In mathematics, a Borwein integral is an integral involving products of sinc(ax), where the sinc function is given by sinc(x) = sin(x)/x for x not equal to 0, and sinc(0) = 1. These integrals are notorious for exhibiting apparent patterns that eventually break down. An example is as follows:
This pattern continues up to
However at the next step the obvious pattern fails:
In general similar integrals have value π/2 whenever the numbers 3, 5, ... are replaced by positive real numbers such that the sum of their reciprocals is less than 1. In the example above, 1/3 + 1/5 + ... + 1/13 < 1, but 1/3 + 1/5 + ... + 1/15 > 1.
An example for a longer series,
is shown in  together with an intuitive mathematical explanation of the reason why the original and the extended series break down.
- Borwein, David; Borwein, Jonathan M. (2001), "Some remarkable properties of sinc and related integrals", The Ramanujan Journal 5 (1): 73–89, doi:10.1023/A:1011497229317, ISSN 1382-4090, MR 1829810
- Baillie, Robert (2011). "Fun With Very Large Numbers". arXiv:1105.3943v1 [math.NT].
- Schmid, Hanspeter (2014), "Two curious integrals and a graphic proof" (PDF), Elemente der Mathematik 69 (1): 11–17, doi:10.4171/EM/239, ISSN 0013-6018