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

john
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Miao
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1 Answers1

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

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Pedro
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  • I don't have that book, and which part of a usual book on Mathematical Analysis could I find the proof in,like Zorich's 《Mathematical Analysis》 or ГригорийМихайлович Фихтенгольц‘s 《Calculus course》. I mean what topic it is included in? – Miao Oct 17 '12 at 23:36
  • I need theorem 233 – PPP Sep 18 '13 at 04:01
  • @LucasZanella You can google the book, it is called "Differential and Integral Calculus" by Edmund Landau. – Pedro Sep 18 '13 at 04:19
  • @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, 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. – Alex Shpilkin Aug 06 '17 at 00:56
  • I like this proof very much. It looks very similar to Herglotz's trick, which is explained in the chapter "Cotangent and the Herglotz trick " from Proofs from the book. Landau uses a limit process (very roughly, considering the values at $(x+k)/2^n$ and letting $n \to \infty$). Herglotz uses only $x/2$ and $(x+1)/2$, I wonder if it's possible to do something similar here. – François Brunault Aug 30 '19 at 20:34