Mittag-Leffler Expansion for Cosecant Function/Real Domain

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

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

Proof
From Half-Range Fourier Cosine Series for $\cos \alpha x$ over $\openint 0 \pi$:


 * $\displaystyle \cos \alpha x \sim \frac {2 \alpha \sin \alpha \pi} \pi \paren {\frac 1 {2 \alpha^2} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {\cos n x} {\alpha^2 - n^2} }$

Setting $x = 0$: