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Statements
πΈ
Ξ½
ΞΌ
β‘
(
x
Β±
i
β’
0
)
=
e
β
Ξ½
β’
Ο
β’
i
/
2
β’
Ο
3
/
2
β’
(
1
-
x
2
)
ΞΌ
/
2
2
Ξ½
+
1
β’
(
x
β’
π
β‘
(
1
2
β’
ΞΌ
-
1
2
β’
Ξ½
+
1
2
,
1
2
β’
Ξ½
+
1
2
β’
ΞΌ
+
1
;
3
2
;
x
2
)
Ξ
β‘
(
1
2
β’
Ξ½
-
1
2
β’
ΞΌ
+
1
2
)
β’
Ξ
β‘
(
1
2
β’
Ξ½
+
1
2
β’
ΞΌ
+
1
2
)
β
i
β’
π
β‘
(
1
2
β’
ΞΌ
-
1
2
β’
Ξ½
,
1
2
β’
Ξ½
+
1
2
β’
ΞΌ
+
1
2
;
1
2
;
x
2
)
Ξ
β‘
(
1
2
β’
Ξ½
-
1
2
β’
ΞΌ
+
1
)
β’
Ξ
β‘
(
1
2
β’
Ξ½
+
1
2
β’
ΞΌ
+
1
)
)
.
associated-Legendre-black-Q
π
π
plus-or-minus
π₯
π
0
superscript
π
minus-or-plus
π
π
π
2
superscript
π
3
2
superscript
1
superscript
π₯
2
π
2
superscript
2
π
1
minus-or-plus
π₯
scaled-hypergeometric-bold-F
1
2
π
1
2
π
1
2
1
2
π
1
2
π
1
3
2
superscript
π₯
2
Euler-Gamma
1
2
π
1
2
π
1
2
Euler-Gamma
1
2
π
1
2
π
1
2
π
scaled-hypergeometric-bold-F
1
2
π
1
2
π
1
2
π
1
2
π
1
2
1
2
superscript
π₯
2
Euler-Gamma
1
2
π
1
2
π
1
Euler-Gamma
1
2
π
1
2
π
1
{\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)=%
\frac{e^{\mp\nu\pi i/2}\pi^{3/2}\left(1-x^{2}\right)^{\mu/2}}{2^{\nu+1}}\left(%
\frac{x\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2},\frac{1}{2}%
\nu+\frac{1}{2}\mu+1;\frac{3}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\nu-%
\frac{1}{2}\mu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+%
\frac{1}{2}\right)}\mp i\frac{\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu,%
\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2};\frac{1}{2};x^{2}\right)}{\Gamma%
\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)\Gamma\left(\frac{1}{2}\nu+\frac{1%
}{2}\mu+1\right)}\right).}}
Ξ
β‘
(
z
)
Euler-Gamma
π§
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
πΈ
Ξ½
ΞΌ
β‘
(
z
)
associated-Legendre-black-Q
π
π
π§
{\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(%
\NVar{z}\right)}}
Ο
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
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)}}
x
π₯
{\displaystyle{\displaystyle x}}
ΞΌ
π
{\displaystyle{\displaystyle\mu}}
Ξ½
π
{\displaystyle{\displaystyle\nu}}
