Integrals

Borwein integral

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 , where the sinc function is given by for not equal to 0, and . 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, 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 below but the fraction involves two 2736 digit integers. The reason the original and the extended series break down has been demonstrated with an intuitive mathematical explanation. 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. (Wikipedia).

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From playlist Integrale

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From playlist Integrale

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From playlist Integrale

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From playlist Integrale

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From playlist Integrale

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Jonathan Borwein | Integral | Random walk | Mathematics | Sinc function