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What is the name of the integrals of this form?

$$\int_{0}^{\infty} \frac{\sin\left(\frac{x}{1}\right)\sin\left(\frac{x}{3}\right)\cdots\sin\left(\frac{x}{2n + 1}\right)}{\left(\frac{x}{1}\right)\left(\frac{x}{3}\right)\cdots\left(\frac{x}{2n + 1}\right)} \:dx$$

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    $\begingroup$ @Travis - Thanks for the edit too :-) $\endgroup$ – user150203 Nov 14 '18 at 1:33
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These are examples of Borwein integrals. See:

Borwein, D.; Borwein, J.M., "Some Remarkable Properties of $\operatorname{Sinc}$ and Related Integrals", Ramanujan J. 5 (2001), 73-89.

John Baez' Azimuth blog has an illuminating post discussing these integrals, and Greg Egan gave an intuitive explanation in terms of Fourier transforms for the pattern-breaking phenomenon often mentioned when these integrals come up.

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  • $\begingroup$ Thanks @Travis! $\endgroup$ – user150203 Nov 14 '18 at 0:42
  • $\begingroup$ Cheers!$\!\!\!$ $\endgroup$ – Travis Nov 14 '18 at 0:47

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