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
We have:

Note that the at $(1)$ is not defined when $e^{i \theta} = 1$.

This happens when $\theta = 2 k \pi$ for $k \in \Z$.

For the given range of $0 < \theta < 2 \pi$ it is therefore seen that $(1)$ does indeed hold.

Then:

Hence the result.