Bid–ask matrix

In mathematics, a Borwein integral is an integral involving products of sinc(ax), where the sinc function is given by sinc(x) = sin(x)/x for x not equal to 0, and sinc(0) = 1. These integrals are notorious for exhibiting apparent patterns that eventually break down. An example is as follows:

$\begin{array}{rl}& \underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin (x)}{x}dx=\pi /2\\ & \underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin (x)}{x}\frac{\sin (x/3)}{x/3}dx=\pi /2\\ & \underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin (x)}{x}\frac{\sin (x/3)}{x/3}\frac{\sin (x/5)}{x/5}dx=\pi /2\end{array}$

This pattern continues up to

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin (x)}{x}\frac{\sin (x/3)}{x/3}\cdots \frac{\sin (x/13)}{x/13}dx=\pi /2$

However at the next step the obvious pattern fails:

$\begin{array}{rl}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin (x)}{x}\frac{\sin (x/3)}{x/3}\cdots \frac{\sin (x/15)}{x/15}dx& =\frac{467807924713440738696537864469}{935615849440640907310521750000}\pi \\ & =\frac{\pi }{2}-\frac{6879714958723010531}{935615849440640907310521750000}\pi \\ & \simeq \frac{\pi }{2}-2.31\times 10^{-11}\end{array}$

In general similar integrals have value π/2 whenever the numbers 3, 5, ... 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.

References

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