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| | Electrical Engineer Kull from Vegreville, has numerous passions that include crosswords, property developers in singapore and aquariums. Has travelled since childhood and has visited many locales, like Tino and Tinetto).<br><br>Here is my homepage ... [http://yourmy.org/profile_info.php?ID=57402&my http://yourmy.org/] |
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| The '''Volkenborn-integral''' is an [[integral]] for p-adic functions.
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| == Definition ==
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| Suppose
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| :<math>f:\Z_p\rightarrow \Bbb Q_p</math>
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| 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
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| :<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>
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| More generally, if
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| :<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>
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| then
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| :<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>
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| This integral was defined by Arnt Volkenborn.
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| == Examples ==
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| :<math> \int_{\Bbb Z_p} 1 \, {\rm d}x = 1 </math>
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| :<math> \int_{\Bbb Z_p} x \, {\rm d}x = -\frac{1}{2} </math>
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| :<math> \int_{\Bbb Z_p} x^2 \, {\rm d}x = \frac{1}{6} </math>
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| :<math> \int_{\Bbb Z_p} x^k \, {\rm d}x = B_k </math> , the k-th [[Bernoulli number]]
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| The above four examples can be easily checked by direct use of the definition and [[Faulhaber's formula]].
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| :<math> \int_{\Bbb Z_p} {x \choose k} \, {\rm d}x = \frac{(-1)^k}{k+1} </math>
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| :<math> \int_{\Bbb Z_p} (1 + a)^x \, {\rm d}x = \frac{\log(1+a)}{a} </math>
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| :<math> \int_{\Bbb Z_p} e^{a x} \, {\rm d}x = \frac{a}{e^a-1} </math>
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| The last two examples can be formally checked by expanding in the [[Taylor series]] and integrating term-wise.
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| :<math> \int_{\Bbb Z_p} \log_p(x+u) \, {\rm d}u = \psi_p(x) </math>
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| with <math> \log_p </math> the p-adic logarithmic function and
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| <math> \psi_p </math> the p-adic [[digamma]] function
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| == Properties ==
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| :<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>
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| From this it follows that the Volkenborn-integral is not translation invariant.
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| If <math> P^t = p^t \Bbb Z_p</math> then
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| :<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>
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| ==See also==
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| * [[P-adic distribution]]
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| == References ==
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| * Arnt Volkenborn: ''Ein p-adisches Integral und seine Anwendungen I.'' In: ''Manuscripta Mathematica.'' Bd. 7, Nr. 4, 1972, [http://eudml.org/doc/154126]
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| * Arnt Volkenborn: ''Ein p-adisches Integral und seine Anwendungen II.'' In: ''Manuscripta Mathematica.'' Bd. 12, Nr. 1, 1974, [http://eudml.org/doc/154225]
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| * Henri Cohen, "Number Theory", Volume II, page 276
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| [[Category:Integrals]]
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| {{analysis-stub}}
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Electrical Engineer Kull from Vegreville, has numerous passions that include crosswords, property developers in singapore and aquariums. Has travelled since childhood and has visited many locales, like Tino and Tinetto).
Here is my homepage ... http://yourmy.org/