Borwein integral
integral with unusual properties
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
.
{\displaystyle {\begin{aligned}\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/15)}{x/15}}\,dx&={\frac {467807924713440738696537864469}{935615849440640907310521750000}}~\pi \\[5pt]&\approx 0.499999999992646859~\pi \\[5pt]&\approx {\frac {\pi }{2}}-2.31\times 10^{-11}.\end{aligned}}}
In general, similar integrals have value π/2 whenever the numbers 3, 5, 7… are replaced by positive real numbers such that the sum of their reciprocals is less than 1.
In the example above, 1/3 + 1/5 + … + 1/13 < 1, but 1/3 + 1/5 + … + 1/15 > 1.
With the inclusion of the additional factor
2
cos
(
x
)
{\displaystyle 2\cos(x)}
, the pattern holds up over a longer series,
∫
0
∞
2
cos
(
x
)
sin
(
x
)
x
sin
(
x
/
3
)
x
/
3
⋯
sin
(
x
/
111
)
x
/
111
d
x
=
π
2
,
{\displaystyle \int _{0}^{\infty }2\cos(x){\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/111)}{x/111}}\,dx={\frac {\pi }{2}},}
but
∫
0
∞
2
cos
(
x
)
sin
(
x
)
x
sin
(
x
/
3
)
x
/
3
⋯
sin
(
x
/
111
)
x
/
111
sin
(
x
/
113
)
x
/
113
d
x
≈
π
2
−
2.3324
×
10
−
138
.
{\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}}\,dx\approx {\frac {\pi }{2}}-2.3324\times 10^{-138}.}
In this case, 1/3 + 1/5 + … + 1/111 < 2, but 1/3 + 1/5 + … + 1/113 > 2.
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