Sum over Integers of Sine of n + alpha of theta over n + alpha

Theorem
For $0 < \theta < 2 \pi$:


 * $\displaystyle \sum_{n \mathop \in \Z} \dfrac {\sin \paren {n + \alpha} \theta} {n + \alpha} = \pi$

Proof
First we establish the following, as they will be needed later.

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


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

From Half-Range Fourier Sine Series for $\sin \alpha x$ over $\openint 0 \pi$:


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