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

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

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


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

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

From the Mittag-Leffler Expansion for the Cosecant Function, we have:

Due to the equality of Lines 3 and 4 above, we now have:

For $\theta = \pi$:
 * $\ds \sum_{n \mathop \in \Z} \dfrac {\map \sin {n + \alpha} \theta} {n + \alpha} = \pi$

To establish this identity for all other values of $\theta$ on the interval $0 < \theta < 2\pi$, we will demonstrate that the sum is a constant function.

We will do this by showing that the derivative of the function is zero everywhere which by Zero Derivative implies Constant Function will complete the proof.

We have:

Next, we utilize the Cosine of Sum angle identity and split the sum.

From Cosine Function is Even and Sine Function is Odd, we have:


 * $\map \cos {-n \theta} = \map \cos {n \theta}$ and
 * $\map \sin {-n \theta} = -\map \sin {n \theta}$

Therefore:

Also see

 * Sum over Integers of Cosine of n + alpha of theta over n + alpha