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Statements
π
β‘
(
a
,
b
;
c
;
z
)
=
e
-
b
β’
Ο
β’
i
β’
Ξ
β‘
(
1
-
b
)
2
β’
Ο
β’
i
β’
Ξ
β‘
(
c
-
b
)
β’
β«
1
(
0
+
)
t
b
-
1
β’
(
1
-
t
)
c
-
b
-
1
(
1
-
z
β’
t
)
a
β’
d
t
,
scaled-hypergeometric-bold-F
π
π
π
π§
superscript
π
π
π
imaginary-unit
Euler-Gamma
1
π
2
π
imaginary-unit
Euler-Gamma
π
π
superscript
subscript
1
limit-from
0
superscript
π‘
π
1
superscript
1
π‘
π
π
1
superscript
1
π§
π‘
π
π‘
{\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=e^{-b\pi\mathrm{i}}%
\frac{\Gamma\left(1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{1}^{(%
0+)}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t,}}
|
ph
β‘
(
1
-
z
)
|
<
Ο
phase
1
π§
{\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
β
β‘
(
c
-
b
)
>
0
π
π
0
{\displaystyle{\displaystyle\Re(c-b)>0}}
|
ph
β‘
(
1
-
z
)
|
<
Ο
ph
1
π§
π
{\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
b
β
1
,
2
,
3
,
β¦
π
1
2
3
β¦
{\displaystyle{\displaystyle b\neq 1,2,3,\dots}}
β
β‘
(
c
-
b
)
>
0
π
π
0
{\displaystyle{\displaystyle\Re(c-b)>0}}
Ξ
β‘
(
z
)
Euler-Gamma
π§
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
Ο
{\displaystyle{\displaystyle\pi}}
d
x
π₯
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
β«
{\displaystyle{\displaystyle\int}}
ph
phase
{\displaystyle{\displaystyle\operatorname{ph}}}
β
β‘
absent
{\displaystyle{\displaystyle\Re}}
π
β‘
(
a
,
b
;
c
;
z
)
scaled-hypergeometric-bold-F
π
π
π
π§
{\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z%
}\right)}}
z
π§
{\displaystyle{\displaystyle z}}
a
π
{\displaystyle{\displaystyle a}}
b
π
{\displaystyle{\displaystyle b}}
c
π
{\displaystyle{\displaystyle c}}
