In mathematics, in the field of p-adic analysis, the Volkenborn integral is a method of integration for p-adic functions.
Definition[edit]
Let :
be a function from the p-adic integers taking values in the p-adic numbers. The Volkenborn integral is defined by the limit, if it exists:
![{\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52acb904ab766977aabb9bd84165fd3a8c6bb818)
More generally, if
![{\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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/409c1ca2f2eecf1bba9bb073a3d6cc6543796537)
then
![{\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4cd54fd096e3ca23da2c6e7683056a7a940cf4b)
This integral was defined by Arnt Volkenborn.
Examples[edit]
![{\displaystyle \int _{\mathbb {Z} _{p}}1\,{\rm {d}}x=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa477f7dda6d295df74ebac824929d2d6b568173)
![{\displaystyle \int _{\mathbb {Z} _{p}}x\,{\rm {d}}x=-{\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a16a2520f0841c025127969cc80911acbf7177)
![{\displaystyle \int _{\mathbb {Z} _{p}}x^{2}\,{\rm {d}}x={\frac {1}{6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af9e1d4923171a5fe1ac572f18584cec6707ff5b)
![{\displaystyle \int _{\mathbb {Z} _{p}}x^{k}\,{\rm {d}}x=B_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/130c16eab8cfa9dd2d9d3bab94b32edfe04bca60)
where
is the k-th Bernoulli number.
The above four examples can be easily checked by direct use of the definition and Faulhaber's formula.
![{\displaystyle \int _{\mathbb {Z} _{p}}{x \choose k}\,{\rm {d}}x={\frac {(-1)^{k}}{k+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc384f008360f9a023ad2dd1b0e38cd140c864e4)
![{\displaystyle \int _{\mathbb {Z} _{p}}(1+a)^{x}\,{\rm {d}}x={\frac {\log(1+a)}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98729e6f63411d1dec849f8ebfaf7a852ef057c4)
![{\displaystyle \int _{\mathbb {Z} _{p}}e^{ax}\,{\rm {d}}x={\frac {a}{e^{a}-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb8248669db8be7fba55e0d96af71af1a24894ed)
The last two examples can be formally checked by expanding in the Taylor series and integrating term-wise.
![{\displaystyle \int _{\mathbb {Z} _{p}}\log _{p}(x+u)\,{\rm {d}}u=\psi _{p}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6bf7cf3be900df83c16bc4b516d745133c0a0d2)
with
the p-adic logarithmic function and
the p-adic digamma function.
Properties[edit]
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b035103062448e8b06f2ae1717d78542113120d9)
From this it follows that the Volkenborn-integral is not translation invariant.
If
then
![{\displaystyle \int _{P^{t}}f(x)\,{\rm {d}}x={\frac {1}{p^{t}}}\int _{\mathbb {Z} _{p}}f(p^{t}x)\,{\rm {d}}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/541d2993c84ae3b46183d8c7301d099e7f73253c)
See also[edit]
References[edit]
- 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