Volkenborn integral
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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 :[math]\displaystyle{ f:\Z_p\to \Complex_p }[/math] 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:
- [math]\displaystyle{ \int_{\Z_p} f(x) \, {\rm d}x = \lim_{n \to \infty} \frac{1}{p^n} \sum_{x=0}^{p^n-1} f(x). }[/math]
More generally, if
- [math]\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\lt n \right\} }[/math]
then
- [math]\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). }[/math]
This integral was defined by Arnt Volkenborn.
Examples
- [math]\displaystyle{ \int_{\Z_p} 1 \, {\rm d}x = 1 }[/math]
- [math]\displaystyle{ \int_{\Z_p} x \, {\rm d}x = -\frac{1}{2} }[/math]
- [math]\displaystyle{ \int_{\Z_p} x^2 \, {\rm d}x = \frac{1}{6} }[/math]
- [math]\displaystyle{ \int_{\Z_p} x^k \, {\rm d}x = B_k }[/math]
where [math]\displaystyle{ B_k }[/math] is the k-th Bernoulli number.
The above four examples can be easily checked by direct use of the definition and Faulhaber's formula.
- [math]\displaystyle{ \int_{\Z_p} {x \choose k} \, {\rm d}x = \frac{(-1)^k}{k+1} }[/math]
- [math]\displaystyle{ \int_{\Z_p} (1 + a)^x \, {\rm d}x = \frac{\log(1+a)}{a} }[/math]
- [math]\displaystyle{ \int_{\Z_p} e^{a x} \, {\rm d}x = \frac{a}{e^a-1} }[/math]
The last two examples can be formally checked by expanding in the Taylor series and integrating term-wise.
- [math]\displaystyle{ \int_{\Z_p} \log_p(x+u) \, {\rm d}u = \psi_p(x) }[/math]
with [math]\displaystyle{ \log_p }[/math] the p-adic logarithmic function and [math]\displaystyle{ \psi_p }[/math] the p-adic digamma function.
Properties
- [math]\displaystyle{ \int_{\Z_p} f(x+m) \, {\rm d}x = \int_{\Z_p} f(x) \, {\rm d}x+ \sum_{x=0}^{m-1} f'(x) }[/math]
From this it follows that the Volkenborn-integral is not translation invariant.
If [math]\displaystyle{ P^t = p^t \Z_p }[/math] then
- [math]\displaystyle{ \int_{P^t} f(x) \, {\rm d}x = \frac{1}{p^t} \int_{\Z_p} f(p^t x) \, {\rm d}x }[/math]
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
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