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:


 * $\displaystyle \pi \csc \pi \lambda = \sum_{n \mathop = 1}^\infty \paren {-1}^n \paren {\frac 1 {n + \lambda} + \frac 1 {n - 1 - \lambda} }$

Proof
Let $\map f x$ be the real function defined on $\openint 0 \pi$ as:


 * $\map f x = \cos \lambda x$

From Half-Range Fourier Cosine Series: $\cos \lambda x$ over $\openint 0 \pi$ its Fourier series can be expressed as:


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

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

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

So, setting $x = 0$:

This expression is fine as is, but to obtain the form we set out to prove, we observe:

and thus:

The result follows.