Borwein integral

Given a sequence of nonzero real numbers, $a\_0,\; a\_1,\; a\_2,\backslash ldots$, a general formula for the integral

: $\backslash int\_0^\backslash infty\; \backslash prod\_\{k=0\}^n\; \backslash frac\{\backslash sin(a\_kx)\}\{a\_kx\}\; \backslash ,\; dx$

can be given. To state the formula, one will need to consider sums involving the $a\_k$. In particular, if $\backslash gamma\; =\; (\backslash gamma\_1,\backslash gamma\_2,\backslash ldots,\backslash gamma\_n)\backslash in\backslash \{\backslash pm\; 1\backslash \}^n$ is an $n$-tuple where each entry is $\backslash pm\; 1$, then we write $b\_\{\backslash gamma\}=a\_0+\backslash gamma\_1a\_1+\backslash gamma\_2a\_2+\backslash cdots+\backslash gamma\_na\_n$, which is a kind of alternating sum of the first few $a\_k$, and we set $\backslash varepsilon\_\backslash gamma=\backslash gamma\_1\backslash gamma\_2\backslash cdots\backslash gamma\_n$, which is either $\backslash pm\; 1$. With this notation, the value for the above integral is

: $\backslash int\_0^\backslash infty\; \backslash prod\_\{k=0\}^n\; \backslash frac\{\backslash sin(a\_kx)\}\{a\_kx\}\; \backslash ,\; dx\; =\; \backslash frac\{\backslash pi\}\{2a\_0\}\; C\_n$

where

: $C\_n\; =\; \backslash frac\; 1\; \{2^nn!\backslash prod\_\{k=1\}^na\_k\}\; \backslash sum\_\{\backslash gamma\backslash in\backslash \{\backslash pm\; 1\backslash \}^n\}\; \backslash varepsilon\_\backslash gamma\; b\_\backslash gamma^n\; \backslash sgn(b\_\backslash gamma)$

In the case when $a\_0\; >\; |a\_1|+|a\_2|+\backslash cdots+|a\_n|$, we have $C\_n=1$.

Furthermore, if there is an $n$ such that for each $k=0,\backslash ldots,n-1$ we have

: $C\_n\; =\; 1\; -\; \backslash frac\{(a\_1+a\_2+\backslash cdots+a\_n-a\_0)^n\}\{2^\{n-1\}n!\; \backslash prod\_\{k=1\}^na\_k\}$

The first example is the case when $a\_k=\backslash frac\{1\}\{2k+1\}$.

Note that if $n=7$ then $a\_7=\backslash frac\{1\}\{15\}$ and $\backslash frac\{1\}\{3\}+\backslash frac\{1\}\{5\}+\backslash frac\{1\}\{7\}+\backslash frac\{1\}\{9\}+\backslash frac\{1\}\{11\}+\backslash frac\{1\}\{13\}\backslash approx\; 0.955$ but $\backslash frac\{1\}\{3\}+\backslash frac\{1\}\{5\}+\backslash frac\{1\}\{7\}+\backslash frac\{1\}\{9\}+\backslash frac\{1\}\{11\}+\backslash frac\{1\}\{13\}+\backslash frac\{1\}\{15\}\backslash approx\; 1.02$, so because $a\_0=1$, we get that

: $\backslash int\_0^\backslash infty\; \backslash frac\{\backslash sin(x)\}\; x\; \backslash frac\{\backslash sin(x/3)\}\{x/3\}\backslash cdots\backslash frac\{\backslash sin(x/13)\}\{x/13\}\; \backslash ,\; dx\; =\; \backslash frac\; \backslash pi\; 2$

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

: $$

\begin{align}

& \int_0^\infty \frac{\sin(x)} x \frac{\sin(x/3)}{x/3}\cdots\frac{\sin(x/15)}{x/15} \, dx \\[5pt]

= {} & \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),

\end{align}

which is equal to the value given previously.

/* This is a sample program to demonstrate for Computer Algebra System "maxima". */
f(n) := if n=1 then sin(x)/x else f(n-2) * (sin(x/n)/(x/n));
for n from 1 thru 15 step 2 do (
print("f(", n, ")=", f(n) ),
print("integral of f for n=", n, " is ", integrate(f(n), x, 0, inf)) );

/* This is also sample program of another problem. */
f(n) := if n=1 then sin(x)/x else f(n-2) * (sin(x/n)/(x/n)); g(n) := 2*cos(x) * f(n);
for n from 1 thru 19 step 2 do (
print("g(", n, ")=", g(n) ),
print("integral of g for n=", n, " is ", integrate(g(n), x, 0, inf)) );

An exact integration method that is efficient for evaluating Borwein-like integrals is discussed here.{{Citation |last1=Jia |last2=Tang |last3=Kempf |title=Integration by differentiation: new proofs, methods and examples |year=2017 |journal=Journal of Physics A |volume=50 |issue=23 |pages=235201|doi=10.1088/1751-8121/aa6f32 |arxiv=1610.09702 |bibcode= 2017JPhA...50w5201J|s2cid=56012760 }} This integration method works by reformulating integration in terms of a series of differentiations and it yields intuition into the unusual behavior of the Borwein integrals. The Integration by Differentiation method is applicable to general integrals, including Fourier and Laplace transforms. It is used in the integration engine of Maple since 2019. The Integration by Differentiation method is independent of the Feynman method that also uses differentiation to integrate.

