I know that $\sin(x)$ can be expressed as an infinite product, and I've seen proofs of it (e.g. Infinite product of sine function). I found How was Euler able to create an infinite product for sinc by using its roots? which discusses how Euler might have found the equation, but I wonder how Euler could have proved it.

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

So how did Euler derive this? I've seen a proof that requires Fourier series (something not know [formally] by Euler, I guess). I also know that this equation can be thought intuitively, and it's really true that it will have the same roots as the sine function, however it's not clear that the entire function converges to the sine function. So, even if Euler guessed it, how was his proof accepted, for this formula to calculate the zeta function for even integers?

$$\zeta(2n) = (-1)^{n+1} \frac{B_{2n}(2\pi )^{2n}}{2(2n)!}$$

I've checked a proof of this result, and it requires the sine infinite product.

Also, the Basel problem (solved by him) used this infinite product too, and he got famous by this proof, so the sine infinite product might have been accepted by the mathematical community at that time.

up vote 5 down vote accepted

Check out this link: Ed Sandifer: How Euler Did It, March 2004

And here: Wikipedia: Basel Problem

The Wikipedia link shows there actually does exist an "elementary" proof of the generalized Basel problem (the evaluation of the Riemann zeta function for positive even integers). Graham, Knuth, and Patashnik, in the text Concrete Mathematics, also give an outline of another elementary proof which is easy to follow, using properties of exponential generating functions and some basic calculus. Although it is not how Euler went about it, the approach certainly would have been within his scope of knowledge.

  • Do you have any information about the first one who proved this? – Lucas Zanella Feb 18 '14 at 4:49
  • @LucasZanella it was probably first proven rigorously by Weierstrass. – Ian Mateus Mar 9 '14 at 21:55

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