Borwein integral

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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 exact answer can be calculated using the general formula provided in the next section, and a representation of it is shown below. Fully expanded, this value turns into a fraction that involves two 2736 digit integers.

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]

General formula[edit]

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.

Infinite products[edit]

While the integral

becomes less than when exceeds 6, it never becomes much less, and in fact Borwein and Bailey[7] have shown

where we can pull the limit out of the integral thanks to the dominated convergence theorem. Similarly, while

becomes less than when exceeds 55, we have

Furthermore, using the Weierstrass factorizations

one can show

and with a change of variables obtain[8]

and[7][9]

References[edit]

  1. ^ 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, S2CID 6515110
  2. ^ Baillie, Robert (2011). "Fun With Very Large Numbers". arXiv:1105.3943 [math.NT].
  3. ^ Hill, Heather M. (September 2019). Random walkers illuminate a math problem (Volume 72, number 9 ed.). American Institute of Physics. pp. 18–19.
  4. ^ 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
  5. ^ Baez, John (September 20, 2018). "Patterns That Eventually Fail". Azimuth. Archived from the original on 2019-05-21.
  6. ^ 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, PMID 31386528, S2CID 184488105
  7. ^ a b Borwein, J. M.; Bailey, D. H. (2003). Mathematics by experiment : plausible reasoning in the 21st century (1st ed.). Wellesley, MA: A K Peters.
  8. ^ Borwein, Jonathan M. (2004). Experimentation in mathematics : computational paths to discovery. David H. Bailey, Roland Girgensohn. Natick, Mass.: AK Peters. ISBN 1-56881-136-5. OCLC 53021555.
  9. ^ Bailey, David H.; Borwein, Jonathan M.; Kapoor, Vishaal; Weisstein, Eric W. (2006-06-01). "Ten Problems in Experimental Mathematics". The American Mathematical Monthly. 113 (6): 481. doi:10.2307/27641975. JSTOR 27641975.

Further reading[edit]

External links[edit]