Volkenborn integral

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:

:$\int_ f(x) \, x = \lim_ \frac \sum_^ f(x).$

More generally, if

:$R_n = \left\$

then

:$\int_K f(x) \, x = \lim_ \frac \sum_ f(x).$

This integral was defined by Arnt Volkenborn.

Examples

:$\int_ 1 \, x = 1$
:$\int_ x \, x = -\frac$
:$\int_ x^2 \, x = \frac$
:$\int_ x^k \, x = B_k$

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.

:$\int_ \, x = \frac$
:$\int_ (1 + a)^x \, x = \frac$
:$\int_ e^ \, x = \frac$

The last two examples can be formally checked by expanding in the Taylor series and integrating term-wise.

:$\int_ \log_p(x+u) \, u = \psi_p(x)$

with $\log_p$ the p-adic logarithmic function and $\psi_p$ the p-adic digamma function.

Properties

:$\int_ f(x+m) \, x = \int_ f(x) \, x+ \sum_^ f'(x)$

From this it follows that the Volkenborn-integral is not translation invariant.

If $P^t = p^t \Z_p$ then

:$\int_ f(x) \, x = \frac \int_ f(p^t x) \, x$

See also

* P-adic distribution

References

* Arnt Volkenborn: ''Ein p-adisches Integral und seine Anwendungen I.'' In: ''Manuscripta Mathematica.'' Bd. 7, Nr. 4, 1972

* Arnt Volkenborn: ''Ein p-adisches Integral und seine Anwendungen II.'' In: ''Manuscripta Mathematica.'' Bd. 12, Nr. 1, 1974


* Henri Cohen, "Number Theory", Volume II, page 276

Integrals

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