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Problem: evaluate that $\int_0^{\infty} \frac{\sin x}{x}\frac{\sin \frac{x}{3}}{\frac{x}{3}} dx = \frac{\pi}{2}$ and prove if a more general case, $\int_0^{\infty} \frac{\sin x}{x}\frac{\sin \frac{x}{3}}{\frac{x}{3}} \frac{\sin \frac{x}{5}}{\frac{x}{5}} ... \frac{\sin \frac{x}{2n+1}}{\frac{x}{2n+1}} dx = \frac{\pi}{2}$ holds?

So it's a rather well known results to prove Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?

and I use a computer program to validate that the general case does appear to hold, but not sure how to prove it..

Chen Chen
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    It stops at $n=15$ or so if i remember correctly. – dezdichado Jun 21 '21 at 00:18
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    Yes that's correct. See this paper for a proof: https://carma.edu.au/resources/db90/pdfs/db90-119.00.pdf – Dave Jun 21 '21 at 00:20
  • I believe they use the n-dimensional convolution theorem for Fourier transforms, and the result that the Fourier transform of a sinc function is just a square wave – Dave Jun 21 '21 at 00:21

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These are Borwein Integrals and the behavior of exactly equaling $\pi/2$ stops at $2n+1 = 15$.

Xoque55
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