Series Expansion for Pi Cosecant of Pi Lambda

Theorem
Let $\lambda \in \R \setminus \Z$ be a real number which is not an integer.

Then:


 * $\displaystyle \pi \csc \pi \lambda = \sum_{n \mathop = 1}^\infty \left({-1}\right)^n \left({\frac 1 {n + \lambda} + \frac 1 {n - 1 - \lambda}}\right)$

Proof
Let $f \left({x}\right)$ be the real function defined on $\left({0 \,.\,.\, \pi}\right)$ as:


 * $f \left({x}\right) = \cos \lambda x$

From Fourier Series: $\cos \lambda x$ over $\left({0 \,.\,.\, \pi}\right)$ its Fourier series can be expressed as:


 * $\displaystyle \cos \lambda x \sim \frac {2 \lambda \sin \lambda \pi} \pi \left({\frac 1 {2 \lambda^2} + \sum_{n \mathop = 1}^\infty \left({-1}\right)^n \frac {\cos n x} {\lambda^2 - n^2} }\right)$

Because of the nature of this expansion, we have that:
 * $f \left({0^+}\right) = f \left({0^-}\right)$

and so the expansion holds for $x = 0$.

So, setting $x = 0$: