Mittag-Leffler Expansion for Cosecant Function/Real Domain

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

Then:
 * $\displaystyle \sum_{n \mathop \ge 1} \dfrac {\paren {-1}^n} {\alpha^2 - n^2} = \dfrac {\pi \alpha \cosec \pi \alpha - 1} {2 \alpha^2}$