Homework Help: Trigonometric integral - help needed

1. Oct 7, 2006

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!

2. Oct 7, 2006

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).

3. Oct 7, 2006

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.

4. Oct 7, 2006

arildno

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

5. Oct 7, 2006

von_biber

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