In der Mathematik bezeichnet Borwein-Integral Integralterme, die Produkte der Sinc-Funktion enthalten.
Folgen dieser Integrale sind bekannt dafür, dass sie scheinbare Muster beinhalten, die sich dann aber als falsch herausstellen.
Die einfachste Folge sind folgende Integrale, die für die ersten sieben Glieder exakt
π
/
2
{\displaystyle \pi /2}
A
1
=
∫
0
∞
sin
(
x
)
x
d
x
=
π
/
2
A
2
=
∫
0
∞
sin
(
x
)
x
sin
(
x
/
3
)
x
/
3
d
x
=
π
/
2
A
3
=
∫
0
∞
sin
(
x
)
x
sin
(
x
/
3
)
x
/
3
sin
(
x
/
5
)
x
/
5
d
x
=
π
/
2
{\displaystyle {\begin{aligned}&A_{1}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{2}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{3}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}{\frac {\sin(x/5)}{x/5}}\,\mathrm {d} x=\pi /2\end{aligned}}}
Dieses Muster wiederholt sich bis
A
7
=
∫
0
∞
sin
(
x
)
x
sin
(
x
/
3
)
x
/
3
⋯
sin
(
x
/
13
)
x
/
13
d
x
=
π
/
2
{\displaystyle A_{7}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/13)}{x/13}}\,\mathrm {d} x=\pi /2}
Das folgende Glied lautet aber:
A
8
=
∫
0
∞
sin
(
x
)
x
sin
(
x
/
3
)
x
/
3
⋯
sin
(
x
/
13
)
x
/
13
sin
(
x
/
15
)
x
/
15
d
x
=
467807924713440738696537864469
935615849440640907310521750000
π
=
π
2
−
6879714958723010531
935615849440640907310521750000
π
≃
π
2
−
2,310
057
⋅
10
−
11
<
π
2
.
{\displaystyle {\begin{aligned}A_{8}&=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/13)}{x/13}}{\frac {\sin(x/15)}{x/15}}\,\mathrm {d} x\\[10pt]&={\frac {467807924713440738696537864469}{935615849440640907310521750000}}\pi \\[10pt]&={\frac {\pi }{2}}-{\frac {6879714958723010531}{935615849440640907310521750000}}\pi \\[10pt]&\simeq {\frac {\pi }{2}}-2{,}310057\cdot 10^{-11}\ {\color {red}<}\ {\frac {\pi }{2}}\ .\end{aligned}}}
Auch die folgenden Glieder
A
9
,
A
10
,
…
{\displaystyle A_{9},A_{10},\ldots }
π
/
2
{\displaystyle \pi /2}
A
∞
{\displaystyle A_{\infty }}
π
/
2
−
3
,
52
⋅
10
−
5
{\displaystyle \,\pi /2-3{,}52\cdot 10^{-5}}
Hinweis
Man beachte
1
/
3
+
1
/
5
+
…
+
1
/
13
=
0,955
1
…
≤
1
{\displaystyle 1/3+1/5+\ldots +1/13=0{,}9551\ldots \leq 1}
1
/
3
+
1
/
5
+
…
+
1
/
13
+
1
/
15
=
1,021
8
…
≰
1
{\displaystyle 1/3+1/5+\ldots +1/13+1/15=1{,}0218\ldots \not \leq 1}
Wem das noch nicht gereicht hat, die Folge behält dieses Verhalten wesentlich länger bei:
A
1
=
∫
0
∞
2
cos
(
x
)
sin
(
x
)
x
d
x
=
π
/
2
A
2
=
∫
0
∞
2
cos
(
x
)
sin
(
x
)
x
sin
(
x
/
3
)
x
/
3
d
x
=
π
/
2
A
3
=
∫
0
∞
2
cos
(
x
)
sin
(
x
)
x
sin
(
x
/
3
)
x
/
3
sin
(
x
/
5
)
x
/
5
d
x
=
π
/
2
{\displaystyle {\begin{aligned}&A_{1\;}=\int _{0}^{\infty }2\cos(x){\frac {\sin(x)}{x}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{2\;}=\int _{0}^{\infty }2\cos(x){\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{3\;}=\int _{0}^{\infty }2\cos(x){\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}{\frac {\sin(x/5)}{x/5}}\,\mathrm {d} x=\pi /2\end{aligned}}}
Dieses Muster wiederholt sich hier bis
A
56
=
∫
0
∞
2
cos
(
x
)
sin
(
x
)
x
sin
(
x
/
3
)
x
/
3
⋯
sin
(
x
/
111
)
x
/
111
d
x
=
π
/
2
,
{\displaystyle A_{56}=\int _{0}^{\infty }2\cos(x){\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/111)}{x/111}}\,\mathrm {d} x=\pi /2,}
aber nicht mehr bei diesem Glied
A
57
=
∫
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
−
2,332
429
⋅
10
−
138
<
π
2
.
{\displaystyle A_{57}=\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}}\,\mathrm {d} x\;\simeq {\frac {\pi }{2}}-2{,}332429\cdot 10^{-138}\ {\color {red}<}\ {\frac {\pi }{2}}\ .}
Hinweis
Man beachte
1
/
1
+
1
/
3
+
…
+
1
/
111
=
2,994
4
…
<
3
{\displaystyle 1/1+1/3+\ldots +1/111=2{,}9944\ldots <3}
1
/
1
+
1
/
3
+
…
+
1
/
111
+
1
/
113
=
3,003
2
…
>
3
{\displaystyle 1/1+1/3+\ldots +1/111+1/113=3{,}0032\ldots >3}
Diese Folge reißt erst nach 14419 Gliedern aus.
