In mathematics, a Borwein integral is an integral whose unusual properties were first presented by mathematicians David Borwein and Jonathan Borwein in 2001. Borwein integrals involve 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 remarkable for exhibiting apparent patterns which, however, 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 term, , the pattern holds up over a longer series,
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.
Given a sequence of real numbers, , a general formula for the integral
can be given. 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
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.
- ^ 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].
- ^ 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