Complex integration

Question

$f(θ)=(1/1+r^2 -2rcosθ)$ , I have calculated this integral:
$\int_{0}^{2\pi} { f(θ) dθ }= (2\pi/{r^2} -1)$ ,How to find:
Integral(θ= 0 ->2pi) (cos(nθ)*f(θ)) dθ

Answer 1

The function defined by $f(\theta)=\dfrac{1}{1+r^2 -2r\cos\theta}$ is called the Poisson Kernel.

You are asked to find the Fourier coefficients $a_n$ of $f$.

In https://en.wikipedia.org/wiki/Poisson_kernel it is explained why the complex Fourier coefficients are $c_n=r^{|n|}$.

All you have to do is to use the (classical) formulas linking coefficients $a_n, b_n$ ($n \in \mathbb{N}$) with coefficients $c_n$ ($n \in \mathbb{Z}$: caution!)