Fourier coefficients of $\cos(x/2)$

Question

Is there a straightforward way to calculate the fourier coefficients of $\cos(x/2)$ in closed form on the interval $[0,2\pi]$? (I mean in terms of the generic $n$)

From a calculation of the first integrals the trend seems $a_n=2\frac{(-1)^{(2k+1)}}{\pi(2n+1)(2n-1)}$ but how do I get it exactly?

Thanks!

Answer 1

Evaluate

$$\frac1{T_0}\int ^{T_0}_0 f(x) e^{-i k \omega _ 0 x}dx,\quad\omega_0=\frac{2\pi}{T_0}$$

Where $T_0$ is the period

$$\frac1{2\pi}\int ^{2\pi}_0 \cos (x/2) e^{-i k x}dx$$