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In mathematics, in the field of [[p-adic analysis]], the '''Volkenborn integral''' is a method of [[integral|integration]] for p-adic functions. |
In mathematics, in the field of [[p-adic analysis]], the '''Volkenborn integral''' is a method of [[integral|integration]] for p-adic functions. |
Revision as of 16:57, 31 May 2020
This article provides insufficient context for those unfamiliar with the subject.(December 2013) |
In mathematics, in the field of p-adic analysis, the Volkenborn integral is a method of integration for p-adic functions.
Definition
Let : 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:
More generally, if
then
This integral was defined by Arnt Volkenborn.
Examples
where is the k-th Bernoulli number.
The above four examples can be easily checked by direct use of the definition and Faulhaber's formula.
The last two examples can be formally checked by expanding in the Taylor series and integrating term-wise.
with the p-adic logarithmic function and the p-adic digamma function
Properties
From this it follows that the Volkenborn-integral is not translation invariant.
If then