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:11.10.E30 - MaRDI portal
Statements
𝐉
ν
(
z
)
=
2
sin
(
1
2
ν
π
)
∑
k
=
0
∞
(
-
1
)
k
J
k
-
1
2
ν
+
1
2
(
1
2
z
)
J
k
+
1
2
ν
+
1
2
(
1
2
z
)
+
2
cos
(
1
2
ν
π
)
∑
′
k
=
0
′
∞
′
(
-
1
)
k
J
k
-
1
2
ν
(
1
2
z
)
J
k
+
1
2
ν
(
1
2
z
)
,
Anger-J
𝜈
𝑧
2
1
2
𝜈
𝜋
superscript
subscript
𝑘
0
superscript
1
𝑘
Bessel-J
𝑘
1
2
𝜈
1
2
1
2
𝑧
Bessel-J
𝑘
1
2
𝜈
1
2
1
2
𝑧
2
1
2
𝜈
𝜋
superscript
subscript
superscript
′
𝑘
0
superscript
1
𝑘
Bessel-J
𝑘
1
2
𝜈
1
2
𝑧
Bessel-J
𝑘
1
2
𝜈
1
2
𝑧
{\displaystyle{\displaystyle{\mathbf{J}_{\nu}\left(z\right)=}\\
{2\sin\left(\tfrac{1}{2}\nu\pi\right)\sum_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}{%
2}\nu+\frac{1}{2}}\left(\tfrac{1}{2}z\right)J_{k+\frac{1}{2}\nu+\frac{1}{2}}%
\left(\tfrac{1}{2}z\right)}+{2\cos\left(\tfrac{1}{2}\nu\pi\right)\sideset{}{{}%
^{\prime}}{\sum}_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}{2}\nu}\left(\tfrac{1}{2}z%
\right)J_{k+\frac{1}{2}\nu}\left(\tfrac{1}{2}z\right)},}}
𝐉
ν
(
z
)
Anger-J
𝜈
𝑧
{\displaystyle{\displaystyle\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)}}
J
ν
(
z
)
Bessel-J
𝜈
𝑧
{\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
z
𝑧
{\displaystyle{\displaystyle z}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
k
𝑘
{\displaystyle{\displaystyle k}}
Identifiers
