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Statements
1
4
π
∫
-
∞
∞
∏
k
=
1
4
Γ
(
a
k
+
i
t
)
Γ
(
a
k
-
i
t
)
Γ
(
2
i
t
)
Γ
(
-
2
i
t
)
d
t
=
∏
1
≤
j
<
k
≤
4
Γ
(
a
j
+
a
k
)
Γ
(
a
1
+
a
2
+
a
3
+
a
4
)
,
1
4
𝜋
superscript
subscript
superscript
subscript
product
𝑘
1
4
Euler-Gamma
subscript
𝑎
𝑘
𝑖
𝑡
Euler-Gamma
subscript
𝑎
𝑘
𝑖
𝑡
Euler-Gamma
2
𝑖
𝑡
Euler-Gamma
2
𝑖
𝑡
𝑡
subscript
product
1
𝑗
𝑘
4
Euler-Gamma
subscript
𝑎
𝑗
subscript
𝑎
𝑘
Euler-Gamma
subscript
𝑎
1
subscript
𝑎
2
subscript
𝑎
3
subscript
𝑎
4
{\displaystyle{\displaystyle\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{%
k=1}^{4}\Gamma\left(a_{k}+it\right)\Gamma\left(a_{k}-it\right)}{\Gamma\left(2%
it\right)\Gamma\left(-2it\right)}\mathrm{d}t=\frac{\prod_{1\leq j<k\leq 4}%
\Gamma\left(a_{j}+a_{k}\right)}{\Gamma\left(a_{1}+a_{2}+a_{3}+a_{4}\right)},}}
ℜ
(
a
k
)
>
0
subscript
𝑎
𝑘
0
{\displaystyle{\displaystyle\Re(a_{k})>0}}
ℜ
(
a
k
)
>
0
subscript
𝑎
𝑘
0
{\displaystyle{\displaystyle\Re(a_{k})>0}}
k
=
1
,
2
,
3
,
4
𝑘
1
2
3
4
{\displaystyle{\displaystyle k=1,2,3,4}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
j
𝑗
{\displaystyle{\displaystyle j}}
k
𝑘
{\displaystyle{\displaystyle k}}
a
𝑎
{\displaystyle{\displaystyle a}}
Identifiers
