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Complex integral

  1. Feb 7, 2007 #1
    1. The problem statement, all variables and given/known data
    2. Relevant equations


    I hope there's someone who can help me with the following:

    I have to calculate the integral over C (the unit cicle) of (z+(1/z))^n dz, where z is a complex number.

    3. The attempt at a solution


    I tried to use the subtitution z=e^(i*theta), so you get
    (z+(1/z))^n dz=(2*i*Sin(theta))^n * i*e^(i*theta) dtheta
    but then I get stuck.
    Is this the right way, and if, how do I proceed. And if it isn't, how should I do it???
     
    Last edited: Feb 7, 2007
  2. jcsd
  3. Feb 7, 2007 #2

    mjsd

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    Homework Helper

    i suppose you integral looks like:
    [tex]\int_{|z|=1} (z+1/z)^n dz = i \int_0^{2\pi} (e^{i\theta}+e^{-i\theta})^n e^{i\theta} d\theta[/tex]

    now did what you did then also try to expand the remaining [tex]e^{i\theta}=\cos \theta +i \sin \theta[/tex], and now you end up with two integrals with just cos and sin.... you can then do the integral for two cases n odd and n even... etc...
     
  4. Feb 7, 2007 #3
    Ok, but then you get:

    (2*cos(theta))^n *e^(i*theta)

    but I don't know how to get rid of the n...

    (Don't know to use latex...)
     
  5. Feb 7, 2007 #4

    mjsd

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    Homework Helper

    i said to use [tex]e^{i\theta}=\cos \theta +i \sin \theta[/tex] to expand the second exponential.. and then multiply out to get something like
    [tex]\cos^{n+1} \theta + \cos^n \theta \sin \theta[/tex] and now you can try integrate these assuming that n is an integer. I am guessing that there will be two cases: n odd an n even
     
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