A
1
=
∫
0
∞
sin
(
x
)
x
d
x
=
π
/
2
A
2
=
∫
0
∞
sin
(
x
)
x
sin
(
x
/
11
)
x
/
11
d
x
=
π
/
2
A
3
=
∫
0
∞
sin
(
x
)
x
sin
(
x
/
11
)
x
/
11
sin
(
x
/
21
)
x
/
21
d
x
=
π
/
2
{\displaystyle {\begin{aligned}&A_{1}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{2}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/11)}{x/11}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{3}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/11)}{x/11}}{\frac {\sin(x/21)}{x/21}}\,\mathrm {d} x=\pi /2\end{aligned}}}
Dieses Muster wiederholt sich bis
A
14418
=
∫
0
∞
sin
(
x
)
x
sin
(
x
/
11
)
x
/
11
⋯
sin
(
x
/
144171
)
x
/
144171
d
x
=
π
/
2
{\displaystyle A_{14418}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/11)}{x/11}}\cdots {\frac {\sin(x/144171)}{x/144171}}\,\mathrm {d} x=\pi /2}
Das folgende Glied lautet aber:
A
14419
=
∫
0
∞
sin
(
x
)
x
sin
(
x
/
11
)
x
/
11
⋯
sin
(
x
/
144171
)
x
/
144171
sin
(
x
/
144181
)
x
/
144181
d
x
<
π
2
.
{\displaystyle A_{14419}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/11)}{x/11}}\cdots {\frac {\sin(x/144171)}{x/144171}}{\frac {\sin(x/144181)}{x/144181}}\,\mathrm {d} x\ {\color {red}<}\ {\frac {\pi }{2}}\ .}
Hinweis
Man beachte
1
/
11
+
1
/
21
+
…
+
1
/
144171
=
0,999
995990
…
≤
1
{\displaystyle 1/11+1/21+\ldots +1/144171=0{,}999995990\ldots \leq 1}
1
/
11
+
1
/
21
+
…
+
1
/
144171
+
1
/
144173
=
1,000
002925
…
≰
1
{\displaystyle 1/11+1/21+\ldots +1/144171+1/144173=1{,}000002925\ldots \not \leq 1}
Für eine Folge reeller Zahlen,
a
0
,
a
1
,
a
2
,
…
{\displaystyle a_{0},a_{1},a_{2},\ldots }
∫
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}}\,\mathrm {d} x}
gegeben werden.[1]
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}}
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}}
b
γ
{\displaystyle b_{\gamma }}
a
k
{\displaystyle a_{k}}
ε
γ
=
γ
1
γ
2
⋯
γ
n
=
±
1
{\displaystyle \varepsilon _{\gamma }=\gamma _{1}\gamma _{2}\cdots \gamma _{n}=\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}}\,\mathrm {d} x={\frac {\pi }{2a_{0}}}C_{n}}
wobei
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 })}
Falls
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}
↑ David Borwein, Jonathan M. Borwein: Some remarkable properties of sinc and related integrals . In: The Ramanujan Journal . Band 5 , 2001, S. 73–89 .
![{\displaystyle {\begin{aligned}&A_{1}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{2}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{3}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}{\frac {\sin(x/5)}{x/5}}\,\mathrm {d} x=\pi /2\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9238d3052bfac5be70f393fa057db468a7c84cf2)
![{\displaystyle {\begin{aligned}A_{8}&=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\cdots {\frac {\sin(x/13)}{x/13}}{\frac {\sin(x/15)}{x/15}}\,\mathrm {d} x\\[10pt]&={\frac {467807924713440738696537864469}{935615849440640907310521750000}}\pi \\[10pt]&={\frac {\pi }{2}}-{\frac {6879714958723010531}{935615849440640907310521750000}}\pi \\[10pt]&\simeq {\frac {\pi }{2}}-2{,}310057\cdot 10^{-11}\ {\color {red}<}\ {\frac {\pi }{2}}\ .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bca7cbac8920432ab16d28e375ef669a3592223e)
![{\displaystyle {\begin{aligned}&A_{1\;}=\int _{0}^{\infty }2\cos(x){\frac {\sin(x)}{x}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{2\;}=\int _{0}^{\infty }2\cos(x){\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{3\;}=\int _{0}^{\infty }2\cos(x){\frac {\sin(x)}{x}}{\frac {\sin(x/3)}{x/3}}{\frac {\sin(x/5)}{x/5}}\,\mathrm {d} x=\pi /2\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc198e30d19e0f6af3e5a2a86db41275316959a5)
![{\displaystyle {\begin{aligned}&A_{1}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{2}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/11)}{x/11}}\,\mathrm {d} x=\pi /2\\[10pt]&A_{3}=\int _{0}^{\infty }{\frac {\sin(x)}{x}}{\frac {\sin(x/11)}{x/11}}{\frac {\sin(x/21)}{x/21}}\,\mathrm {d} x=\pi /2\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae8ba2a6ade06c32dccef019a722cb6430a630fb)
