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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.

Mikhail Katz
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  • It follows from the Euler product for $\sin$: $$\sin \pi z = \pi z \prod_{n=1}^\infty \left(1-\frac{z^2}{n^2}\right)$$ But you'd obviously want to prove that, too. – Thomas Andrews May 11 '14 at 14:04
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    See http://math.stackexchange.com/questions/157372/infinite-product-of-sine-function and http://math.stackexchange.com/questions/786046/is-there-an-elementary-proof-for-eulers-product-for-sine/786231#786231 and replace $z$ by $iz$. – user91500 May 11 '14 at 14:29

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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

Mikhail Katz
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