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Math Help - working out residue

  1. #1
    Newbie
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    Sep 2007
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    working out residue

    Hi
    im stuck halfway through this problem.
    The question is: if a>0, integral from 0 to 2pi of
    I= [a dX]/[a^2+ sin^2(X)]=pi/(1+a^2)^.5

    i let z=e^iX. then did some working and have
    I=2/i integral az dz/(z-g)(z-f)(z-k)(z-L) where g=a-(a^2+1)^.5
    f=a+(a^2+1)^.5 k==-a+(a^2+1)^.5 L=-a-(a^2+1)^.5

    i dont know how to work out the residue. could someone show me how using laurent series.

    Thanks
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  2. #2
    Global Moderator

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    I am not sure you are doing this right:

    \int_0^{2\pi} \frac{adx}{a^2+\sin ^2 x} = \oint_{|z|=1} \frac{a}{a^2 + \left[ \frac{1}{2i}(z+z^{-1}) \right]^2} \cdot \frac{1}{iz} \cdot dz
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  3. #3
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    Sep 2007
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    Thanks. yes thats the question.

    i multiplied the denominator through by z^2 then did some fiddeling and got 4 factors on the denominator.

    Its to complicated to use taylors theory to calculate the residue, i need some help calculating the residue via laurents series?
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