How to Evaluate this Integral of a Green's Function

Question

Let $n\in\mathbb{Z}$ and $z\in\mathbb{C}-\mathbb{R}$ be given. I have the following integral: $$ \frac{1}{2\pi}\int_0^{2\pi}\frac{e^{i k n}}{2\left[1-\cos\left(k\right)\right]-z}dk$$which I am not sure how to solve, because the boundaries of integration are not infinite, so I'm not sure how contour integration and residues would lend themselves to this problem...

Answer 1

Let $I(z)=\frac1{2\pi}\int_0^{2\pi}\frac{e^{ikn}}{2(1-\cos(k))-z}\,dk$.

Enforcing the substitution $w=e^{ik}$ so that $dk=\frac{1}{iw}\,dw$, $I(z)$ can be written

$$I(z)=\frac{1}{2\pi i}\oint_{|w|=1} \frac{w^n}{-w^2+(2-z)w-1}\,dw \tag 1$$

The integral in $(1)$ can be evaluated using the residue theorem.

Can you finish now?