Volkenborn integral - Alchetron, The Free Social Encyclopedia

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 → C 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:

∫ Z p f ( x ) d x = lim n → ∞ 1 p n ∑ x = 0 p n − 1 f ( x ) .

More generally, if

R n = { x = ∑ i = r n − 1 b i x i | b i = 0 , … , p − 1 for r < n }

then

∫ K f ( x ) d x = lim n → ∞ 1 p n ∑ x ∈ R n ∩ K f ( x ) .

This integral was defined by Arnt Volkenborn.

∫ Z p 1 d x = 1 ∫ Z p x d x = − 1 2 ∫ Z p x 2 d x = 1 6 ∫ Z p x k d x = B k , the k-th Bernoulli number

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

∫ Z p ( x k ) d x = ( − 1 ) k k + 1 ∫ Z p ( 1 + a ) x d x = log ⁡ ( 1 + a ) a ∫ Z p e a x d x = a e a − 1

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

∫ Z p log p ⁡ ( x + u ) d u = ψ p ( x )

with log p the p-adic logarithmic function and ψ p the p-adic digamma function

∫ Z p f ( x + m ) d x = ∫ Z p f ( x ) d x + ∑ x = 0 m − 1 f ′ ( x )

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

If P t = p t Z p then

∫ P t f ( x ) d x = 1 p t ∫ Z p f ( p t x ) d x

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