In mathematics, in the field of

p-adic analysis In mathematics, ''p''-adic analysis is a branch of number theory that studies functions of ''p''-adic numbers. Along with the more classical fields of real and complex analysis, which deal, respectively, with functions on the real and complex ...

, the Volkenborn integral is a method of

integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...

for p-adic functions.

Definition

Let :

f:\Z_p\to \Complex_p

be a function from the

p-adic In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infin ...

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 function ...

. The above four examples can be easily checked by direct use of the definition and

Faulhaber's formula In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the ''p''-th powers of the first ''n'' positive integers \sum_^ k^p = 1^p + 2^p + 3^p + \cdots + n^p as a polynomial in&n ...

. :

\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(z) = \frac\ln\Gamma(z) = \frac. It is the first of the polygamma functions. This function is Monotonic function, strictly increasing a ...

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

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 University of Cologne () is a university in Cologne, Germany. It was established in 1388. It closed in 1798 before being re-established in 1919. It is now one of the largest universities in Germany with around 45,187 students. The Universit ...

. The Volkenborn integral allows integration of all locally analytic functions, such as

Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ...

. It is used in the computation of the generalized p-adic Bernoulli numbers (like in the examples above) and other p-adic functions.

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

Definition

Examples

Properties

Origin

See also

References