Borwein integral

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 remarkable for exhibiting apparent patterns which, however, eventually break down. An example is as follows,

${\begin{aligned}&\int _{0}^{\infty }{\frac {\sin(x)}{x}}\,dx=\pi /2\\[10pt]&\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\,dx=\pi /2\\[10pt]&\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}{\frac {\sin(x/5)}{x/5}}\,dx=\pi /2\end{aligned}}$

This pattern continues up to

${\displaystyle \int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/13)}{x/13}}\,dx=\pi /2~.}$

Nevertheless, at the next step the obvious pattern fails,

${\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/15)}{x/15}}\,dx&={\frac {467807924713440738696537864469}{935615849440640907310521750000}}~\pi \\&={\frac {\pi }{2}}-{\frac {6879714958723010531}{935615849440640907310521750000}}~\pi \\&\simeq {\frac {\pi }{2}}-2.31\times 10^{-11}~.\end{aligned}}}$

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.

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. In this case, 1/3 + 1/5 + … + 1/111 < 2, but 1/3 + 1/5 + … + 1/113 > 2.

General Formula

Given a sequence of real numbers, ${\displaystyle a_{0},a_{1},a_{2},...}$, a general formula for the integral

${\displaystyle \int _{0}^{\infty }\prod _{k=0}^{n}{\frac {\sin(a_{k}x)}{a_{k}x}}dx}$

can be given.^[1] To state the formula, we will need to consider sums involving the $a_{k}$. In particular, if ${\displaystyle \gamma =(\gamma _{1},\gamma _{2},...,\gamma _{n})\in \{\pm 1\}^{n}}$ is an $n$-tuple where each entry is $\pm 1$, then we write ${\displaystyle b_{\gamma }=a_{0}+\gamma _{1}a_{1}+\gamma _{2}a_{2}+\cdots +\gamma _{n}a_{n}}$, which is a kind of alternating sum of the first few $a_{k}$, and we set ${\displaystyle \epsilon _{\gamma }=\gamma _{1}\gamma _{2}\cdots \gamma _{n}}$, which is either $\pm 1$. With this notation, the value for the above integral is

${\displaystyle \int _{0}^{\infty }\prod _{k=0}^{n}{\frac {\sin(a_{k}x)}{a_{k}x}}dx={\frac {\pi }{2a_{0}}}C_{n}}$

where

${\displaystyle C_{n}={\frac {1}{2^{n}n!\prod _{k=1}^{n}a_{k}}}\sum _{\gamma \in \{\pm 1\}^{n}}\epsilon _{\gamma }b_{\gamma }^{n}\operatorname {sgn}(b_{\gamma })}$

In the case when ${\displaystyle a_{0}>|a_{1}|+|a_{2}|+\cdots +|a_{n}|}$, we have ${\displaystyle C_{n}=1}$.

Furthermore, if there is an $n$ so that for each ${\displaystyle k=0,...,n-1}$ we have ${\displaystyle 0<a_{n}<2a_{k}}$ and ${\displaystyle a_{1}+a_{2}+\cdots +a_{n-1}<a_{0}<a_{1}+a_{2}+\cdots +a_{n-1}+a_{n}}$, which means that $n$ is the first value when the partial sum of the first $n$ elements of the sequence exceed $a_{0}$, then ${\displaystyle C_{k}=1}$ for each ${\displaystyle k=0,...,n-1}$ but

${\displaystyle C_{n}=1-{\frac {(a_{1}+a_{2}+\cdots +a_{n}-a_{0})^{n}}{2^{n-1}n!\prod _{k=1}^{n}a_{k}}}}$

The first example is the case when ${\displaystyle a_{k}={\frac {1}{2k+1}}}$. Note that if ${\displaystyle n=7}$ then ${\displaystyle a_{7}={\frac {1}{15}}}$ and ${\displaystyle {\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{9}}+{\frac {1}{11}}+{\frac {1}{13}}\approx 0.955}$ but ${\displaystyle {\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{9}}+{\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{15}}\approx 1.02}$, so because ${\displaystyle a_{0}=1}$, we get that

${\displaystyle \int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/13)}{x/13}}\,dx={\frac {\pi }{2}}}$

which remains true if we remove any of the products, but that

${\displaystyle \int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/15)}{x/15}}\,dx={\frac {\pi }{2}}\left(1-{\frac {(3^{-1}+5^{-1}+7^{-1}+9^{-1}+11^{-1}+13^{-1}+15^{-1}-1)^{7}}{2^{6}\cdot 7!\cdot (1/3\cdot 1/5\cdot 1/7\cdot 1/9\cdot 1/11\cdot 1/13\cdot 1/15)}}\right)}$

which is equal to the value given previously.

References

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:1105.3943 [math.NT].
3. Schmid, Hanspeter (2014), "Two curious integrals and a graphic proof" (PDF), Elemente der Mathematik, 69 (1): 11–17, doi:10.4171/EM/239, ISSN 0013-6018