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In mathematics, a **Borwein integral** is an integral whose unusual properties were first presented by mathematicians David Borwein and Jonathan Borwein in 2001.^{[1]} Borwein integrals involve products of , where the sinc function is given by for not equal to 0, and .^{[1]}^{[2]}

These integrals are remarkable for exhibiting apparent patterns that eventually break down. The following is an example.

This pattern continues up to

At the next step the obvious pattern fails,

In general, similar integrals have value π/2 whenever the numbers 3, 5, 7… 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.

With the inclusion of the additional factor , the pattern holds up over a longer series,^{[3]}

but

In this case, 1/3 + 1/5 + … + 1/111 < 2, but 1/3 + 1/5 + … + 1/113 > 2.

The reason the original and the extended series break down has been demonstrated with an intuitive mathematical explanation.^{[4]}^{[5]} In particular, a random walk reformulation with a causality argument sheds light on the pattern breaking and opens the way for a number of generalizations.^{[6]}

Given a sequence of nonzero real numbers, , a general formula for the integral

can be given.^{[1]} To state the formula, one will need to consider sums involving the . In particular, if is an -tuple where each entry is , then we write , which is a kind of alternating sum of the first few , and we set , which is either . With this notation, the value for the above integral is

where

In the case when , we have .

Furthermore, if there is an such that for each we have and , which means that is the first value when the partial sum of the first elements of the sequence exceed , then for each but

The first example is the case when .

Note that if then and but , so because , we get that

which remains true if we remove any of the products, but that

which is equal to the value given previously.

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^{a}^{b}^{c}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:1105.3943 [math.NT].**^**Hill, Heather M. (September 2019).*Random walkers illuminate a math problem*(Volume 72, number 9 ed.). American Institute of Physics. pp. 18–19.**^**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**^**Baez, John (September 20, 2018). "Patterns That Eventually Fail".*Azimuth*. Archived from the original on 2019-05-21.**^**Satya Majumdar; Emmanuel Trizac (2019), "When random walkers help solving intriguing integrals",*Physical Review Letters*,**123**(2): 020201, arXiv:1906.04545, Bibcode:2019arXiv190604545M, doi:10.1103/PhysRevLett.123.020201, ISSN 1079-7114

- Patterns That Eventually Fail, 20 September 2018
- Breakdown, 2 February 2012
- Illusive patterns in math explained by ideas in physics, 19 July 2019
- (video) When random walkers help solving intriguing integrals 19 July 2019
- (video) Convolution, Fourier Transforms and Sinc Integrals 16 September 2020