- By integrating the function \[z^n (z - a)^{-1} (z - a^{-1})^{-1}\] around the unit circle and applying the residue theorem, evaluate \[\int_{0}^{2\pi} \frac{\cos n\theta}{1 - 2a\cos\theta + a^2}\, d\theta,\tag{1.1}\] where $a$ is real, $a > 1$, and $n$ is a non-negative integer.
- Obtain the same result using Cauchy’s integral formula instead of the residue theorem.
Contour Integration via Residue Theorem and Cauchy’s Integral Formula
Source: J. E. Santos, Cambridge Part IB Complex Methods, Example Sheet 2, Question 9.