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
{\displaystyle \operatorname {sinc} (ax)}
, where the sinc function is given by
[1][2] These integrals are remarkable for exhibiting apparent patterns that eventually break down.
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.
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] 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
, which is a kind of alternating sum of the first few
With this notation, the value for the above integral is where In the case when
elements of the sequence exceed
, we get that which remains true if we remove any of the products, but that which is equal to the value given previously.
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.
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.
exceeds 55, we have Furthermore, using the Weierstrass factorizations one can show and with a change of variables obtain[9] and[8][10] 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