Borwein integral

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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.[1][2] These integrals are notorious for exhibiting apparent patterns that eventually break down. An example is as follows:


\begin{align}
& \int_0^\infty \frac{\sin(x)}{x} \, dx=\pi/2 \\[10pt]
& \int_0^\infty \frac{\sin(x)}{x}\frac{\sin(x/3)}{x/3} \, dx = \pi/2 \\[10pt]
& \int_0^\infty \frac{\sin(x)}{x}\frac{\sin(x/3)}{x/3}\frac{\sin(x/5)}{x/5} \, dx = \pi/2
\end{align}

This pattern continues up to

\int_0^\infty \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{align}
\int_0^\infty \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{align}

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.

An example for a longer series,

\int_0^\infty 2 \cos(x) \frac{\sin(x)}{x}\frac{\sin(x/3)}{x/3}\cdots\frac{\sin(x/111)}{x/111} \, dx = \pi/2,

but

\int_0^\infty 2 \cos(x) \frac{\sin(x)}{x}\frac{\sin(x/3)}{x/3}\cdots\frac{\sin(x/111)}{x/111}\frac{\sin(x/113)}{x/113} \, dx < \pi/2,

is shown in [3] together with an intuitive mathematical explanation of the reason why the original and the extended series break down.

References[edit]

  1. ^ 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 
  2. ^ Baillie, Robert (2011). "Fun With Very Large Numbers". arXiv:1105.3943v1 [math.NT].
  3. ^ Schmid, Hanspeter (2014), "Two curious integrals and a graphic proof", Elemente der Mathematik 69 (1): 11–17, doi:10.4171/EM/239, ISSN 0013-6018