Multidimensional Integrals
Let
V
n
be the simplex:
t
1
+
t
2
+
…
+
t
n
≤
1
,
t
k
≥
0
. Then for
ℜ
z
k
>
0
,
k
=
1
,
2
,
…
,
n
+
1
,
∫
V
n
t
1
z
1
-
1
t
2
z
2
-
1
⋯
t
n
z
n
-
1
ⅆ
t
1
ⅆ
t
2
⋯
ⅆ
t
n
=
Γ
(
z
1
)
Γ
(
z
2
)
⋯
Γ
(
z
n
)
Γ
(
1
+
z
1
+
z
2
+
…
+
z
n
)
∫
V
n
(
1
-
∑
k
=
1
n
t
k
)
z
n
+
1
-
1
∏
k
=
1
n
t
k
z
k
-
1
ⅆ
t
k
=
Γ
(
z
1
)
Γ
(
z
2
)
⋯
Γ
(
z
n
+
1
)
Γ
(
z
1
+
z
2
+
…
+
z
n
+
1
)
Selberg-type Integrals
Δ
(
t
1
,
t
2
,
…
,
t
n
)
=
∏
1
≤
j
<
k
≤
n
(
t
j
-
t
k
)
Then
∫
[
0
,
1
]
n
t
1
t
2
⋯
t
m
|
Δ
(
t
1
,
…
,
t
n
)
|
2
c
∏
k
=
1
n
t
k
a
-
1
(
1
-
t
k
)
b
-
1
ⅆ
t
k
=
1
(
Γ
(
1
+
c
)
)
n
∏
k
=
1
m
a
+
(
n
-
k
)
c
a
+
b
+
(
2
n
-
k
-
1
)
c
∏
k
=
1
n
Γ
(
a
+
(
n
-
k
)
c
)
Γ
(
b
+
(
n
-
k
)
c
)
Γ
(
1
+
k
c
)
Γ
(
a
+
b
+
(
2
n
-
k
-
1
)
c
)
provided that
ℜ
a
,
ℜ
b
>
0
,
ℜ
c
>
-
min
(
1
n
,
ℜ
a
(
n
-
1
)
,
ℜ
b
(
n
-
1
)
)
Secondly,
∫
[
0
,
∞
)
n
t
1
t
2
⋯
t
m
|
Δ
(
t
1
,
…
,
t
n
)
|
2
c
∏
k
=
1
n
t
k
a
-
1
ⅇ
-
t
k
ⅆ
t
k
=
∏
k
=
1
m
(
a
+
(
n
-
k
)
c
)
∏
k
=
1
n
Γ
(
a
+
(
n
-
k
)
c
)
Γ
(
1
+
k
c
)
(
Γ
(
1
+
c
)
)
n
when
ℜ
a
>
0
,
ℜ
c
>
-
min
(
1
n
,
ℜ
a
(
n
-
1
)
)
.
Thirdly,
1
(
2
π
)
n
2
∫
(
-
∞
,
∞
)
n
|
Δ
(
t
1
,
…
,
t
n
)
|
2
c
∏
k
=
1
n
exp
(
-
1
2
t
k
2
)
ⅆ
t
k
=
∏
k
=
1
n
Γ
(
1
+
k
c
)
(
Γ
(
1
+
c
)
)
n
Dyson's Integral
1
(
2
π
)
n
∫
[
-
π
,
π
]
n
∏
1
≤
j
<
k
≤
n
|
ⅇ
ⅈ
θ
j
-
ⅇ
ⅈ
θ
k
|
2
b
ⅆ
θ
1
⋯
ⅆ
θ
n
=
Γ
(
1
+
b
n
)
(
Γ
(
1
+
b
)
)
n
ℜ
b
>
1
n
.
