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Statements
F
3
(
α
,
α
′
;
β
,
β
′
;
γ
;
x
,
y
)
=
Γ
(
γ
)
Γ
(
β
)
Γ
(
β
′
)
Γ
(
γ
-
β
-
β
′
)
∬
Δ
u
β
-
1
v
β
′
-
1
(
1
-
u
-
v
)
γ
-
β
-
β
′
-
1
(
1
-
u
x
)
α
(
1
-
v
y
)
α
′
d
u
d
v
,
Appell-F-3
𝛼
superscript
𝛼
′
𝛽
superscript
𝛽
′
𝛾
𝑥
𝑦
Euler-Gamma
𝛾
Euler-Gamma
𝛽
Euler-Gamma
superscript
𝛽
′
Euler-Gamma
𝛾
𝛽
superscript
𝛽
′
subscript
double-integral
Δ
superscript
𝑢
𝛽
1
superscript
𝑣
superscript
𝛽
′
1
superscript
1
𝑢
𝑣
𝛾
𝛽
superscript
𝛽
′
1
superscript
1
𝑢
𝑥
𝛼
superscript
1
𝑣
𝑦
superscript
𝛼
′
𝑢
𝑣
{\displaystyle{\displaystyle{F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{%
\prime};\gamma;x,y\right)=\frac{\Gamma\left(\gamma\right)}{\Gamma\left(\beta%
\right)\Gamma\left(\beta^{\prime}\right)\Gamma\left(\gamma-\beta-\beta^{\prime%
}\right)}\iint_{\Delta}\frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u-v)^{\gamma-%
\beta-\beta^{\prime}-1}}{(1-ux)^{\alpha}(1-vy)^{\alpha^{\prime}}}\mathrm{d}u%
\mathrm{d}v,}}
ℜ
(
γ
-
β
-
β
′
)
>
0
𝛾
𝛽
superscript
𝛽
′
0
{\displaystyle{\displaystyle\Re(\gamma-\beta-\beta^{\prime})>0}}
ℜ
β
>
0
𝛽
0
{\displaystyle{\displaystyle\Re\beta>0}}
ℜ
β
′
>
0
superscript
𝛽
′
0
{\displaystyle{\displaystyle\Re\beta^{\prime}>0}}
F
3
(
α
,
α
′
;
β
,
β
′
;
γ
;
x
,
y
)
Appell-F-3
𝛼
superscript
𝛼
′
𝛽
superscript
𝛽
′
𝛾
𝑥
𝑦
{\displaystyle{\displaystyle{F_{3}}\left(\NVar{\alpha},\NVar{\alpha^{\prime}};%
\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};\NVar{x},\NVar{y}\right)}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
Identifiers
