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Statements
P
ν
-
μ
(
x
)
=
2
1
/
2
Γ
(
μ
+
1
2
)
(
x
2
-
1
)
μ
/
2
π
1
/
2
Γ
(
ν
+
μ
+
1
)
Γ
(
μ
-
ν
)
∫
0
∞
cosh
(
(
ν
+
1
2
)
t
)
(
x
+
cosh
t
)
μ
+
(
1
/
2
)
d
t
,
Legendre-P-first-kind
𝜇
𝜈
𝑥
superscript
2
1
2
Euler-Gamma
𝜇
1
2
superscript
superscript
𝑥
2
1
𝜇
2
superscript
𝜋
1
2
Euler-Gamma
𝜈
𝜇
1
Euler-Gamma
𝜇
𝜈
superscript
subscript
0
𝜈
1
2
𝑡
superscript
𝑥
𝑡
𝜇
1
2
𝑡
{\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{2^{1/2}\Gamma%
\left(\mu+\frac{1}{2}\right)\left(x^{2}-1\right)^{\mu/2}}{\pi^{1/2}\Gamma\left%
(\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\*\int_{0}^{\infty}\frac{\cosh%
\left(\left(\nu+\frac{1}{2}\right)t\right)}{(x+\cosh t)^{\mu+(1/2)}}\mathrm{d}%
t,}}
ℜ
(
μ
-
ν
)
>
0
𝜇
𝜈
0
{\displaystyle{\displaystyle\Re(\mu-\nu)>0}}
ν
+
μ
≠
-
1
,
-
2
,
-
3
,
…
𝜈
𝜇
1
2
3
…
{\displaystyle{\displaystyle\nu+\mu\neq-1,-2,-3,\dots}}
ℜ
(
μ
-
ν
)
>
0
𝜇
𝜈
0
{\displaystyle{\displaystyle\Re(\mu-\nu)>0}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
P
ν
μ
(
z
)
Legendre-P-first-kind
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
cosh
z
𝑧
{\displaystyle{\displaystyle\cosh\NVar{z}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
x
𝑥
{\displaystyle{\displaystyle x}}
μ
𝜇
{\displaystyle{\displaystyle\mu}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
