Mittag-Leffler Expansion for Cotangent Function/Real Domain

Theorem
Let $\alpha \in \R$ be a real number which is specifically not an integer.

For $0 < \theta < 2 \pi$:
 * $\displaystyle \dfrac 1 \alpha + \sum_{n \mathop \ge 1} \dfrac {2 \alpha} {\alpha^2 - n^2} = \pi \cot \pi \alpha$