
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/(zg)(zf)(zk)(zL) 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

I am not sure you are doing this right:
$\displaystyle \int_0^{2\pi} \frac{adx}{a^2+\sin ^2 x} = \oint_{z=1} \frac{a}{a^2 + \left[ \frac{1}{2i}(z+z^{1}) \right]^2} \cdot \frac{1}{iz} \cdot dz$

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?