I have gotten stuck on this question:
$$f(l,q)=\int_{-\pi}^{\pi} \frac{e^{-i l \theta}}{1-q \cos \theta} d\theta$$ where l is an integer and q is a complex number with |q|<1
I am supposed to introduce a suitable complex integration variable to produce a contour integral for f, so I could solve it using the residue theorem. I have tried $z=e^{i \theta}$ and $z=q e^{i \theta}$ but so far have only produced extremely messy results. If someone could let me know if I have gone about this wrong or give me a hint that would be great. (Please no actual full answers for the problem as this is an assignment) Thanks!