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Statements
ζ
(
s
,
a
)
=
Γ
(
1
-
s
)
2
π
i
∫
-
∞
(
0
+
)
e
a
z
z
s
-
1
1
-
e
z
d
z
,
Hurwitz-zeta
𝑠
𝑎
Euler-Gamma
1
𝑠
2
𝜋
𝑖
superscript
subscript
limit-from
0
superscript
𝑒
𝑎
𝑧
superscript
𝑧
𝑠
1
1
superscript
𝑒
𝑧
𝑧
{\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{\Gamma\left(1-s\right)%
}{2\pi i}\int_{-\infty}^{(0+)}\frac{e^{az}z^{s-1}}{1-e^{z}}\mathrm{d}z,}}
ℜ
a
>
0
𝑎
0
{\displaystyle{\displaystyle\Re a>0}}
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)}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
a
𝑎
{\displaystyle{\displaystyle a}}
s
𝑠
{\displaystyle{\displaystyle s}}
z
𝑧
{\displaystyle{\displaystyle z}}
