0
$\begingroup$

Possible Duplicate:
infinite product of sine function

Here is an other one which is more or less what Euler did in one of his proofs.

The function sinx where x∈R is zero exactly at x=nπ for each integer n. If we factorized it as an infinite product we get

How to prove $ \sin x=...(1+\frac{x}{3\pi})(1+\frac{x}{2\pi})(1+\frac{x}{\pi})x(1-\frac{x}{3\pi})(1-\frac{x}{2\pi})(1-\frac{x}{\pi})... $

1 Answers 1

1

Courtesy of Edmund Landau, from his Differential and Integral Calculus.

enter image description here enter image description here enter image description here enter image description here

enter image description here

  • 0
    @Miao (Probably very late, but) Fichtenholz has a (IMO much nicer) proof of this fact in Vol. 2 Art. 408. When reading it, it helps to remember that sinc x = sin x / x appears in the [Whittaker interpolation formula](https://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula), and that the analogous interpolation formula for functions on a circle involves things like sin(2n+1)x / sin x instead, compare also [this summation formula](https://math.stackexchange.com/questions/17966/how-can-we-sum-up-sin-and-cos-series-when-the-angles-are-in-arithmetic-pro).2017-08-06