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

\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,

{\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, $a_{0},a_{1},a_{2},...$ , a general formula for the integral

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

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

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

where

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

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

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

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

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

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

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

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

which is equal to the value given previously.

References

1 2 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
↑ Baillie, Robert (2011). "Fun With Very Large Numbers". arXiv:1105.3943 [math.NT].
↑ 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

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