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Statements
∑
n
=
0
∞
(
-
1
)
n
(
n
+
a
)
s
=
1
Γ
(
s
)
∫
0
∞
x
s
-
1
e
-
a
x
1
+
e
-
x
d
x
=
2
-
s
(
ζ
(
s
,
1
2
a
)
-
ζ
(
s
,
1
2
(
1
+
a
)
)
)
,
superscript
subscript
𝑛
0
superscript
1
𝑛
superscript
𝑛
𝑎
𝑠
1
Euler-Gamma
𝑠
superscript
subscript
0
superscript
𝑥
𝑠
1
superscript
𝑒
𝑎
𝑥
1
superscript
𝑒
𝑥
𝑥
superscript
2
𝑠
Hurwitz-zeta
𝑠
1
2
𝑎
Hurwitz-zeta
𝑠
1
2
1
𝑎
{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=%
\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}%
\mathrm{d}x=2^{-s}\left(\zeta\left(s,\tfrac{1}{2}a\right)-\zeta\left(s,\tfrac{%
1}{2}(1+a)\right)\right),}}
ℜ
a
>
0
𝑎
0
{\displaystyle{\displaystyle\Re a>0}}
ℜ
s
>
0
𝑠
0
{\displaystyle{\displaystyle\Re s>0}}
ℜ
a
=
0
𝑎
0
{\displaystyle{\displaystyle\Re a=0}}
ℑ
a
≠
0
𝑎
0
{\displaystyle{\displaystyle\Im a\neq 0}}
0
<
ℜ
s
<
1
0
𝑠
1
{\displaystyle{\displaystyle 0<\Re s<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}}}
ℑ
absent
{\displaystyle{\displaystyle\Im}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
a
𝑎
{\displaystyle{\displaystyle a}}
s
𝑠
{\displaystyle{\displaystyle s}}
Identifiers
