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:10.9.E13 - MaRDI portal
Statements
(
z
+
ζ
z
-
ζ
)
1
2
ν
J
ν
(
(
z
2
-
ζ
2
)
1
2
)
=
1
π
∫
0
π
e
ζ
cos
θ
cos
(
z
sin
θ
-
ν
θ
)
d
θ
-
sin
(
ν
π
)
π
∫
0
∞
e
-
ζ
cosh
t
-
z
sinh
t
-
ν
t
d
t
,
superscript
𝑧
𝜁
𝑧
𝜁
1
2
𝜈
Bessel-J
𝜈
superscript
superscript
𝑧
2
superscript
𝜁
2
1
2
1
𝜋
superscript
subscript
0
𝜋
superscript
𝑒
𝜁
𝜃
𝑧
𝜃
𝜈
𝜃
𝜃
𝜈
𝜋
𝜋
superscript
subscript
0
superscript
𝑒
𝜁
𝑡
𝑧
𝑡
𝜈
𝑡
𝑡
{\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}%
\nu}J_{\nu}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^%
{\pi}e^{\zeta\cos\theta}\cos\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta%
-\frac{\sin\left(\nu\pi\right)}{\pi}\int_{0}^{\infty}e^{-\zeta\cosh t-z\sinh t%
-\nu t}\mathrm{d}t,}}
ℜ
(
z
+
ζ
)
>
0
𝑧
𝜁
0
{\displaystyle{\displaystyle\Re(z+\zeta)>0}}
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}}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
cosh
z
𝑧
{\displaystyle{\displaystyle\cosh\NVar{z}}}
sinh
z
𝑧
{\displaystyle{\displaystyle\sinh\NVar{z}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
z
𝑧
{\displaystyle{\displaystyle z}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
