Nella matematica , un integrale di Borwein è un integrale che coinvolge prodotti di
s
i
n
c
(
a
x
)
{\displaystyle \mathrm {sinc} (ax)}
funzione sinc è data da
s
i
n
c
(
x
)
=
sin
(
x
)
x
{\displaystyle \mathrm {sinc} (x)={\frac {\sin(x)}{x}}}
x
≠
0
{\displaystyle x\neq 0}
s
i
n
c
(
0
)
=
1
{\displaystyle \mathrm {sinc} (0)=1}
[ 1] [ 2]
Questi integrali sono importanti per esibire schemi apparenti che, tuttavia, alla fine falliscono. Un esempio è ciò che segue,
∫
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}}}
Questo schema continua fino a
∫
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}}.}
Tuttavia, al passo successivo lo schema evidente fallisce,
∫
0
∞
sin
(
x
)
x
sin
(
x
/
3
)
x
/
3
⋯
sin
(
x
/
15
)
x
/
15
d
x
=
467807924713440738696537864469
935615849440640907310521750000
π
=
π
2
−
6879714958723010531
935615849440640907310521750000
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]&={\frac {\pi }{2}}-{\frac {6879714958723010531}{935615849440640907310521750000}}~\pi \\[5pt]&\simeq {\frac {\pi }{2}}-2.31\times 10^{-11}.\end{aligned}}}
In generale, integrali analoghi valgono
π
/
2
{\displaystyle \pi /2}
3
,
5
,
7
,
…
{\displaystyle 3,5,7,\ldots }
Nell'esempio precedente,
1
3
+
1
5
+
…
+
1
13
<
1
{\displaystyle {\frac {1}{3}}+{\frac {1}{5}}+\ldots +{\frac {1}{13}}<1}
1
3
+
1
5
+
…
+
1
15
>
1
{\displaystyle {\frac {1}{3}}+{\frac {1}{5}}+\ldots +{\frac {1}{15}}>1}
L'esempio con una serie più estesa
∫
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}},}
con tuttavia
∫
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
,
{\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<{\frac {\pi }{2}},}
è mostrato in [ 3]
1
3
+
1
5
+
…
+
1
111
<
2
{\displaystyle {\frac {1}{3}}+{\frac {1}{5}}+\ldots +{\frac {1}{111}}<2}
1
3
+
1
5
+
…
+
1
113
>
2
{\displaystyle {\frac {1}{3}}+{\frac {1}{5}}+\ldots +{\frac {1}{113}}>2}
Data una sequenza di numeri reali,
a
0
,
a
1
,
a
2
,
…
{\displaystyle a_{0},a_{1},a_{2},\ldots }
[ 1]
∫
0
∞
∏
k
=
0
n
sin
(
a
k
x
)
a
k
x
d
x
{\displaystyle \int _{0}^{\infty }\prod _{k=0}^{n}{\frac {\sin(a_{k}x)}{a_{k}x}}\,dx}
Per affermare la formula, serve considerare delle somme che coinvolgono
a
k
{\displaystyle a_{k}}
γ
=
(
γ
1
,
γ
2
,
…
,
γ
n
)
∈
{
±
1
}
n
{\displaystyle \gamma =(\gamma _{1},\gamma _{2},\ldots ,\gamma _{n})\in \{\pm 1\}^{n}}
n
{\displaystyle n}
±
1
{\displaystyle \pm 1}
b
γ
=
a
0
+
γ
1
a
1
+
γ
2
a
2
+
⋯
+
γ
n
a
n
{\displaystyle b_{\gamma }=a_{0}+\gamma _{1}a_{1}+\gamma _{2}a_{2}+\cdots +\gamma _{n}a_{n}}
a
k
{\displaystyle a_{k}}
ε
γ
=
γ
1
γ
2
⋯
γ
n
{\displaystyle \varepsilon _{\gamma }=\gamma _{1}\gamma _{2}\cdots \gamma _{n}}
±
1
{\displaystyle \pm 1}
∫
0
∞
∏
k
=
0
n
sin
(
a
k
x
)
a
k
x
d
x
=
π
2
a
0
C
n
{\displaystyle \int _{0}^{\infty }\prod _{k=0}^{n}{\frac {\sin(a_{k}x)}{a_{k}x}}\,dx={\frac {\pi }{2a_{0}}}C_{n}}
dove
C
n
=
1
2
n
n
!
