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Statements
π
β‘
(
a
,
b
1
2
;
z
)
=
2
a
+
b
-
(
3
/
2
)
Ο
β’
Ξ
β‘
(
a
+
1
2
)
β’
Ξ
β‘
(
b
+
1
2
)
β’
(
z
-
1
)
(
-
a
-
b
+
(
1
/
2
)
)
/
2
β’
(
e
Β±
Ο
β’
i
β’
(
a
+
b
-
(
1
/
2
)
)
β’
P
a
-
b
-
(
1
/
2
)
-
a
-
b
+
(
1
/
2
)
β‘
(
-
z
)
+
P
a
-
b
-
(
1
/
2
)
-
a
-
b
+
(
1
/
2
)
β‘
(
z
)
)
,
scaled-hypergeometric-bold-F
π
π
1
2
π§
superscript
2
π
π
3
2
π
Euler-Gamma
π
1
2
Euler-Gamma
π
1
2
superscript
π§
1
π
π
1
2
2
superscript
π
plus-or-minus
π
imaginary-unit
π
π
1
2
Legendre-P-first-kind
π
π
1
2
π
π
1
2
π§
Legendre-P-first-kind
π
π
1
2
π
π
1
2
π§
{\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{1}{2}};z\right)=%
\frac{2^{a+b-(\ifrac{3}{2})}}{\pi}\Gamma\left(a+\tfrac{1}{2}\right)\Gamma\left%
(b+\tfrac{1}{2}\right)\*(z-1)^{(-a-b+(\ifrac{1}{2}))/2}\*\left(e^{\pm\pi%
\mathrm{i}(a+b-(\ifrac{1}{2}))}P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}%
\left(-\sqrt{z}\right)+P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt%
{z}\right)\right),}}
0
<
|
ph
β‘
z
|
<
Ο
0
phase
π§
π
{\displaystyle{\displaystyle 0<|\operatorname{ph}z|<\pi}}
a
,
b
β
-
1
2
,
-
3
2
,
-
5
2
,
β¦
formulae-sequence
π
π
1
2
3
2
5
2
β¦
{\displaystyle{\displaystyle a,b\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},%
\ldots}}
0
<
|
ph
β‘
z
|
<
Ο
0
ph
π§
π
{\displaystyle{\displaystyle 0<|\operatorname{ph}z|<\pi}}
Ξ
β‘
(
z
)
Euler-Gamma
π§
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
P
Ξ½
ΞΌ
β‘
(
z
)
Legendre-P-first-kind
π
π
π§
{\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
Ο
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
ph
phase
{\displaystyle{\displaystyle\operatorname{ph}}}
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}}
