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Statements
n
∫
0
a
ζ
′
(
1
-
n
,
x
)
d
x
=
ζ
′
(
-
n
,
a
)
-
ζ
′
(
-
n
)
+
B
n
+
1
-
B
n
+
1
(
a
)
n
(
n
+
1
)
,
𝑛
superscript
subscript
0
𝑎
diffop
Hurwitz-zeta
1
1
𝑛
𝑥
𝑥
diffop
Hurwitz-zeta
1
𝑛
𝑎
diffop
Riemann-zeta
1
𝑛
Bernoulli-number-B
𝑛
1
Bernoulli-polynomial-B
𝑛
1
𝑎
𝑛
𝑛
1
{\displaystyle{\displaystyle n\int_{0}^{a}\zeta'\left(1-n,x\right)\mathrm{d}x=%
\zeta'\left(-n,a\right)-\zeta'\left(-n\right)+\frac{B_{n+1}-B_{n+1}\left(a%
\right)}{n(n+1)},}}
ℜ
a
>
0
𝑎
0
{\displaystyle{\displaystyle\Re a>0}}
n
=
1
,
2
,
…
𝑛
1
2
…
{\displaystyle{\displaystyle n=1,2,\dots}}
ℜ
a
>
0
𝑎
0
{\displaystyle{\displaystyle\Re a>0}}
B
n
Bernoulli-number-B
𝑛
{\displaystyle{\displaystyle B_{\NVar{n}}}}
B
n
(
x
)
Bernoulli-polynomial-B
𝑛
𝑥
{\displaystyle{\displaystyle B_{\NVar{n}}\left(\NVar{x}\right)}}
ζ
(
s
,
a
)
Hurwitz-zeta
𝑠
𝑎
{\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
ζ
(
s
)
Riemann-zeta
𝑠
{\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
a
𝑎
{\displaystyle{\displaystyle a}}
Identifiers
