User:PetrusJohannes/sandbox

The Volkenborn-integral is an integral for p-adic functions.

Definition
Suppose
 * $$f:\Z_p\rightarrow \mathbb Q_p$$

is a function from the p-adic integers to the p-adic rationals, then, under certain conditions, the Volkenborn-Integral is defined by
 * $$ \int_{\mathbb Z_p} f(x) \, {\rm d}x = \lim_{n \to \infty} \frac{1}{p^n} \sum_{x=0}^{p^n-1} f(x). $$

Examples

 * $$ \int_{\mathbb Z_p} 1 \, {\rm d}x = 1 $$
 * $$ \int_{\mathbb Z_p} x \, {\rm d}x = -\frac{1}{2} $$
 * $$ \int_{\mathbb Z_p} x^2 \, {\rm d}x = \frac{1}{6} $$
 * $$ \int_{\mathbb Z_p} x^k \, {\rm d}x = B_k $$, the k-th Bernoulli number
 * $$ \int_{\mathbb Z_p} {x \choose k} \, {\rm d}x = \frac{(-1)^k}{k+1} $$
 * $$ \int_{\mathbb Z_p} (1 + a)^x \, {\rm d}x = \frac{\log(1+a)}{a} $$

Properties

 * $$ \int_{\mathbb Z_p} f(x+m) \, {\rm d}x = \int_{\mathbb Z_p} f(x) \, {\rm d}x+ \sum_{x=0}^{m-1} f'(x)$$