Contour Integration with $\cos(n \theta)$ I need to compute the following real integral using complex numbers. I'm unsure how to handle the numerator so that the ensuing calculations do not become too unwieldily. $\int_{0}^{2\pi} \frac{ \cos n\theta}{\cos(\theta)-a} d\theta$ Thanks!

Let $I(a)$ be given by

$$I(a)=\int_0^{2\pi} \frac{\cos (n\theta)}{cos \theta -a}\,d\theta$$

Using Euler's Formula, we can write $I(a)$ as

$$I(a)=\text{Re}\left(\int_0^{2\pi} \frac{e^{in\theta}}{cos \theta -a}\,d\theta\right) \tag 1$$

Now, letting $z=e^{i\theta}$ in $(1)$ reveals

$$\begin{align} I(a)&=\text{Re}\left(-2i\oint_{|z|=1}\frac{z^n}{z^2-2az+1}\,dz\right)\\\\\\\ &=4\pi \text{Res}\left(\frac{z^n}{z^2-2az+1},z=a-\sqrt{a^2-1}\right)\\\\\\\ &=-2\pi \frac{\left(a-\sqrt{a^2-1}\right)^n}{\sqrt{a^2-1}} \end{align}$$