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.E49 - MaRDI portal
Statements
∫
0
∞
t
μ
-
1
e
-
a
t
J
ν
(
b
t
)
d
t
=
(
1
2
b
)
ν
a
μ
+
ν
Γ
(
μ
+
ν
)
𝐅
(
μ
+
ν
2
,
μ
+
ν
+
1
2
;
ν
+
1
;
-
b
2
a
2
)
,
superscript
subscript
0
superscript
𝑡
𝜇
1
superscript
𝑒
𝑎
𝑡
Bessel-J
𝜈
𝑏
𝑡
𝑡
superscript
1
2
𝑏
𝜈
superscript
𝑎
𝜇
𝜈
Euler-Gamma
𝜇
𝜈
scaled-hypergeometric-bold-F
𝜇
𝜈
2
𝜇
𝜈
1
2
𝜈
1
superscript
𝑏
2
superscript
𝑎
2
{\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu-1}e^{-at}J_{\nu}\left(bt%
\right)\mathrm{d}t=\frac{(\tfrac{1}{2}b)^{\nu}}{a^{\mu+\nu}}\Gamma\left(\mu+%
\nu\right)\*\mathbf{F}\left(\frac{\mu+\nu}{2},\frac{\mu+\nu+1}{2};\nu+1;-\frac%
{b^{2}}{a^{2}}\right),}}
ℜ
(
μ
+
ν
)
>
0
,
ℜ
(
a
±
i
b
)
>
0
formulae-sequence
𝜇
𝜈
0
plus-or-minus
𝑎
𝑖
𝑏
0
{\displaystyle{\displaystyle\Re(\mu+\nu)>0,\Re(a\pm ib)>0}}
J
ν
(
z
)
Bessel-J
𝜈
𝑧
{\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
𝐅
(
a
,
b
;
c
;
z
)
scaled-hypergeometric-bold-F
𝑎
𝑏
𝑐
𝑧
{\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z%
}\right)}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
