Uit Wikipedia, de vrije encyclopedie
In de wiskunde is een Borwein-integraal een integraal over producten van
sin
(
a
x
)
a
x
{\displaystyle {\frac {\sin(ax)}{ax}}}
a
{\displaystyle a}
David Borwein en Jonathan Borwein de onderstaande relaties in 2001 ontdekten.
∫
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
sin
(
x
7
)
x
7
d
x
=
π
2
∫
0
∞
sin
(
x
)
x
sin
(
x
3
)
x
3
sin
(
x
5
)
x
5
sin
(
x
7
)
x
7
sin
(
x
9
)
x
9
d
x
=
π
2
∫
0
∞
sin
(
x
)
x
sin
(
x
3
)
x
3
sin
(
x
5
)
x
5
sin
(
x
7
)
x
7
sin
(
x
9
)
x
9
sin
(
x
11
)
x
11
d
x
=
π
2
∫
0
∞
sin
(
x
)
x
sin
(
x
3
)
x
3
sin
(
x
5
)
x
5
sin
(
x
7
)
x
7
sin
(
x
9
)
x
9
sin
(
x
11
)
x
11
sin
(
x
13
)
x
13
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({\frac {x}{3}})}{\frac {x}{3}}}\,\mathrm {d} x={\frac {\pi }{2}}\\[10pt]&\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin({\frac {x}{3}})}{\frac {x}{3}}}{\frac {\sin({\frac {x}{5}})}{\frac {x}{5}}}{\frac {\sin({\frac {x}{7}})}{\frac {x}{7}}}\,\mathrm {d} x={\frac {\pi }{2}}\\[10pt]&\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin({\frac {x}{3}})}{\frac {x}{3}}}{\frac {\sin({\frac {x}{5}})}{\frac {x}{5}}}{\frac {\sin({\frac {x}{7}})}{\frac {x}{7}}}{\frac {\sin({\frac {x}{9}})}{\frac {x}{9}}}\,\mathrm {d} x={\frac {\pi }{2}}\\[10pt]&\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin({\frac {x}{3}})}{\frac {x}{3}}}{\frac {\sin({\frac {x}{5}})}{\frac {x}{5}}}{\frac {\sin({\frac {x}{7}})}{\frac {x}{7}}}{\frac {\sin({\frac {x}{9}})}{\frac {x}{9}}}{\frac {\sin({\frac {x}{11}})}{\frac {x}{11}}}\,\mathrm {d} x={\frac {\pi }{2}}\\[10pt]&\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin({\frac {x}{3}})}{\frac {x}{3}}}{\frac {\sin({\frac {x}{5}})}{\frac {x}{5}}}{\frac {\sin({\frac {x}{7}})}{\frac {x}{7}}}{\frac {\sin({\frac {x}{9}})}{\frac {x}{9}}}{\frac {\sin({\frac {x}{11}})}{\frac {x}{11}}}{\frac {\sin({\frac {x}{13}})}{\frac {x}{13}}}\,\mathrm {d} x={\frac {\pi }{2}}\end{aligned}}}
∫
0
∞
sin
(
x
)
x
sin
(
x
3
)
x
3
sin
(
x
5
)
x
5
sin
(
x
7
)
x
7
sin
(
x
9
)
x
9
sin
(
x
11
)
x
11
sin
(
x
13
)
x
13
sin
(
x
15
)
x
15
d
x
=
{\displaystyle \int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin({\frac {x}{3}})}{\frac {x}{3}}}{\frac {\sin({\frac {x}{5}})}{\frac {x}{5}}}{\frac {\sin({\frac {x}{7}})}{\frac {x}{7}}}{\frac {\sin({\frac {x}{9}})}{\frac {x}{9}}}{\frac {\sin({\frac {x}{11}})}{\frac {x}{11}}}{\frac {\sin({\frac {x}{13}})}{\frac {x}{13}}}{\frac {\sin({\frac {x}{15}})}{\frac {x}{15}}}\,\mathrm {d} x=}
=
467807924713440738696537864469
935615849440640907310521750000
π
=
{\displaystyle ={\frac {467807924713440738696537864469}{935615849440640907310521750000}}\pi =}
=
(
1
2
−
6879714958723010531
