Borwein integral

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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.[1][2]

These integrals are remarkable for exhibiting apparent patterns which, however, eventually break down. An example is as follows,

This pattern continues up to

Nevertheless, 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.

An example for a longer series,

but

is shown in [3] together with an intuitive mathematical explanation of the reason why the original and the extended series break down. In this case, 1/3 + 1/5 + … + 1/111 < 2, but 1/3 + 1/5 + … + 1/113 > 2.

References[edit]

  1. ^ 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 
  2. ^ Baillie, Robert (2011). "Fun With Very Large Numbers". arXiv:1105.3943free to read [math.NT]. 
  3. ^ 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