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 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]


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




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.

/* This is a sample program to demonstrate for Computer Algebra System "maxima". */
f(n) := if n=1 then sin(x)/x else f(n-2) * (sin(x/n)/(x/n));
for n from 1 thru 15 step 2 do (
  print("f(", n, ")=", f(n) ),
  print("integral of f for n=", n, " is ", integrate(f(n), x, 0, inf)) );
/* This is also sample program of another problem. */
f(n) := if n=1 then sin(x)/x else f(n-2) * (sin(x/n)/(x/n)); g(n) := 2*cos(x) * f(n);
for n from 1 thru 19 step 2 do (
  print("g(", n, ")=", g(n) ),
  print("integral of g for n=", n, " is ", integrate(g(n), x, 0, inf)) );

Method to solve Borwein integrals edit

An exact integration method that is efficient for evaluating Borwein-like integrals is discussed here.[7] This integration method works by reformulating integration in terms of a series of differentiations and it yields intuition into the unusual behavior of the Borwein integrals. The Integration by Differentiation method is applicable to general integrals, including Fourier and Laplace transforms. It is used in the integration engine of Maple since 2019. The Integration by Differentiation method is independent of the Feynman method that also uses differentiation to integrate.

Infinite products edit

While the integral


becomes less than   when   exceeds 6, it never becomes much less, and in fact Borwein and Bailey[8] 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[9]




Probabilistic formulation edit

Schmuland[11] has given appealing probabilistic formulations of the infinite product Borwein integrals. For example, consider the random harmonic series


where one flips independent fair coins to choose the signs. This series converges almost surely, that is, with probability 1. The probability density function of the result is a well-defined function, and value of this function at 2 is close to 1/8. However, it is closer to


Schmuland's explanation is that this quantity is   times


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 (2019). "Random walkers illuminate a math problem". Physics Today. doi:10.1063/PT.6.1.20190808a. S2CID 202930808.
  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:2019PhRvL.123b0201M, doi:10.1103/PhysRevLett.123.020201, ISSN 1079-7114, PMID 31386528, S2CID 184488105
  7. ^ Jia; Tang; Kempf (2017), "Integration by differentiation: new proofs, methods and examples", Journal of Physics A, 50 (23): 235201, arXiv:1610.09702, Bibcode:2017JPhA...50w5201J, doi:10.1088/1751-8121/aa6f32, S2CID 56012760
  8. ^ 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. OCLC 1064987843.
  9. ^ 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.
  10. ^ 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. hdl:1959.13/928097. JSTOR 27641975.
  11. ^ Schmuland, Byron (2003). "Random Harmonic Series". The American Mathematical Monthly. 110 (5): 407–416. doi:10.2307/3647827. JSTOR 3647827.

External links edit