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