Borwein integral

In mathematics, a Borwein integral is an integral whose unusual properties were first presented by mathematicians David Borwein and Jonathan Borwein in 2001. Borwein integrals involve products of sinc ⁡ ( a x ) {\displaystyle \operatorname {sinc} (ax)} , where the sinc function is given by sinc ⁡ ( x ) = sin ⁡ ( x ) / x {\displaystyle \operatorname {sinc} (x)=\sin(x)/x} for x {\displaystyle x} not equal to 0, and sinc ⁡ ( 0 ) = 1 {\displaystyle \operatorname {sinc} (0)=1} . These integrals are remarkable for exhibiting apparent patterns that eventually break down. The following is an example. ∫ 0 ∞ sin ⁡ ( x ) x d x = π 2 ∫ 0 ∞ sin ⁡ ( x ) x sin ⁡ ( x / 3 ) x / 3 d x = π 2 ∫ 0 ∞ sin ⁡ ( x ) x sin ⁡ ( x / 3 ) x / 3 sin ⁡ ( x / 5 ) x / 5 d x = π 2 {\displaystyle {\begin{aligned}&\int _{0}^{\infty }{\frac {\sin(x)}{x}}\,dx={\frac {\pi }{2}}\\[10pt]&\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\,dx={\frac {\pi }{2}}\\[10pt]&\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}{\frac {\sin(x/5)}{x/5}}\,dx={\frac {\pi }{2}}\end{aligned}}} This pattern continues up to ∫ 0 ∞ sin ⁡ ( x ) x sin ⁡ ( x / 3 ) x / 3 ⋯ sin ⁡ ( x / 13 ) x / 13 d x = π 2 . {\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}}.} At the next step the pattern fails, ∫ 0 ∞ sin ⁡ ( x ) x sin ⁡ ( x / 3 ) x / 3 ⋯ sin ⁡ ( x / 15 ) x / 15 d x = 467807924713440738696537864469 935615849440640907310521750000 π ≈ 0.499999999992646859 π ≈ π 2 − 2.31 × 10 − 11 ....