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