From quite some time I'm struggling on proving the Fourier series expansion of $ \cos(\alpha n) $:
$$\cos(\alpha \theta) = \dfrac{\sin(\alpha \pi)}{\alpha \pi}\left(1 + 2\alpha^2\sum_{m = 1}^{+\infty}\dfrac{(-1)^{m-1}\cos(m\theta)}{m^2 - \alpha^2}\right)$$
Note: Here $\alpha \in \mathbb{C}- \mathbb{Z}$
I have no idea where to start. Any help would be appreciated. Thanks for reading.