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Evalutaion of a real integral using the residue theorem

  1. Nov 1, 2011 #1
    Evaluate the following integral using the residue theorem:

    gif.latex?%5Cdpi{120}%20%5Cint_{0}^{%5Cpi%20}%5Csin^{2n}(x)%20dx.gif

    Any hint?
     
  2. jcsd
  3. Nov 1, 2011 #2

    lurflurf

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    2i sin(x)=eix-e-ix
    and
    (n+1)undu=dun+1
    with u=eix is
    (n+1)einxdeix=dei(n+1)x
     
  4. Nov 2, 2011 #3
    Sorry, I can't see how to use these facts. Can you give any further hint, please?
    Besides, what contour should be chosen?

    Thanks
     
    Last edited: Nov 2, 2011
  5. Nov 2, 2011 #4

    lurflurf

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    contour is unit circle
    let
    z=ei x
    dx=dz/(i z)
    sin(x)=((z-1/z)/(2i))
    sin2n(x)=((z-1/z)/(2i))2n

    [tex]\int_0^\pi \sin^{2n}(x) dx=\frac{1}{2}\int_{-\pi}^\pi \sin^{2n}(x) dx=\oint_{|z|=1}\left( \frac{z-\frac{1}{z}}{2i}\right)^{2n}\frac{dz}{2i z}[/tex]
     
  6. Nov 3, 2011 #5
    Thanks! :smile:
     
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