While the integral

: $$

\begin{align}

\int_0^\infty \prod_{k = 0}^n \frac{\sin(x/(2k+1))}{x/(2k+1)} \, dx

\end{align}

becomes less than $\backslash frac\{\backslash pi\}\{2\}$ when $n$ exceeds 6, it never becomes much less, and in fact Borwein and Bailey{{Cite book |last1=Borwein |first1=J. M. |title=Mathematics by experiment : plausible reasoning in the 21st century |last2=Bailey |first2=D. H. |publisher=A K Peters |year=2003 |isbn= |edition=1st |location=Wellesley, MA |oclc=1064987843 }} have shown

: $$

\begin{align}

\int_0^\infty \prod_{k = 0}^\infty \frac{\sin(x/(2k+1))}{x/(2k+1)} \, dx &=

\int_0^\infty \lim_{n \to \infty} \prod_{k = 0}^n \frac{\sin(x/(2k+1))}{x/(2k+1)} \, dx \\[5pt]

&= \lim_{n \to \infty} \int_0^\infty \prod_{k = 0}^n \frac{\sin(x/(2k+1))}{x/(2k+1)} \, dx \\[5pt]

&\approx \frac{\pi}{2} - 0.0000352

\end{align}

where we can pull the limit out of the integral thanks to the dominated convergence theorem. Similarly, while

: $$

\int_0^\infty 2 \cos x \prod_{k = 0}^n \frac{\sin(x/(2k+1))}{x/(2k+1)} \, dx

becomes less than $\backslash frac\{\backslash pi\}\{2\}$ when $n$ exceeds 55, we have

: $$

\int_0^\infty 2 \cos x \prod_{k = 0}^n\frac{\sin(x/(2k+1))}{x/(2k+1)} \, dx \approx \frac{\pi}{2} - 2.9629 \cdot 10^{-42}

Furthermore, using the Weierstrass factorizations

:$\backslash frac\{\backslash sin\; x\}\{x\}\; =\; \backslash prod\_\{n\; =\; 1\}^\backslash infty\; \backslash left(\; 1\; -\; \backslash frac\{x^2\}\{\backslash pi^2\; n^2\}\; \backslash right)\; \backslash qquad\; \backslash cos\; x\; =\; \backslash prod\_\{n\; =\; 0\}^\backslash infty\; \backslash left(\; 1\; -\; \backslash frac\{4x^2\}\{\backslash pi^2\; (2n+1)^2\}\; \backslash right)$

one can show

:$$

\prod_{n = 0}^\infty \frac{\sin (2x/(2n+1))}{2x/(2n+1)} =

\prod_{n = 1}^\infty \cos\left(\frac{x}{n}\right)

and with a change of variables obtain{{Cite book |last=Borwein |first=Jonathan M. |title=Experimentation in mathematics : computational paths to discovery |date=2004 |publisher=AK Peters |others=David H. Bailey, Roland Girgensohn |isbn=1-56881-136-5 |location=Natick, Mass. |oclc=53021555}}

:$$

\int_0^\infty \prod_{n=1}^\infty \cos\left(\frac{x}{n} \right) \, d x = \frac{1}{2} \int_0^\infty \prod_{n = 0}^\infty \frac{\sin (x/(2n+1))}{x/(2n+1)} \,d x \approx \frac{\pi}{4} - 0.0000176

and{{Cite journal |last1=Bailey |first1=David H. |last2=Borwein |first2=Jonathan M. |last3=Kapoor |first3=Vishaal |last4=Weisstein |first4=Eric W. |date=2006-06-01 |title=Ten Problems in Experimental Mathematics |journal=The American Mathematical Monthly |language=en |volume=113 |issue=6 |pages=481 |doi=10.2307/27641975|jstor=27641975 |url=https://digital.library.unt.edu/ark:/67531/metadc879570/ |hdl=1959.13/928097 |hdl-access=free }}

:$$

\int_0^\infty \cos(2x) \prod_{n=1}^\infty \cos\left(\frac{x}{n} \right) \, d x = \frac{1}{2}

\int_0^\infty \cos(x) \prod_{n = 0}^\infty \frac{\sin (x/(2n+1))}{x/(2n+1)} \,d x \approx \frac{\pi}{8} - 7.4073 \cdot 10^{-43}

Schmuland{{Cite journal |last=Schmuland |first=Byron |date=2003 |title=Random Harmonic Series |journal=The American Mathematical Monthly |volume=110 |issue=5 |pages=407–416 |doi=10.2307/3647827|jstor=3647827 }} has given appealing probabilistic formulations of the infinite product Borwein integrals. For example, consider the random harmonic series

:$\backslash pm\; 1\; \backslash pm\; \backslash frac\{1\}\{2\}\; \backslash pm\; \backslash frac\{1\}\{3\}\; \backslash pm\; \backslash frac\{1\}\{4\}\; \backslash pm\; \backslash frac\{1\}\{5\}\; \backslash pm\; \backslash cdots$

where one flips independent fair coins to choose the signs. This series converges almost surely, that is, with probability 1. The probability density function of the result is a well-defined function, and value of this function at 2 is close to 1/8. However, it is closer to

:$0.124999999999999999999999999999999999999999764\; \backslash ldots$

Schmuland's explanation is that this quantity is $1/\backslash pi$ times

:$\backslash int\_0^\backslash infty\; \backslash cos(2x)\; \backslash prod\_\{n=1\}^\backslash infty\; \backslash cos\backslash left(\backslash frac\{x\}\{n\}\; \backslash right)\; \backslash ,\; d\; x\; \backslash approx\; \backslash frac\{\backslash pi\}\{8\}\; -\; 7.4073\; \backslash cdot\; 10^\{-43\}$

{{reflist}}

• {{MathWorld|title=Infinite Cosine Product Integral|urlname=InfiniteCosineProductIntegral|author=Weisstein, Eric W.|access-date=10 January 2023}}

Category:Integrals