Trigonometric Integral Help: Solving for a<1 Using Complex Contour Method

[von_biber]

need some assistance with the following integral:

\int_0^{2\pi} cosx/(a-cosx), a-parameter (say a>0)

i've converted it into a complex contour integral over z=e^(ix):

~ \int_{|z|=1} dz (z^2+1)/[z(z^2-2az+1)]

which is easily evaluated for a>1. my question regards a<1 - i am not sure how to solve it in this case, because the the 2 poles

z_1=a+Sqrt[a^2-1], z_2=a-Sqrt[a^2-1]

are exactly on the unit circle and off the real axis. thanks for any suggestions!

 

[arildno]

Hmm..is it really necessay to venture out in complexity here?
Rational expressions in the trig. functions are easily handled by the transformation u=tan(x/2).

 

[von_biber]

it get's a bit complicated with this tan(x/2) substitution, with complex substitution it's very easily evaluated for a>1 and i'd like to find out what happens for a<1.

 

[arildno]

Well, for a<1, the integral will readily diverge, so no wonder if you get into some problems.

 

[von_biber]

anyway, what is the principal value for a<1?
thanks for your help, btw. i really appreciate it.