935615849440640907310521750000
)
π
≃
{\displaystyle =\left({\tfrac {1}{2}}-{\frac {6879714958723010531}{935615849440640907310521750000}}\right)\pi \,\simeq }
≃
(
1
2
−
7
,
35
×
10
−
12
)
π
{\displaystyle \simeq \left({\tfrac {1}{2}}-7{,}35\times 10^{-12}\right)\pi }
In het algemeen hebben alle integralen van deze vorm als uitkomst
π
2
{\displaystyle {\frac {\pi }{2}}}
reële getallen waarvan de som van de reciproque waarden kleiner is dan 1. In het bovenstaande voorbeeld wijkt het bij 15 dus af omdat
1
3
+
1
5
+
1
7
+
…
+
1
15
>
1
{\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{5}}+{\tfrac {1}{7}}+\ldots +{\tfrac {1}{15}}>1}
1
3
+
1
5
+
1
7
+
…
+
1
13
<
1
{\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{5}}+{\tfrac {1}{7}}+\ldots +{\tfrac {1}{13}}<1}
Het verhaal gaat dat, nadat David en Jonathan Borwein deze numerieke curiositeit gevonden hadden, ze verifieerden dat het computerprogramma Maple alle waarden van deze integralen correct berekende en ze de waarde van de laatste integraal bij wijze van grap als een bug in de software rapporteerden. Later verklaarde Maple-informaticus Jacques Carette dat hij minstens drie dagen had besteed om de bug te traceren voordat hij doorhad dat Borwein hem beet had.
![{\displaystyle {\begin{aligned}&\int _{0}^{\infty }{\frac {\sin(x)}{x}}\,dx={\frac {\pi }{2}}\\[10pt]&\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin({\frac {x}{3}})}{\frac {x}{3}}}\,\mathrm {d} x={\frac {\pi }{2}}\\[10pt]&\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin({\frac {x}{3}})}{\frac {x}{3}}}{\frac {\sin({\frac {x}{5}})}{\frac {x}{5}}}{\frac {\sin({\frac {x}{7}})}{\frac {x}{7}}}\,\mathrm {d} x={\frac {\pi }{2}}\\[10pt]&\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin({\frac {x}{3}})}{\frac {x}{3}}}{\frac {\sin({\frac {x}{5}})}{\frac {x}{5}}}{\frac {\sin({\frac {x}{7}})}{\frac {x}{7}}}{\frac {\sin({\frac {x}{9}})}{\frac {x}{9}}}\,\mathrm {d} x={\frac {\pi }{2}}\\[10pt]&\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin({\frac {x}{3}})}{\frac {x}{3}}}{\frac {\sin({\frac {x}{5}})}{\frac {x}{5}}}{\frac {\sin({\frac {x}{7}})}{\frac {x}{7}}}{\frac {\sin({\frac {x}{9}})}{\frac {x}{9}}}{\frac {\sin({\frac {x}{11}})}{\frac {x}{11}}}\,\mathrm {d} x={\frac {\pi }{2}}\\[10pt]&\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin({\frac {x}{3}})}{\frac {x}{3}}}{\frac {\sin({\frac {x}{5}})}{\frac {x}{5}}}{\frac {\sin({\frac {x}{7}})}{\frac {x}{7}}}{\frac {\sin({\frac {x}{9}})}{\frac {x}{9}}}{\frac {\sin({\frac {x}{11}})}{\frac {x}{11}}}{\frac {\sin({\frac {x}{13}})}{\frac {x}{13}}}\,\mathrm {d} x={\frac {\pi }{2}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b879c49651f316f851716b06c1e4f34777c731ca)
