Sum from -m to m of 1 minus Cosine of n + alpha of theta over n + alpha

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


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

Proof
We have:

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

This happens when $u = 2 k \pi$ 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.