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Statements
Φ
(
z
,
s
,
a
)
=
1
2
a
-
s
+
∫
0
∞
z
x
(
a
+
x
)
s
d
x
-
2
∫
0
∞
sin
(
x
ln
z
-
s
arctan
(
x
/
a
)
)
(
a
2
+
x
2
)
s
/
2
(
e
2
π
x
-
1
)
d
x
,
Lerch-Phi
𝑧
𝑠
𝑎
1
2
superscript
𝑎
𝑠
superscript
subscript
0
superscript
𝑧
𝑥
superscript
𝑎
𝑥
𝑠
𝑥
2
superscript
subscript
0
𝑥
𝑧
𝑠
𝑥
𝑎
superscript
superscript
𝑎
2
superscript
𝑥
2
𝑠
2
superscript
𝑒
2
𝜋
𝑥
1
𝑥
{\displaystyle{\displaystyle\Phi\left(z,s,a\right)=\frac{1}{2}a^{-s}+\int_{0}^%
{\infty}\frac{z^{x}}{(a+x)^{s}}\mathrm{d}x-2\int_{0}^{\infty}\frac{\sin\left(x%
\ln z-s\operatorname{arctan}\left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2%
\pi x}-1)}\mathrm{d}x,}}
ℜ
s
>
0
𝑠
0
{\displaystyle{\displaystyle\Re s>0}}
ℜ
s
>
1
𝑠
1
{\displaystyle{\displaystyle\Re s>1}}
|
z
|
=
1
,
ℜ
a
>
0
formulae-sequence
𝑧
1
𝑎
0
{\displaystyle{\displaystyle|z|=1,\Re a>0}}
ℜ
s
>
0
𝑠
0
{\displaystyle{\displaystyle\Re s>0}}
|
z
|
<
1
𝑧
1
{\displaystyle{\displaystyle|z|<1}}
ℜ
s
>
1
𝑠
1
{\displaystyle{\displaystyle\Re s>1}}
|
z
|
=
1
,
ℜ
a
>
0
formulae-sequence
𝑧
1
𝑎
0
{\displaystyle{\displaystyle|z|=1,\Re a>0}}
Φ
(
z
,
s
,
a
)
Lerch-Phi
𝑧
𝑠
𝑎
{\displaystyle{\displaystyle\Phi\left(\NVar{z},\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}}}
ln
z
𝑧
{\displaystyle{\displaystyle\ln\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}}
z
𝑧
{\displaystyle{\displaystyle z}}
Identifiers
