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 :f:\Z_p\to \Complex_p be a function from the
p-adic In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensi ...
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 In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
. 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 In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
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 In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac. It is the first of the polygamma functions. It is strictly increasing and strict ...
.


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 {{analysis-stub