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Statements
F
4
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γ
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γ
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;
x
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1
-
y
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,
y
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1
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x
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)
=
Γ
(
γ
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Γ
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γ
′
)
Γ
(
α
)
Γ
(
β
)
Γ
(
γ
-
α
)
Γ
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γ
′
-
β
)
∫
0
1
∫
0
1
u
α
-
1
v
β
-
1
(
1
-
u
)
γ
-
α
-
1
(
1
-
v
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γ
′
-
β
-
1
(
1
-
u
x
)
γ
+
γ
′
-
α
-
1
(
1
-
v
y
)
γ
+
γ
′
-
β
-
1
(
1
-
u
x
-
v
y
)
α
+
β
-
γ
-
γ
′
+
1
d
u
d
v
,
Appell-F-4
𝛼
𝛽
𝛾
superscript
𝛾
′
𝑥
1
𝑦
𝑦
1
𝑥
Euler-Gamma
𝛾
Euler-Gamma
superscript
𝛾
′
Euler-Gamma
𝛼
Euler-Gamma
𝛽
Euler-Gamma
𝛾
𝛼
Euler-Gamma
superscript
𝛾
′
𝛽
superscript
subscript
0
1
superscript
subscript
0
1
superscript
𝑢
𝛼
1
superscript
𝑣
𝛽
1
superscript
1
𝑢
𝛾
𝛼
1
superscript
1
𝑣
superscript
𝛾
′
𝛽
1
superscript
1
𝑢
𝑥
𝛾
superscript
𝛾
′
𝛼
1
superscript
1
𝑣
𝑦
𝛾
superscript
𝛾
′
𝛽
1
superscript
1
𝑢
𝑥
𝑣
𝑦
𝛼
𝛽
𝛾
superscript
𝛾
′
1
𝑢
𝑣
{\displaystyle{\displaystyle{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x%
(1-y),y(1-x)\right)=\frac{\Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}%
\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)\Gamma\left(\gamma-%
\alpha\right)\Gamma\left(\gamma^{\prime}-\beta\right)}\int_{0}^{1}\!\!\!\int_{%
0}^{1}\frac{u^{\alpha-1}v^{\beta-1}(1-u)^{\gamma-\alpha-1}(1-v)^{\gamma^{%
\prime}-\beta-1}}{(1-ux)^{\gamma+\gamma^{\prime}-\alpha-1}(1-vy)^{\gamma+%
\gamma^{\prime}-\beta-1}(1-ux-vy)^{\alpha+\beta-\gamma-\gamma^{\prime}+1}}%
\mathrm{d}u\mathrm{d}v,}}
ℜ
γ
>
ℜ
α
>
0
𝛾
𝛼
0
{\displaystyle{\displaystyle\Re\gamma>\Re\alpha>0}}
ℜ
γ
′
>
ℜ
β
>
0
superscript
𝛾
′
𝛽
0
{\displaystyle{\displaystyle\Re\gamma^{\prime}>\Re\beta>0}}
F
4
(
α
,
β
;
γ
,
γ
′
missing
missing
;
missing
,
missing
)
Appell-F-4
𝛼
𝛽
𝛾
superscript
𝛾
′
missing
missing
{\displaystyle{\displaystyle{F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{%
\gamma},\NVar{\gamma^{\prime}}{missing}{missing};missing,missing\right)\)\@add@PDF@RDFa@triples\end{document}}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
