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.E71 - MaRDI portal
Statements
∫
0
∞
J
μ
(
a
t
)
J
ν
(
b
t
)
J
ν
(
c
t
)
t
1
-
μ
d
t
=
(
b
c
)
μ
-
1
(
sin
ϕ
)
μ
-
1
2
(
2
π
)
1
2
a
μ
𝖯
ν
-
1
2
1
2
-
μ
(
cos
ϕ
)
,
superscript
subscript
0
Bessel-J
𝜇
𝑎
𝑡
Bessel-J
𝜈
𝑏
𝑡
Bessel-J
𝜈
𝑐
𝑡
superscript
𝑡
1
𝜇
𝑡
superscript
𝑏
𝑐
𝜇
1
superscript
italic-ϕ
𝜇
1
2
superscript
2
𝜋
1
2
superscript
𝑎
𝜇
Ferrers-Legendre-P-first-kind
1
2
𝜇
𝜈
1
2
italic-ϕ
{\displaystyle{\displaystyle\int_{0}^{\infty}J_{\mu}\left(at\right)J_{\nu}%
\left(bt\right)J_{\nu}\left(ct\right)t^{1-\mu}\mathrm{d}t=\frac{(bc)^{\mu-1}(%
\sin\phi)^{\mu-\frac{1}{2}}}{(2\pi)^{\frac{1}{2}}a^{\mu}}\mathsf{P}^{\frac{1}{%
2}-\mu}_{\nu-\frac{1}{2}}(\cos\phi),}}
ℜ
μ
>
-
1
2
,
ℜ
ν
>
-
1
,
|
b
-
c
|
<
a
<
b
+
c
,
cos
ϕ
=
(
b
2
+
c
2
-
a
2
)
/
(
2
b
c
)
formulae-sequence
formulae-sequence
𝜇
1
2
formulae-sequence
𝜈
1
𝑏
𝑐
𝑎
𝑏
𝑐
italic-ϕ
superscript
𝑏
2
superscript
𝑐
2
superscript
𝑎
2
2
𝑏
𝑐
{\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2},\Re\nu>-1,|b-c|<a<b+c,\cos%
\phi=(b^{2}+c^{2}-a^{2})/(2bc)}}
J
ν
(
z
)
Bessel-J
𝜈
𝑧
{\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
𝖯
ν
μ
(
x
)
Ferrers-Legendre-P-first-kind
𝜇
𝜈
𝑥
{\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}%
\right)}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
