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Statements
1
Γ
(
s
)
∫
0
∞
x
s
-
1
e
-
a
x
2
cosh
x
d
x
=
4
-
s
(
ζ
(
s
,
1
4
+
1
4
a
)
-
ζ
(
s
,
3
4
+
1
4
a
)
)
,
1
Euler-Gamma
𝑠
superscript
subscript
0
superscript
𝑥
𝑠
1
superscript
𝑒
𝑎
𝑥
2
𝑥
𝑥
superscript
4
𝑠
Hurwitz-zeta
𝑠
1
4
1
4
𝑎
Hurwitz-zeta
𝑠
3
4
1
4
𝑎
{\displaystyle{\displaystyle\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}%
\frac{x^{s-1}e^{-ax}}{2\cosh x}\mathrm{d}x=4^{-s}\left(\zeta\left(s,\tfrac{1}{%
4}+\tfrac{1}{4}a\right)-\zeta\left(s,\tfrac{3}{4}+\tfrac{1}{4}a\right)\right),}}
ℜ
s
>
0
𝑠
0
{\displaystyle{\displaystyle\Re s>0}}
ℜ
a
>
-
1
𝑎
1
{\displaystyle{\displaystyle\Re a>-1}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
ζ
(
s
,
a
)
Hurwitz-zeta
𝑠
𝑎
{\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
cosh
z
𝑧
{\displaystyle{\displaystyle\cosh\NVar{z}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
x
𝑥
{\displaystyle{\displaystyle x}}
a
𝑎
{\displaystyle{\displaystyle a}}
s
𝑠
{\displaystyle{\displaystyle s}}
Identifiers