∏
k
=
1
n
a
k
∑
γ
∈
{
±
1
}
n
ε
γ
b
γ
n
sgn
(
b
γ
)
{\displaystyle C_{n}={\frac {1}{2^{n}n!\prod _{k=1}^{n}a_{k}}}\sum _{\gamma \in \{\pm 1\}^{n}}\varepsilon _{\gamma }b_{\gamma }^{n}\operatorname {sgn}(b_{\gamma })}
Nel caso in cui
a
0
>
|
a
1
|
+
|
a
2
|
+
⋯
+
|
a
n
|
{\displaystyle a_{0}>|a_{1}|+|a_{2}|+\cdots +|a_{n}|}
C
n
=
1
{\displaystyle C_{n}=1}
Inoltre, se esiste un
n
{\displaystyle n}
k
=
0
,
…
,
n
−
1
{\displaystyle k=0,\ldots ,n-1}
0
<
a
n
<
2
a
k
{\displaystyle 0<a_{n}<2a_{k}}
a
1
+
a
2
+
⋯
+
a
n
−
1
<
a
0
<
a
1
+
a
2
+
⋯
+
a
n
−
1
+
a
n
{\displaystyle a_{1}+a_{2}+\cdots +a_{n-1}<a_{0}<a_{1}+a_{2}+\cdots +a_{n-1}+a_{n}}
n
{\displaystyle n}
n
{\displaystyle n}
a
0
{\displaystyle a_{0}}
C
k
=
1
{\displaystyle C_{k}=1}
k
=
0
,
…
,
n
−
1
{\displaystyle k=0,\ldots ,n-1}
C
n
=
1
−
(
a
1
+
a
2
+
⋯
+
a
n
−
a
0
)
n
2
n
−
1
n
!
∏
k
=
1
n
a
k
{\displaystyle C_{n}=1-{\frac {(a_{1}+a_{2}+\cdots +a_{n}-a_{0})^{n}}{2^{n-1}n!\prod _{k=1}^{n}a_{k}}}}
Il primo esempio è il caso in cui
a
k
=
1
2
k
+
1
{\displaystyle a_{k}={\frac {1}{2k+1}}}
n
=
7
{\displaystyle n=7}
a
7
=
1
15
{\displaystyle a_{7}={\frac {1}{15}}}
1
3
+
1
5
+
1
7
+
1
9
+
1
11
+
1
13
≈
0.955
{\displaystyle {\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{9}}+{\frac {1}{11}}+{\frac {1}{13}}\approx 0.955}
1
3
+
1
5
+
1
7
+
1
9
+
1
11
+
1
13
+
1
15
≈
1.02
{\displaystyle {\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{9}}+{\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{15}}\approx 1.02}
a
0
=
1
{\displaystyle a_{0}=1}
∫
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}}}
che rimane vera se si toglie qualunque fattore, tuttavia
∫
0
∞
sin
(
x
)
x
sin
(
x
/
3
)
x
/
3
⋯
sin
(
x
/
15
)
x
/
15
d
x
=
π
2
(
1
−
(
3
−
1
+
5
−
1
+
7
−
1
+
9
−
1
+
11
−
1
+
13
−
1
+
15
−
1
−
1
)
7
2
6
⋅
7
!
⋅
(
1
/
3
⋅
1
/
5
⋅
1
/
7
⋅
1
/
9
⋅
1
/
11
⋅
1
/
13
⋅
1
/
15
)
)
{\displaystyle {\begin{aligned}&\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{aligned}}}
che è uguale al valore dato precedentemente.
Fu schedato come bug per il supporto Maple . Ci sono voluti tre giorni allo sviluppatore Jacques Carette per capire che non fosse un errore [ 4]
^ a b Borwein, David; Borwein, Jonathan M. (2001), "Some remarkable properties of sinc and related integrals", The Ramanujan Journal , 5 (1): 73–89, doi:10.1023/A:1011497229317 , ISSN 1382-4090 , MR 1829810
^ Baillie, Robert (2011). "Fun With Very Large Numbers". arXiv:1105.3943 .
^ Schmid, Hanspeter (2014), "Two curious integrals and a graphic proof" (PDF), Elemente der Mathematik , 69 (1): 11–17, doi:10.4171/EM/239 , ISSN 0013-6018
^ https://mathoverflow.net/questions/11517/computer-algebra-errors/11607#comment28278_11607 
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9670ee100344ef5d0de572a51754e9a34b5aa47)
![{\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]&={\frac {\pi }{2}}-{\frac {6879714958723010531}{935615849440640907310521750000}}~\pi \\[5pt]&\simeq {\frac {\pi }{2}}-2.31\times 10^{-11}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78124d3ca5316f86182a6af0825a587bfdaa0b72)
![{\displaystyle {\begin{aligned}&\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12a2ad2632cf40d6ce99f63e35a5a1b31dae0368)
