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:24.7.E1 - MaRDI portal
Statements
B
2
n
=
(
-
1
)
n
+
1
4
n
1
-
2
1
-
2
n
∫
0
∞
t
2
n
-
1
e
2
π
t
+
1
d
t
=
(
-
1
)
n
+
1
2
n
1
-
2
1
-
2
n
∫
0
∞
t
2
n
-
1
e
-
π
t
sech
(
π
t
)
d
t
,
Bernoulli-number-B
2
𝑛
superscript
1
𝑛
1
4
𝑛
1
superscript
2
1
2
𝑛
superscript
subscript
0
superscript
𝑡
2
𝑛
1
superscript
𝑒
2
𝜋
𝑡
1
𝑡
superscript
1
𝑛
1
2
𝑛
1
superscript
2
1
2
𝑛
superscript
subscript
0
superscript
𝑡
2
𝑛
1
superscript
𝑒
𝜋
𝑡
𝜋
𝑡
𝑡
{\displaystyle{\displaystyle B_{2n}=(-1)^{n+1}\frac{4n}{1-2^{1-2n}}\int_{0}^{%
\infty}\frac{t^{2n-1}}{e^{2\pi t}+1}\mathrm{d}t=(-1)^{n+1}\frac{2n}{1-2^{1-2n}%
}\int_{0}^{\infty}t^{2n-1}e^{-\pi t}\operatorname{sech}\left(\pi t\right)%
\mathrm{d}t,}}
B
n
Bernoulli-number-B
𝑛
{\displaystyle{\displaystyle B_{\NVar{n}}}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
sech
z
𝑧
{\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}
∫
{\displaystyle{\displaystyle\int}}
n
𝑛
{\displaystyle{\displaystyle n}}
t
𝑡
{\displaystyle{\displaystyle t}}
Identifiers
