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Raise complex number using De Moivre - integer only?

  1. Mar 27, 2012 #1
    This is probably a silly question, but it is not really clear to me whether De Moivre's theorem of raising a complex number to the nth power only work if n is an integer value?
    E.g. if I try to raise (2-2i) to the power of 3.01 then my manual calculation get a different result than my calculator, but for all integer values it works.
     
  2. jcsd
  3. Mar 27, 2012 #2

    cepheid

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    Well, at least according to the Wikipedia article on the subject, 'n' must be an integer:

    http://en.wikipedia.org/wiki/De_Moivre's_formula
     
  4. Mar 27, 2012 #3
    >>Well, at least according to the Wikipedia article on the subject, 'n' must be an integer:
    argh, I missed that. Thanks
    Do you know how I can raise a complex number by a real number?
     
  5. Mar 27, 2012 #4
    I may be wrong but in mathematics, n is used to denote an integer by convention.
     
  6. Mar 27, 2012 #5
    >>I may be wrong but in mathematics, n is used to denote an integer by convention.
    I think you are right.
     
  7. Mar 27, 2012 #6
    You can raise it to any power by using this method:

    [itex]z^w = e^{ln|z^w|} = e ^{w \cdot ln|z|}[/itex] and simplify from there. From the top of my head, I believe this formula gives an infinite number of solutions depending on how you choose your branch of the logarithm.
     
  8. Mar 28, 2012 #7

    HallsofIvy

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    No, the power does not have to be an integer. DeMoivre's formula is very commonly used, for example, to find roots, the "1/n" power.
     
  9. Mar 28, 2012 #8
    After some more Bing'ing I found this
    http://www.suitcaseofdreams.net/De_Moivre_formula.htm

    For r = 1 we obtain De Moivre’s formula for fractional powers:
    (cos+i sin)p/q = cos((p/q))+i sin((p/q)). (1.27)


    So the decimal power should work as well, right?. I will try and convert the decimal to a fraction, but I dont see why it would make a difference.

    When I tried it worked correct for quadrant I and II, but I could not get it right for III and IV.

    I will keep playing with this.
     
  10. Mar 29, 2012 #9
    I played some more with this and it was all just a matter of adjusting θ depending of the quadrant.
    For quadrant II
    θ=(atan(im(z)/re(z))+pi)*n, where n is any number
    And for III
    θ=(atan(im(z)/re(z))-pi)*n
     
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