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Statements
ζ
(
s
,
a
)
=
1
2
a
-
s
+
a
1
-
s
s
-
1
+
2
∫
0
∞
sin
(
s
arctan
(
x
/
a
)
)
(
a
2
+
x
2
)
s
/
2
(
e
2
π
x
-
1
)
d
x
,
Hurwitz-zeta
𝑠
𝑎
1
2
superscript
𝑎
𝑠
superscript
𝑎
1
𝑠
𝑠
1
2
superscript
subscript
0
𝑠
𝑥
𝑎
superscript
superscript
𝑎
2
superscript
𝑥
2
𝑠
2
superscript
𝑒
2
𝜋
𝑥
1
𝑥
{\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1%
-s}}{s-1}+2\int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}\left(x/a%
\right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\mathrm{d}x,}}
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}}
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}}}
∫
{\displaystyle{\displaystyle\int}}
arctan
z
𝑧
{\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
x
𝑥
{\displaystyle{\displaystyle x}}
a
𝑎
{\displaystyle{\displaystyle a}}
s
𝑠
{\displaystyle{\displaystyle s}}
Identifiers
