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Statements
F
(
x
,
s
)
=
Γ
(
1
-
s
)
(
2
π
)
1
-
s
(
e
π
i
(
1
-
s
)
/
2
ζ
(
1
-
s
,
x
)
+
e
π
i
(
s
-
1
)
/
2
ζ
(
1
-
s
,
1
-
x
)
)
,
periodic-zeta
𝑥
𝑠
Euler-Gamma
1
𝑠
superscript
2
𝜋
1
𝑠
superscript
𝑒
𝜋
𝑖
1
𝑠
2
Hurwitz-zeta
1
𝑠
𝑥
superscript
𝑒
𝜋
𝑖
𝑠
1
2
Hurwitz-zeta
1
𝑠
1
𝑥
{\displaystyle{\displaystyle F\left(x,s\right)=\frac{\Gamma\left(1-s\right)}{(%
2\pi)^{1-s}}\*\left(e^{\pi i(1-s)/2}\zeta\left(1-s,x\right)+e^{\pi i(s-1)/2}%
\zeta\left(1-s,1-x\right)\right),}}
ℜ
s
>
1
𝑠
1
{\displaystyle{\displaystyle\Re s>1}}
0
<
x
<
1
0
𝑥
1
{\displaystyle{\displaystyle 0<x<1}}
ℜ
s
>
1
𝑠
1
{\displaystyle{\displaystyle\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)}}
π
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
F
(
x
,
s
)
periodic-zeta
𝑥
𝑠
{\displaystyle{\displaystyle F\left(\NVar{x},\NVar{s}\right)}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
x
𝑥
{\displaystyle{\displaystyle x}}
s
𝑠
{\displaystyle{\displaystyle s}}
Identifiers
