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:10.22.E75 - MaRDI portal
Statements
∫
0
∞
Y
ν
(
a
t
)
J
ν
(
b
t
)
J
ν
(
c
t
)
t
1
+
ν
d
t
=
{
-
(
a
b
c
)
ν
(
-
A
)
-
ν
-
1
2
π
1
2
2
ν
+
1
Γ
(
1
2
-
ν
)
,
0
<
a
<
|
b
-
c
|
,
0
,
|
b
-
c
|
<
a
<
b
+
c
,
(
a
b
c
)
ν
(
-
A
)
-
ν
-
1
2
π
1
2
2
ν
+
1
Γ
(
1
2
-
ν
)
,
a
>
b
+
c
.
superscript
subscript
0
Bessel-Y-Weber
𝜈
𝑎
𝑡
Bessel-J
𝜈
𝑏
𝑡
Bessel-J
𝜈
𝑐
𝑡
superscript
𝑡
1
𝜈
𝑡
cases
superscript
𝑎
𝑏
𝑐
𝜈
superscript
𝐴
𝜈
1
2
superscript
𝜋
1
2
superscript
2
𝜈
1
Euler-Gamma
1
2
𝜈
0
𝑎
𝑏
𝑐
0
𝑏
𝑐
𝑎
𝑏
𝑐
superscript
𝑎
𝑏
𝑐
𝜈
superscript
𝐴
𝜈
1
2
superscript
𝜋
1
2
superscript
2
𝜈
1
Euler-Gamma
1
2
𝜈
𝑎
𝑏
𝑐
{\displaystyle{\displaystyle\int_{0}^{\infty}Y_{\nu}\left(at\right)J_{\nu}%
\left(bt\right)J_{\nu}\left(ct\right)t^{1+\nu}\mathrm{d}t=\begin{cases}-\dfrac%
{(abc)^{\nu}(-A)^{-\nu-\frac{1}{2}}}{\pi^{\frac{1}{2}}2^{\nu+1}\Gamma\left(%
\frac{1}{2}-\nu\right)},&0<a<|b-c|,\\
0,&|b-c|<a<b+c,\\
\dfrac{(abc)^{\nu}(-A)^{-\nu-\frac{1}{2}}}{\pi^{\frac{1}{2}}2^{\nu+1}\Gamma%
\left(\frac{1}{2}-\nu\right)},&a>b+c.\end{cases}}}
J
ν
(
z
)
Bessel-J
𝜈
𝑧
{\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
Y
ν
(
z
)
Bessel-Y-Weber
𝜈
𝑧
{\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
A
𝐴
{\displaystyle{\displaystyle A}}
Identifiers
