z^n = r^n cis (nθ)


    This is what i have done:
    z^3=i
    z^3=i cis(0)
    z^3=cis(π/2+2kπ),k=0,1,2
    z=cis(π/6+2kπ/3),k=0,1,2
    z=cis(π/6),cis(π/6+2π/3),cis(π/6+4π/3)
    z=√3/2+0.5i,-√3/2+0.5i,- √3/2-0.5i

    but the 3rd solution is incorrect, should be -i. what have i done wrong?
     
  2. jcsd
  3. Mar 29, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    welcome to pf!

    hi lll030lll! welcome to pf! :smile:
    dunno, but π/6+4π/3 = 9π/6 :redface:

    (personally, i find degrees easier … 30°, 30° ± 120° :wink:)
     
  4. Mar 29, 2012 #3
    Re: Patterns found in complex numbers URGENT!!!!

    anyway, got it right in using a+bi=re^iθ
     
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