# working out residue

• Sep 28th 2007, 08:25 PM
jules
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
• Sep 29th 2007, 04:29 PM
ThePerfectHacker
I am not sure you are doing this right:

$\displaystyle \int_0^{2\pi} \frac{adx}{a^2+\sin ^2 x} = \oint_{|z|=1} \frac{a}{a^2 + \left[ \frac{1}{2i}(z+z^{-1}) \right]^2} \cdot \frac{1}{iz} \cdot dz$
• Sep 29th 2007, 09:50 PM
jules
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?