Deprecated : $wgMWOAuthSharedUserIDs=false is deprecated, set $wgMWOAuthSharedUserIDs=true, $wgMWOAuthSharedUserSource='local' instead [Called from MediaWiki\HookContainer\HookContainer::run in /var/www/html/w/includes/HookContainer/HookContainer.php at line 135] in /var/www/html/w/includes/Debug/MWDebug.php on line 372 DLMF:18.17.E7 - MaRDI portal
Statements
(
P
n
(
x
)
)
2
+
4
π
-
2
(
𝖰
n
(
x
)
)
2
=
4
π
-
2
∫
1
∞
Q
n
(
x
2
+
(
1
-
x
2
)
t
)
(
t
2
-
1
)
-
1
2
d
t
,
superscript
Legendre-spherical-polynomial
𝑛
𝑥
2
4
superscript
𝜋
2
superscript
shorthand-Ferrers-Legendre-Q-first-kind
𝑛
𝑥
2
4
superscript
𝜋
2
superscript
subscript
1
shorthand-Legendre-Q-second-kind
𝑛
superscript
𝑥
2
1
superscript
𝑥
2
𝑡
superscript
superscript
𝑡
2
1
1
2
𝑡
{\displaystyle{\displaystyle\left(P_{n}\left(x\right)\right)^{2}+4\pi^{-2}%
\left(\mathsf{Q}_{n}\left(x\right)\right)^{2}=4\pi^{-2}\*\int_{1}^{\infty}Q_{n%
}\left(x^{2}+(1-x^{2})t\right)(t^{2}-1)^{-\frac{1}{2}}\mathrm{d}t,}}
-
1
<
x
<
1
1
𝑥
1
{\displaystyle{\displaystyle-1<x<1}}
P
n
(
x
)
Legendre-spherical-polynomial
𝑛
𝑥
{\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
𝖰
ν
(
x
)
=
𝖰
ν
0
(
x
)
shorthand-Ferrers-Legendre-Q-first-kind
𝜈
𝑥
Ferrers-Legendre-Q-first-kind
0
𝜈
𝑥
{\displaystyle{\displaystyle\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=%
\mathsf{Q}^{0}_{\nu}\left(x\right)}}
Q
ν
(
z
)
=
Q
ν
0
(
z
)
shorthand-Legendre-Q-second-kind
𝜈
𝑧
Legendre-Q-second-kind
0
𝜈
𝑧
{\displaystyle{\displaystyle Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}%
\left(z\right)}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
Identifiers
