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Statements
P
ν
-
μ
(
x
)
=
(
2
μ
π
)
1
/
2
1
Γ
(
μ
+
1
)
(
1
-
α
1
+
α
)
(
ν
/
2
)
+
(
1
/
4
)
(
1
-
α
2
)
-
μ
/
2
(
α
2
+
η
2
α
2
(
x
2
-
1
)
+
1
)
1
/
4
K
ν
+
1
2
(
μ
η
)
(
1
+
O
(
1
μ
)
)
,
Legendre-P-first-kind
𝜇
𝜈
𝑥
superscript
2
𝜇
𝜋
1
2
1
Euler-Gamma
𝜇
1
superscript
1
𝛼
1
𝛼
𝜈
2
1
4
superscript
1
superscript
𝛼
2
𝜇
2
superscript
superscript
𝛼
2
superscript
𝜂
2
superscript
𝛼
2
superscript
𝑥
2
1
1
1
4
modified-Bessel-second-kind
𝜈
1
2
𝜇
𝜂
1
Big-O
1
𝜇
{\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\left(\frac{2\mu}{%
\pi}\right)^{1/2}\frac{1}{\Gamma\left(\mu+1\right)}\left(\frac{1-\alpha}{1+%
\alpha}\right)^{(\nu/2)+(1/4)}\*\left(1-\alpha^{2}\right)^{-\mu/2}\left(\frac{%
\alpha^{2}+\eta^{2}}{\alpha^{2}\left(x^{2}-1\right)+1}\right)^{1/4}\*K_{\nu+%
\frac{1}{2}}\left(\mu\eta\right)\left(1+O\left(\frac{1}{\mu}\right)\right),}}
O
(
x
)
Big-O
𝑥
{\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
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}}
K
ν
(
z
)
modified-Bessel-second-kind
𝜈
𝑧
{\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
x
𝑥
{\displaystyle{\displaystyle x}}
μ
𝜇
{\displaystyle{\displaystyle\mu}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
α
𝛼
{\displaystyle{\displaystyle\alpha}}
η
𝜂
{\displaystyle{\displaystyle\eta}}
Identifiers
