Borwein integral

In mathematics , Borwein integral denotes integral terms that contain products of the sinc function . These integrals are known to contain apparent patterns that turn out to be false. An example is the following:

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

This pattern repeats itself up to

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

But then the next step is:

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

An example of a longer episode is

${\displaystyle \int _{0}^{\infty }2\cos(x){\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/111)}{x/111}}\,\mathrm {d} x=\pi /2,}$

but

${\displaystyle \int _{0}^{\infty }2\cos(x){\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/111)}{x/111}}{\frac {\sin(x/113)}{x/113}}\,\mathrm {d} x<\pi /2.}$

General formula

For a sequence of real numbers, can be a closed form of ${\displaystyle a_{0},a_{1},a_{2},\ldots }$

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

are given. The closed form deals with sums of the . For an n-tuple let . One such is an "alternating sum" of the first . Set . Then ${\displaystyle a_{k}}$ ${\displaystyle \gamma =(\gamma _{1},\gamma _{2},\ldots ,\gamma _{n})\in \{\pm 1\}^{n}}$ ${\displaystyle b_{\gamma }:=a_{0}+\gamma _{1}a_{1}+\gamma _{2}a_{2}+\cdots +\gamma _{n}a_{n}}$ ${\displaystyle b_{\gamma }}$ ${\displaystyle a_{k}}$ ${\displaystyle \varepsilon _{\gamma }=\gamma _{1}\gamma _{2}\cdots \gamma _{n}=\pm 1}$

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

in which

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

If applies . ${\displaystyle a_{0}>|a_{1}|+|a_{2}|+\cdots +|a_{n}|}$ ${\displaystyle C_{n}=1}$

Individual evidence

1. ↑ David Borwein, Jonathan M. Borwein: Some remarkable properties of sinc and related integrals . tape 5 , 2001, p. 73-89 .