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Statements
𝑸
ν
μ
(
x
)
=
(
π
2
)
1
/
2
(
e
μ
)
ν
+
(
1
/
2
)
(
1
-
α
1
+
α
)
μ
/
2
(
1
-
α
2
)
-
(
ν
/
2
)
-
(
1
/
4
)
(
α
2
+
η
2
α
2
(
x
2
-
1
)
+
1
)
1
/
4
I
ν
+
1
2
(
μ
η
)
(
1
+
O
(
1
μ
)
)
,
associated-Legendre-black-Q
𝜇
𝜈
𝑥
superscript
𝜋
2
1
2
superscript
𝑒
𝜇
𝜈
1
2
superscript
1
𝛼
1
𝛼
𝜇
2
superscript
1
superscript
𝛼
2
𝜈
2
1
4
superscript
superscript
𝛼
2
superscript
𝜂
2
superscript
𝛼
2
superscript
𝑥
2
1
1
1
4
modified-Bessel-first-kind
𝜈
1
2
𝜇
𝜂
1
Big-O
1
𝜇
{\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\left(%
\frac{\pi}{2}\right)^{1/2}\left(\frac{e}{\mu}\right)^{\nu+(1/2)}\left(\frac{1-%
\alpha}{1+\alpha}\right)^{\mu/2}\*\left(1-\alpha^{2}\right)^{-(\nu/2)-(1/4)}%
\left(\frac{\alpha^{2}+\eta^{2}}{\alpha^{2}\left(x^{2}-1\right)+1}\right)^{1/4%
}\*I_{\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)}}
𝑸
ν
μ
(
z
)
associated-Legendre-black-Q
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(%
\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
I
ν
(
z
)
modified-Bessel-first-kind
𝜈
𝑧
{\displaystyle{\displaystyle I_{\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}}
