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:8.14.E6 - MaRDI portal
Statements
∫
0
∞
x
a
-
1
e
-
s
x
Γ
(
b
,
x
)
d
x
=
Γ
(
a
+
b
)
a
(
1
+
s
)
a
+
b
F
(
1
,
a
+
b
;
1
+
a
;
s
/
(
1
+
s
)
)
,
superscript
subscript
0
superscript
𝑥
𝑎
1
superscript
𝑒
𝑠
𝑥
incomplete-Gamma
𝑏
𝑥
𝑥
Euler-Gamma
𝑎
𝑏
𝑎
superscript
1
𝑠
𝑎
𝑏
Gauss-hypergeometric-F
1
𝑎
𝑏
1
𝑎
𝑠
1
𝑠
{\displaystyle{\displaystyle\int_{0}^{\infty}x^{a-1}e^{-sx}\Gamma\left(b,x%
\right)\mathrm{d}x=\frac{\Gamma\left(a+b\right)}{a(1+s)^{a+b}}\*F\left(1,a+b;1%
+a;s/(1+s)\right),}}
ℜ
s
>
-
1
𝑠
1
{\displaystyle{\displaystyle\Re s>-1}}
ℜ
(
a
+
b
)
>
0
𝑎
𝑏
0
{\displaystyle{\displaystyle\Re(a+b)>0}}
ℜ
a
>
0
𝑎
0
{\displaystyle{\displaystyle\Re a>0}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
F
(
a
,
b
;
c
;
z
)
Gauss-hypergeometric-F
𝑎
𝑏
𝑐
𝑧
{\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
Γ
(
a
,
z
)
incomplete-Gamma
𝑎
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{a},\NVar{z}\right)}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
x
𝑥
{\displaystyle{\displaystyle x}}
a
𝑎
{\displaystyle{\displaystyle a}}
Identifiers
