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How to prove that $$\frac{\sin \pi x}{\pi x}=\prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2}\right)$$

I tried it with the Taylor series of $\sin(x)$ but I failed.

Is there any help?

Davide Giraudo
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mnsh
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    This is a prime example of the Weierstrass Factorization Theorem. http://en.wikipedia.org/wiki/Weierstrass_factorization_theorem, although you may be looking for a more elementary proof than this. – Eric Auld Jul 19 '13 at 00:47
  • @EricAuld thanks but it is not clear for me and it is better if there is another one :) – mnsh Jul 19 '13 at 00:55
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    You should check out Journey Through Genius by William Dunham. He goes through the intuition behind this factorization when Euler (?) did it but when it was first realized, there was little to no rigor to the arguments. The "proof" he presents that Euler used is good enough to get the idea, though. – Cameron Williams Jul 19 '13 at 00:55
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    You should probably read this excellent article :-)

    http://cornellmath.wordpress.com/2007/07/13/eulers-nonstandard-nonsense/

     – TenaliRaman Jul 19 '13 at 01:38
  • I saw the beautiful and interesting answer in Journey Through Genius but is there another answer ? – mnsh Jul 19 '13 at 02:42
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    Euler argued that because $\frac{\sin \pi x}{\pi x}$ has roots at $x = \pm1, \pm2, \pm3, ...$, the Factor theorem allows you to write $\frac{\sin \pi x}{\pi x} = (1 - x)(1 + x)(2 - x)(2 + x)...(n - x)(n + x)... = (1 - x^2)(2^2 - x^2)...(n^2 - x^2)... = \prod_{n=1}^{\infty}(1-\frac{x^2}{n^2})$. – Matthew Hampsey Jul 19 '13 at 02:47
  • Have you tried taking logs and differentiating ? See what happens... I don't know if this will lead anywhere; just a thought. – pshmath0 Jul 19 '13 at 10:21
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    • – Start wearing purple Jul 19 '13 at 10:27