en>Sgiitm: rewording

2014-07-31T02:54:20Z

rewording

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	~~The '''Volkenborn-integral''' is an [[integral]] for p-adic functions.~~

	~~== Definition ==~~
	~~Suppose~~
	~~:<math>f:\Z_p\rightarrow \Bbb Q_p</math>~~
	~~is a function~~ from ~~the [[P-adic number\|p-adic]] integers to the p-adic rationals~~, ~~then, under certain conditions, the Volkenborn-Integral is defined by~~
	~~:<math> \int_{\Bbb Z_p} f(x) \, {\rm d}x = \lim_{n \to \infty} \frac{1}{p^n} \sum_{x=0}^{p^n-1} f(x). </math>~~

	~~More generally, if~~
	~~:<math> R_n = \left\{ x = \sum_{i=r}^{n-1} b_i x^i \| b_i=0, \ldots~~, ~~p-1 \text{ for } r<n \right\} </math>~~
	~~then~~
	~~:<math> \int_K f(x) \, {\rm d}x = \lim_{n \to \infty} \frac{1}{p^n} \sum_{x \~~in ~~R_n \cap K} f(x). </math>~~

	~~This integral was defined by Arnt Volkenborn.~~

	~~== Examples ==~~
	~~:<math> \int_{\Bbb Z_p} 1 \, {\rm d}x = 1 </math>~~
	~~:<math> \int_{\Bbb Z_p} x \, {\rm d}x = -\frac{1}{2} </math>~~
	~~:<math> \int_{\Bbb Z_p} x^2 \, {\rm d}x = \frac{1}{6} </math>~~
	~~:<math> \int_{\Bbb Z_p} x^k \, {\rm d}x = B_k </math> , the k-th [[Bernoulli number]]~~

	~~The above four examples can be easily checked by direct use of the definition~~ and ~~[[Faulhaber's formula]]~~.
	~~:<math> \int_{\Bbb Z_p} {x \choose k} \, {\rm d}x = \frac{(-1)^k}{k+1} </math>~~
	~~:<math> \int_{\Bbb Z_p} (1 + a)^x \, {\rm d}x = \frac{\log(1+a)}{a} </math>~~
	~~:<math> \int_{\Bbb Z_p} e^{a x} \, {\rm d}x = \frac{a}{e^a-1} </math>~~
	~~The last two examples can be formally checked by expanding in the [[Taylor series]]~~ and ~~integrating term-wise.~~

	~~:<math> \int_{\Bbb Z_p} \log_p(x+u) \~~, ~~{\rm d}u = \psi_p(x) </math>~~
	~~with <math> \log_p </math> the p-adic logarithmic function~~ and
	~~<math> \psi_p </math> the p-adic [[digamma]] function~~


	~~== Properties ==~~
	~~:<math> \int_{\Bbb Z_p} f(x+m) \, {\rm d}x = \int_{\Bbb Z_p} f(x) \, {\rm d}x+ \sum_{x=0}^{m-1} f'(x~~)~~</math>~~
	~~From this it follows that the Volkenborn-integral is not translation invariant~~.

	If <~~math~~> ~~P^t = p^t \Bbb Z_p~~<~~/math~~> ~~then~~
	~~:<math> \int_{P^t} f(x) \, {\rm d}x = \frac{1}{p^t} \int_{\Bbb Z_p} f(p^t x) \, {\rm d}x</math>~~


	~~==See also==~~
	* [[P-adic distribution]]

	~~== References ==~~
	* Arnt Volkenborn: ''Ein p-adisches Integral und seine Anwendungen I.'' In: ''Manuscripta Mathematica.~~'' Bd~~. ~~7, Nr~~. ~~4, 1972,~~ [http://~~eudml~~.org/~~doc/154126]~~
	* Arnt Volkenborn: ''Ein p-adisches Integral und seine Anwendungen II.~~'' In: ''Manuscripta Mathematica.'' Bd. 12, Nr. 1, 1974, [~~http://~~eudml~~.org/~~doc/154225]~~
	* Henri Cohen, "Number Theory", Volume II, page 276

	~~[[Category:Integrals]~~]


	~~{{analysis-stub}}~~

en>Sgiitm: /* Equivalence of the two representations */

2013-12-22T00:06:43Z

Equivalence of the two representations

New page

{{multiple issues|
{{Orphan|date=December 2013}}
{{context|date=December 2013}}
}}

The '''Volkenborn-integral''' is an [[integral]] for p-adic functions.

