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:14.20.E6 - MaRDI portal
Statements
P
-
1
2
+
i
τ
-
μ
(
x
)
=
i
e
-
μ
π
i
sinh
(
τ
π
)
|
Γ
(
μ
+
1
2
+
i
τ
)
|
2
(
Q
-
1
2
+
i
τ
μ
(
x
)
-
Q
-
1
2
-
i
τ
μ
(
x
)
)
,
Legendre-P-first-kind
𝜇
1
2
𝑖
𝜏
𝑥
𝑖
superscript
𝑒
𝜇
𝜋
𝑖
𝜏
𝜋
superscript
Euler-Gamma
𝜇
1
2
𝑖
𝜏
2
Legendre-Q-second-kind
𝜇
1
2
𝑖
𝜏
𝑥
Legendre-Q-second-kind
𝜇
1
2
𝑖
𝜏
𝑥
{\displaystyle{\displaystyle P^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\frac%
{ie^{-\mu\pi i}}{\sinh\left(\tau\pi\right)\left|\Gamma\left(\mu+\frac{1}{2}+i%
\tau\right)\right|^{2}}\*\left(Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)-Q^{%
\mu}_{-\frac{1}{2}-i\tau}\left(x\right)\right),}}
τ
≠
0
𝜏
0
{\displaystyle{\displaystyle\tau\neq 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)}}
Q
ν
μ
(
z
)
Legendre-Q-second-kind
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
sinh
z
𝑧
{\displaystyle{\displaystyle\sinh\NVar{z}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
x
𝑥
{\displaystyle{\displaystyle x}}
τ
𝜏
{\displaystyle{\displaystyle\tau}}
μ
𝜇
{\displaystyle{\displaystyle\mu}}
Identifiers
