Infinite Product Identity for Hyperbolic Sine

Question

Prove $\prod_{n\in\mathbb{N}\backslash\left\{ 0\right\} }\left(1+\left(\frac{\alpha}{\pi n}\right)^{2}\right)=\frac{\sinh\left(\alpha\right)}{\alpha}$.

I saw this formula in a book and have no idea how to approach it.

Answer 1

This is not a trivial exercise. One thing to point out is that the lefthandside and the righthandside have the same zeros, but from this to a full solution there is quite a road. This equation was originally proved by Euler. He gave several proofs, one of which is examined in this article. Other proofs rely on techniques developed by Weierstrass a century and a half later; see http://en.wikipedia.org/wiki/Infinite_product#Product_representations_of_functions