== Definition ==
Suppose
:<math>f:\Z_p\rightarrow \Bbb Q_p</math>
is a function from the [[P-adic number|p-adic]] integers to the p-adic rationals, then, under certain conditions, the Volkenborn-Integral is defined by
:<math> \int_{\Bbb Z_p} f(x) \, {\rm d}x = \lim_{n \to \infty} \frac{1}{p^n} \sum_{x=0}^{p^n-1} f(x). </math>

More generally, if
:<math> R_n = \left\{ x = \sum_{i=r}^{n-1} b_i x^i | b_i=0, \ldots, p-1 \text{ for } r<n \right\} </math>
then
:<math> \int_K f(x) \, {\rm d}x = \lim_{n \to \infty} \frac{1}{p^n} \sum_{x \in R_n \cap K} f(x). </math>

This integral was defined by Arnt Volkenborn.

== Examples ==
:<math> \int_{\Bbb Z_p} 1 \, {\rm d}x = 1 </math>
:<math> \int_{\Bbb Z_p} x \, {\rm d}x = -\frac{1}{2} </math>
:<math> \int_{\Bbb Z_p} x^2 \, {\rm d}x = \frac{1}{6} </math>
:<math> \int_{\Bbb Z_p} x^k \, {\rm d}x = B_k </math> , the k-th [[Bernoulli number]]

The above four examples can be easily checked by direct use of the definition and [[Faulhaber's formula]].
:<math> \int_{\Bbb Z_p} {x \choose k} \, {\rm d}x = \frac{(-1)^k}{k+1} </math>
:<math> \int_{\Bbb Z_p} (1 + a)^x \, {\rm d}x = \frac{\log(1+a)}{a} </math>
:<math> \int_{\Bbb Z_p} e^{a x} \, {\rm d}x = \frac{a}{e^a-1} </math>
The last two examples can be formally checked by expanding in the [[Taylor series]] and integrating term-wise.

:<math> \int_{\Bbb Z_p} \log_p(x+u) \, {\rm d}u = \psi_p(x) </math>
with <math> \log_p </math> the p-adic logarithmic function and
<math> \psi_p </math> the p-adic [[digamma]] function

== Properties ==
:<math> \int_{\Bbb Z_p} f(x+m) \, {\rm d}x = \int_{\Bbb Z_p} f(x) \, {\rm d}x+ \sum_{x=0}^{m-1} f'(x)</math>
From this it follows that the Volkenborn-integral is not translation invariant.

If <math> P^t = p^t \Bbb Z_p</math> then
:<math> \int_{P^t} f(x) \, {\rm d}x = \frac{1}{p^t} \int_{\Bbb Z_p} f(p^t x) \, {\rm d}x</math>

==See also==
* [[P-adic distribution]]

== References ==
* Arnt Volkenborn: ''Ein p-adisches Integral und seine Anwendungen I.'' In: ''Manuscripta Mathematica.'' Bd. 7, Nr. 4, 1972, [http://eudml.org/doc/154126]
* Arnt Volkenborn: ''Ein p-adisches Integral und seine Anwendungen II.'' In: ''Manuscripta Mathematica.'' Bd. 12, Nr. 1, 1974, [http://eudml.org/doc/154225]
* Henri Cohen, "Number Theory", Volume II, page 276

[[Category:Integrals]]

{{analysis-stub}}

Solid partition - Revision history

en>Sgiitm: rewording

en>Sgiitm: /* Equivalence of the two representations */