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\to \Complex_p 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:
\int_{\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 = \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
\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
\int_{\Z_p} 1 \, {\rm d}x = 1\int_{\Z_p} x \, {\rm d}x = -\frac{1}{2}\int_{\Z_p} x^2 \, {\rm d}x = \frac{1}{6}\int_{\Z_p} x^k \, {\rm d}x = B_k
where B_k is the k-th Bernoulli number.
The above four examples can be easily checked by direct use of the definition and Faulhaber's formula.
\int_{\Z_p} {x \choose k} \, {\rm d}x = \frac{(-1)^k}{k+1}\int_{\Z_p} (1 + a)^x \, {\rm d}x = \frac{\log(1+a)}{a}\int_{\Z_p} e^{a x} \, {\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.
\int_{\Z_p} \log_p(x+u) \, {\rm d}u = \psi_p(x)
with \log_p the p-adic logarithmic function and \psi_p the p-adic digamma function.
Properties
\int_{\Z_p} f(x+m) \, {\rm d}x = \int_{\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 then
\int_{P^t} f(x) \, {\rm d}x = \frac{1}{p^t} \int_{\Z_p} f(p^t x) \, {\rm d}x
Origin
The idea of integrating p-adic functions was initially proposed by F. Thomas and François Bruhat. However, the definition of their translation-invariant p-adic integral proved too restrictive for analytical and number-theoretical purposes. Arnt Volkenborn developed the generalized p-adic integral, later named after him, in his 1971 dissertation at the University of Cologne. The Volkenborn integral allows integration of all locally analytic functions, such as Laurent series. It is used in the computation of the generalized p-adic Bernoulli numbers (like in the examples above) and other p-adic functions.