- Source: Selberg integral
In mathematics, the Selberg integral is a generalization of Euler beta function to n dimensions introduced by Atle Selberg.
Selberg's integral formula
When
R
e
(
α
)
>
0
,
R
e
(
β
)
>
0
,
R
e
(
γ
)
>
−
min
(
1
n
,
R
e
(
α
)
n
−
1
,
R
e
(
β
)
n
−
1
)
{\displaystyle Re(\alpha )>0,Re(\beta )>0,Re(\gamma )>-\min \left({\frac {1}{n}},{\frac {Re(\alpha )}{n-1}},{\frac {Re(\beta )}{n-1}}\right)}
, we have
S
n
(
α
,
β
,
γ
)
=
∫
0
1
⋯
∫
0
1
∏
i
=
1
n
t
i
α
−
1
(
1
−
t
i
)
β
−
1
∏
1
≤
i
<
j
≤
n
|
t
i
−
t
j
|
2
γ
d
t
1
⋯
d
t
n
=
∏
j
=
0
n
−
1
Γ
(
α
+
j
γ
)
Γ
(
β
+
j
γ
)
Γ
(
1
+
(
j
+
1
)
γ
)
Γ
(
α
+
β
+
(
n
+
j
−
1
)
γ
)
Γ
(
1
+
γ
)
{\displaystyle {\begin{aligned}S_{n}(\alpha ,\beta ,\gamma )&=\int _{0}^{1}\cdots \int _{0}^{1}\prod _{i=1}^{n}t_{i}^{\alpha -1}(1-t_{i})^{\beta -1}\prod _{1\leq i
Selberg's formula implies Dixon's identity for well poised hypergeometric series, and some special cases of Dyson's conjecture. This is a corollary of Aomoto.
Aomoto's integral formula
Aomoto proved a slightly more general integral formula. With the same conditions as Selberg's formula,
∫
0
1
⋯
∫
0
1
(
∏
i
=
1
k
t
i
)
∏
i
=
1
n
t
i
α
−
1
(
1
−
t
i
)
β
−
1
∏
1
≤
i
<
j
≤
n
|
t
i
−
t
j
|
2
γ
d
t
1
⋯
d
t
n
{\displaystyle \int _{0}^{1}\cdots \int _{0}^{1}\left(\prod _{i=1}^{k}t_{i}\right)\prod _{i=1}^{n}t_{i}^{\alpha -1}(1-t_{i})^{\beta -1}\prod _{1\leq i
=
S
n
(
α
,
β
,
γ
)
∏
j
=
1
k
α
+
(
n
−
j
)
γ
α
+
β
+
(
2
n
−
j
−
1
)
γ
.
{\displaystyle =S_{n}(\alpha ,\beta ,\gamma )\prod _{j=1}^{k}{\frac {\alpha +(n-j)\gamma }{\alpha +\beta +(2n-j-1)\gamma }}.}
A proof is found in Chapter 8 of Andrews, Askey & Roy (1999).
Mehta's integral
When
R
e
(
γ
)
>
−
1
/
n
{\displaystyle Re(\gamma )>-1/n}
,
1
(
2
π
)
n
/
2
∫
−
∞
∞
⋯
∫
−
∞
∞
∏
i
=
1
n
e
−
t
i
2
/
2
∏
1
≤
i
<
j
≤
n
|
t
i
−
t
j
|
2
γ
d
t
1
⋯
d
t
n
=
∏
j
=
1
n
Γ
(
1
+
j
γ
)
Γ
(
1
+
γ
)
.
{\displaystyle {\frac {1}{(2\pi )^{n/2}}}\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }\prod _{i=1}^{n}e^{-t_{i}^{2}/2}\prod _{1\leq i
It is a corollary of Selberg, by setting
α
=
β
{\displaystyle \alpha =\beta }
, and change of variables with
t
i
=
1
+
t
i
′
/
2
α
2
{\displaystyle t_{i}={\frac {1+t'_{i}/{\sqrt {2\alpha }}}{2}}}
, then taking
α
→
∞
{\displaystyle \alpha \to \infty }
.
This was conjectured by Mehta & Dyson (1963), who were unaware of Selberg's earlier work.
It is the partition function for a gas of point charges moving on a line that are attracted to the origin.
Macdonald's integral
Macdonald (1982) conjectured the following extension of Mehta's integral to all finite root systems, Mehta's original case corresponding to the An−1 root system.
1
(
2
π
)
n
/
2
∫
⋯
∫
|
∏
r
2
(
x
,
r
)
(
r
,
r
)
|
γ
e
−
(
x
1
2
+
⋯
+
x
n
2
)
/
2
d
x
1
⋯
d
x
n
=
∏
j
=
1
n
Γ
(
1
+
d
j
γ
)
Γ
(
1
+
γ
)
{\displaystyle {\frac {1}{(2\pi )^{n/2}}}\int \cdots \int \left|\prod _{r}{\frac {2(x,r)}{(r,r)}}\right|^{\gamma }e^{-(x_{1}^{2}+\cdots +x_{n}^{2})/2}dx_{1}\cdots dx_{n}=\prod _{j=1}^{n}{\frac {\Gamma (1+d_{j}\gamma )}{\Gamma (1+\gamma )}}}
The product is over the roots r of the roots system and the numbers dj are the degrees of the generators of the ring of invariants of the reflection group.
Opdam (1989) gave a uniform proof for all crystallographic reflection groups. Several years later he proved it in full generality, making use of computer-aided calculations by Garvan.
References
Kata Kunci Pencarian:
- Daftar topik teori bilangan
- Teori bilangan
- Hipotesis Riemann
- Selberg integral
- Rankin–Selberg method
- Atle Selberg
- Selberg trace formula
- Elliptic integral
- Riemann hypothesis
- Maass–Selberg relations
- List of eponyms of special functions
- Dyson conjecture
- Prime number theorem
