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Statements
∫
0
a
x
n
ψ
(
x
)
d
x
=
(
-
1
)
n
-
1
ζ
′
(
-
n
)
+
(
-
1
)
n
h
(
n
)
B
n
+
1
n
+
1
-
∑
k
=
0
n
(
-
1
)
k
(
n
k
)
h
(
k
)
B
k
+
1
(
a
)
k
+
1
a
n
-
k
+
∑
k
=
0
n
(
-
1
)
k
(
n
k
)
ζ
′
(
-
k
,
a
)
a
n
-
k
,
superscript
subscript
0
𝑎
superscript
𝑥
𝑛
digamma
𝑥
𝑥
superscript
1
𝑛
1
diffop
Riemann-zeta
1
𝑛
superscript
1
𝑛
ℎ
𝑛
Bernoulli-number-B
𝑛
1
𝑛
1
superscript
subscript
𝑘
0
𝑛
superscript
1
𝑘
binomial
𝑛
𝑘
ℎ
𝑘
Bernoulli-number-B
𝑘
1
𝑎
𝑘
1
superscript
𝑎
𝑛
𝑘
superscript
subscript
𝑘
0
𝑛
superscript
1
𝑘
binomial
𝑛
𝑘
diffop
Hurwitz-zeta
1
𝑘
𝑎
superscript
𝑎
𝑛
𝑘
{\displaystyle{\displaystyle\int_{0}^{a}x^{n}\psi\left(x\right)\mathrm{d}x=(-1%
)^{n-1}\zeta'\left(-n\right)+(-1)^{n}h(n)\frac{B_{n+1}}{n+1}-\sum_{k=0}^{n}(-1%
)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}h(k)\frac{B_{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}%
^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}\zeta'\left(-k,a\right)a^{n-k},}}
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}}}}
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)}}
(
m
n
)
binomial
𝑚
𝑛
{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
ψ
(
z
)
digamma
𝑧
{\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
k
𝑘
{\displaystyle{\displaystyle k}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
a
𝑎
{\displaystyle{\displaystyle a}}
h
(
n
)
ℎ
𝑛
{\displaystyle{\displaystyle h(n)}}
Identifiers
