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 whenever the numbers are replaced by positive real numbers such that the sum of their reciprocals is less than 1. In the example above, but With the inclusion of the additional factor , the pattern holds up over a longer series, but In this case, but . 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. 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. General formula Given a sequence of nonzero real numbers, , a general formula for the integral can be given. 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 While the integral becomes less than when exceeds 6, it never becomes much less, and in fact Borwein and Bailey have shown where we can pull the limit