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Statements
ζ
(
s
,
a
)
=
1
a
s
(
1
2
+
a
s
-
1
)
-
s
(
s
+
1
)
2
∫
0
∞
B
~
2
(
x
)
-
B
2
(
x
+
a
)
s
+
2
d
x
,
Hurwitz-zeta
𝑠
𝑎
1
superscript
𝑎
𝑠
1
2
𝑎
𝑠
1
𝑠
𝑠
1
2
superscript
subscript
0
periodic-Bernoulli-polynomial-B
2
𝑥
Bernoulli-number-B
2
superscript
𝑥
𝑎
𝑠
2
𝑥
{\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{a^{s}}\left(\frac{1%
}{2}+\frac{a}{s-1}\right)-\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}%
_{2}\left(x\right)-B_{2}}{(x+a)^{s+2}}\mathrm{d}x,}}
ℜ
s
>
-
1
𝑠
1
{\displaystyle{\displaystyle\Re s>-1}}
ℜ
s
>
-
1
𝑠
1
{\displaystyle{\displaystyle\Re s>-1}}
a
>
0
𝑎
0
{\displaystyle{\displaystyle a>0}}
B
n
Bernoulli-number-B
𝑛
{\displaystyle{\displaystyle B_{\NVar{n}}}}
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}}}
∫
{\displaystyle{\displaystyle\int}}
B
~
n
(
x
)
periodic-Bernoulli-polynomial-B
𝑛
𝑥
{\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
x
𝑥
{\displaystyle{\displaystyle x}}
a
𝑎
{\displaystyle{\displaystyle a}}
s
𝑠
{\displaystyle{\displaystyle s}}
Identifiers
