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

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

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


 * $\ds \dfrac 1 \alpha + \sum_{n \mathop \ge 1} \dfrac {2 \alpha} {\alpha^2 - n^2} = \sum_{n \mathop \in \Z} \dfrac {\cos \paren {n + \alpha} \theta} {n + \alpha}$

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

We have:

Setting $\theta = 0$:

To establish this identity for all other values of $\theta$ on the interval $0 \le \theta < 2\pi$, 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:

Then:

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 Sine of n + alpha of theta over n + alpha