# What is the term for this family of improper integrals?

What is the name of the integrals of this form?

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

• @Travis - Thanks for the edit too :-) – user150203 Nov 14 '18 at 1:33

## 1 Answer

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.

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