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In mathematics, in the field of p-adic analysis , the Volkenborn integral is a method of integration for p-adic functions.
Definition [ ] Let :
f
:
Z
p
→
C
p
{\displaystyle f:\mathbb {Z} _{p}\to \mathbb {C} _{p}}
p-adic integers taking values in the p-adic numbers. The Volkenborn integral is defined by the limit, if it exists:
∫
Z
p
f
(
x
)
d
x
=
lim
n
→
∞
1
p
n
∑
x
=
0
p
n
−
1
f
(
x
)
.
{\displaystyle \int _{\mathbb {Z} _{p}}f(x)\,{\rm {d}}x=\lim _{n\to \infty }{\frac {1}{p^{n}}}\sum _{x=0}^{p^{n}-1}f(x).}
More generally, if
R
n
=
{
x
=
∑
i
=
r
n
−
1
b
i
x
i
|
b
i
=
0
,
…
,
p
−
1
for
r
<
n
}
{\displaystyle R_{n}=\left\{\left.x=\sum _{i=r}^{n-1}b_{i}x^{i}\right|b_{i}=0,\ldots ,p-1{\text{ for }}r<n\right\}}
then
∫
K
f
(
x
)
d
x
=
lim
n
→
∞
1
p
n
∑
x
∈
R
n
∩
K
f
(
x
)
.
{\displaystyle \int _{K}f(x)\,{\rm {d}}x=\lim _{n\to \infty }{\frac {1}{p^{n}}}\sum _{x\in R_{n}\cap K}f(x).}
This integral was defined by Arnt Volkenborn.
Examples [ ]
∫
Z
p
1
d
x
=
1
{\displaystyle \int _{\mathbb {Z} _{p}}1\,{\rm {d}}x=1}
∫
Z
p
x
d
x
=
−
1
2
{\displaystyle \int _{\mathbb {Z} _{p}}x\,{\rm {d}}x=-{\frac {1}{2}}}
∫
Z
p
x
2
d
x
=
1
6
{\displaystyle \int _{\mathbb {Z} _{p}}x^{2}\,{\rm {d}}x={\frac {1}{6}}}
∫
Z
p
x
k
d
x
=
B
k
{\displaystyle \int _{\mathbb {Z} _{p}}x^{k}\,{\rm {d}}x=B_{k}}
where
B
k
{\displaystyle B_{k}}
Bernoulli number .
The above four examples can be easily checked by direct use of the definition and Faulhaber's formula .
∫
Z
p
(
x
k
)
d
x
=
(
−
1
)
k
k
+
1
{\displaystyle \int _{\mathbb {Z} _{p}}{x \choose k}\,{\rm {d}}x={\frac {(-1)^{k}}{k+1}}}
∫
Z
p
(
1
+
a
)
x
d
x
=
log
(
1
+
a
)
a
{\displaystyle \int _{\mathbb {Z} _{p}}(1+a)^{x}\,{\rm {d}}x={\frac {\log(1+a)}{a}}}
∫
Z
p
e
a
x
d
x
=
a
e
a
−
1
{\displaystyle \int _{\mathbb {Z} _{p}}e^{ax}\,{\rm {d}}x={\frac {a}{e^{a}-1}}}
The last two examples can be formally checked by expanding in the Taylor series and integrating term-wise.
∫
Z
p
log
p
(
x
+
u
)
d
u
=
ψ
p
(
x
)
{\displaystyle \int _{\mathbb {Z} _{p}}\log _{p}(x+u)\,{\rm {d}}u=\psi _{p}(x)}
with
log
p
{\displaystyle \log _{p}}
ψ
p
{\displaystyle \psi _{p}}
digamma function .
Properties [ ]
∫
Z
p
f
(
x
+
m
)
d
x
=
∫
Z
p
f
(
x
)
d
x
+
∑
x
=
0
m
−
1
f
′
(
x
)
{\displaystyle \int _{\mathbb {Z} _{p}}f(x+m)\,{\rm {d}}x=\int _{\mathbb {Z} _{p}}f(x)\,{\rm {d}}x+\sum _{x=0}^{m-1}f'(x)}
From this it follows that the Volkenborn-integral is not translation invariant.
If
P
t
=
p
t
Z
p
{\displaystyle P^{t}=p^{t}\mathbb {Z} _{p}}
∫
P
t
f
(
x
)
d
x
=
1
p
t
∫
Z
p
f
(
p
t
x
)
d
x
{\displaystyle \int _{P^{t}}f(x)\,{\rm {d}}x={\frac {1}{p^{t}}}\int _{\mathbb {Z} _{p}}f(p^{t}x)\,{\rm {d}}x}
See also [ ] References [ ] Arnt Volkenborn: Ein p-adisches Integral und seine Anwendungen I. In: Manuscripta Mathematica. Bd. 7, Nr. 4, 1972, [1]
Arnt Volkenborn: Ein p-adisches Integral und seine Anwendungen II. In: Manuscripta Mathematica. Bd. 12, Nr. 1, 1974, [2]
Henri Cohen, "Number Theory", Volume II, page 276 
