Sum from -m to m of Sine of n + alpha of theta over n + alpha

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


 * $\displaystyle \sum_{n \mathop = -m}^m \dfrac {\sin \paren {n + \alpha} \theta} {n + \alpha} = \int_0^\theta \map \cos {\alpha \theta} \dfrac {\sin \paren {m + \frac 1 2} \theta \rd \theta} {\sin \frac 1 2 \theta}$

Proof
See Sum of Sines of Arithmetic Sequence of Angles.

We have: