Summary
A Borwein integral is a definite integral involving products of sinc functions, where
$\operatorname{sinc}(x)=\dfrac{\sin x}{x}$.
They were highlighted by mathematicians David and Jonathan Borwein because they exhibit a striking phenomenon: a simple numerical pattern remains true for several successive integrals and then suddenly fails by an extremely tiny amount. For example,
$\int_0^\infty \frac{\sin x}{x}\frac{\sin(x/3)}{x/3}\frac{\sin(x/5)}{x/5}\cdots\frac{\sin(x/13)}{x/13},dx=\frac{\pi}{2}$.
The same value occurs when fewer factors are included. However, after adding the next factor,
$\dfrac{\sin(x/15)}{x/15}$,
the result becomes slightly smaller than $\dfrac{\pi}{2}$, differing by only about
$2.31\times10^{-11}$.
The explanation comes from a general condition. For integrals of the form
$\int_0^\infty \prod_{k=0}^{n}\frac{\sin(a_kx)}{a_kx},dx$,
the value remains
$\frac{\pi}{2a_0}$
when $a_0$ is larger than the sum of the magnitudes of the remaining parameters.
In the classical example, this corresponds to the reciprocal sum
$\frac13+\frac15+\frac17+\cdots+\frac1{13}<1$.
Adding $\frac1{15}$ makes the sum exceed $1$, causing the pattern to break. The deviation is initially extremely small because the correction term involves a high power of the amount by which the threshold has been exceeded.
Related versions containing $2\cos x$ maintain the apparent pattern much longer. In one famous case, the breakdown occurring after the factor involving $113$ is only about
$2.3\times10^{-138}$.
Borwein integrals also connect several areas of mathematics. They can be studied through Fourier analysis, exact integration methods, infinite products, and probability. One particularly elegant interpretation connects them with random walks and the random harmonic series
$\pm1\pm\frac12\pm\frac13\pm\frac14\pm\cdots$,
where the signs are chosen independently at random.
These interpretations help explain why the apparently perfect pattern eventually fails and make Borwein integrals a famous example of how convincing numerical evidence can conceal a subtle mathematical threshold.
Key takeaways
ARTICLE
A Borwein integral is a definite integral involving products of sinc functions, where
$\operatorname{sinc}(x)=\dfrac{\sin x}{x}$.
They were highlighted by mathematicians David and Jonathan Borwein because they exhibit a striking phenomenon: a simple numerical pattern remains true for several successive integrals and then suddenly fails by an extremely tiny amount. For example,
$\int_0^\infty \frac{\sin x}{x}\frac{\sin(x/3)}{x/3}\frac{\sin(x/5)}{x/5}\cdots\frac{\sin(x/13)}{x/13},dx=\frac{\pi}{2}$.
The same value occurs when fewer factors are included. However, after adding the next factor,
$\dfrac{\sin(x/15)}{x/15}$,
the result becomes slightly smaller than $\dfrac{\pi}{2}$, differing by only about
$2.31\times10^{-11}$.
The explanation comes from a general condition. For integrals of the form
$\int_0^\infty \prod_{k=0}^{n}\frac{\sin(a_kx)}{a_kx},dx$,
the value remains
$\frac{\pi}{2a_0}$
when $a_0$ is larger than the sum of the magnitudes of the remaining parameters.
In the classical example, this corresponds to the reciprocal sum
$\frac13+\frac15+\frac17+\cdots+\frac1{13}<1$.
Adding $\frac1{15}$ makes the sum exceed $1$, causing the pattern to break. The deviation is initially extremely small because the correction term involves a high power of the amount by which the threshold has been exceeded.
Related versions containing $2\cos x$ maintain the apparent pattern much longer. In one famous case, the breakdown occurring after the factor involving $113$ is only about
$2.3\times10^{-138}$.
Borwein integrals also connect several areas of mathematics. They can be studied through Fourier analysis, exact integration methods, infinite products, and probability. One particularly elegant interpretation connects them with random walks and the random harmonic series
$\pm1\pm\frac12\pm\frac13\pm\frac14\pm\cdots$,
where the signs are chosen independently at random.
These interpretations help explain why the apparently perfect pattern eventually fails and make Borwein integrals a famous example of how convincing numerical evidence can conceal a subtle mathematical threshold.
Key takeaways
- Borwein integrals are products of sinc functions with unexpectedly simple values.
- Many consecutive examples equal exactly $\frac{\pi}{2}$ before the pattern suddenly fails.
- The critical point occurs when a certain sum of parameters crosses a threshold.
- They are a classic lesson in experimental mathematics: even an extremely convincing numerical pattern is not necessarily a theorem.
ARTICLE
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│ KONSTANTINOS MICHAILIDIS │
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│ KONSTANTINOS MICHAILIDIS │
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