Wave function renormalization - Revision history

70.247.169.157: link

2014-05-20T02:20:37Z

link

← Older revision		Revision as of 04:20, 20 May 2014
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	~~In [[mathematics]], the '''Nörlund–Rice integral''', sometimes called '''Rice's method''', relates the ''n''th [[forward difference]] of a function to a [[line integral]] on the [[complex plane]]~~. ~~As such, it commonly appears in the theory of [[finite differences]],~~ and ~~also has been applied in [[computer science]] and [[graph theory]]~~ to ~~estimate [[binary tree]] lengths. It is named in honour of [[Niels Erik Nørlund]]~~ and ~~[[Stephen O. Rice]]. Nørlund~~'~~s contribution was~~ to ~~define the integral; Rice's contribution was to demonstrate its utility by applying [[Method of steepest descent\|saddle-point technique]]s to its evaluation.~~		Greetings! I am Myrtle Shroyer. The favorite hobby for my children and me is to perform baseball and I'm trying to make it a occupation. Since she was 18 she's been working as a receptionist but her promotion by no means comes. For a while I've been in South Dakota and my mothers and fathers reside nearby.<br><br>My webpage - [http://xrambo.com/blog/192034 at home std test]

	~~==Definition==~~
	~~The ''n''th [[forward difference]] of~~ a ~~function ''f''(''x'') is given by~~

	~~:<math>\Delta^n[f](x)= \sum_{k=0}^n {n \choose k} (-1)^{n-k} f(x+k)</math>~~

	~~where <math>{n \choose k}</math> is the [[binomial coefficient]]~~.

	~~The Nörlund–Rice integral is given by~~

	~~:<math>\sum_{k=\alpha}^n {n \choose k} (-1)^{n-k} f(k) =~~
	~~\frac{n!}{2\pi i}~~
	~~\oint_\gamma \frac{f(z)}{z(z-1)(z-2)\cdots(z-n)}\, \mathrm{d}z</math>~~

	~~where~~ ''f'' is understood to be [[meromorphic]], α is an integer, <math>0\leq \alpha \leq n</math>, and the contour of integration is understood to circle the [[pole (complex analysis)\|poles]] located at the integers α, ..., ''n'', but none of the poles of ''f''. The integral may also be written as

	~~:<math>\sum_{k=\alpha}^n {n \choose k} (-1)^{k} f(k) =~~
	~~-\frac{1}{2\pi i}~~
	~~\oint_\gamma B(n+1, -z) f(z)\, \mathrm{d}z</math>~~

	~~where ''B''(''~~a'',''b'') is the Euler [[beta function]]. If the function <math>f(z)</math> is [[polynomially bounded]] on the right hand side of the complex plane, then the contour may be extended to infinity on the right hand side, allowing the transform to be written as

	~~:<math>\sum_{k=\alpha}^n {n \choose k} (-1)^{n-k} f(k) =~~
	~~\frac{-n!}{2\pi i}~~
	~~\int_{c-i\infty}^{c+i\infty} \frac{f(z)}{z(z-1)(z-2)\cdots(z-n)}\, \mathrm{d}z</math>~~

	~~where the constant ''c'' is to the left of α.~~

	~~==Poisson–Mellin–Newton cycle==~~
	~~The Poisson–Mellin–Newton cycle, noted by Flajolet et al. in 1985, is the observation that the resemblance of the Nørlund–Rice integral to the [[Mellin transform]] is not accidental,~~ but ~~is related~~ by means ~~of the [[binomial transform]] and the [[Newton series]]~~. ~~In this cycle, let <math>\{f_n\}</math> be~~ a ~~[[sequence]],~~ and ~~let ''g''(''t'') be the corresponding [[Poisson generating function]], that is, let~~

	~~:<math>g(t) = e^{-t} \sum_{n=0}^\infty f_n t^n~~.<~~/math~~>

	~~Taking its Mellin transform~~

	:<~~math~~>~~\phi(s)=\int_0^\infty g(t) t^{s~~-~~1}\, \mathrm{d}t,</math>~~

	~~one can then regain the original sequence by means of the Nörlund–Rice integral:~~

	~~:<math>f_n = \frac{(-1)^n }{2\pi i}~~
	~~\int_\gamma~~
	~~\frac {\phi(s)}{\Gamma(-s)} \frac{n!}{s(s-1)\cdots (s-n)}\, \mathrm{d}s</math>~~

	~~where Γ is the [[gamma function]].~~

	~~==Riesz mean==~~
	A closely related integral frequently occurs in the discussion of [[Riesz mean]]s. Very roughly, it can be said to be related to the Nörlund–Rice integral in the same way that [[Perron's formula]] is related to the Mellin transform: rather than dealing with infinite series, it deals with finite series.

	~~==Utility==~~
	The integral representation for these types of series is interesting because the integral can often be evaluated using [[asymptotic expansion]] or [[Method of steepest descent\|saddle-point]] techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large ''n''.

	~~==See also==~~
	* [[Table of Newtonian series]]
	* [[List of factorial and binomial topics]]

	~~==References==~~
	* Niels Erik Nørlund, ''Vorlesungen uber Differenzenrechnung'', (1954) Chelsea Publishing Company, New York.
	* Donald E. Knuth, ''The Art of Computer Programming'', (1973), Vol. 3 Addison-Wesley.
	* Philippe Flajolet and Robert Sedgewick, "[http://~~www-rocq.inria~~.fr/~~algo~~/~~flajolet/Publications/FlSe95.pdf Mellin transforms and asymptotics: Finite differences and Rice's integrals]", ''Theoretical Computer Science'' '''144''' (1995) pp 101–124.~~
	* Peter Kirschenhofer, "[http://www.combinatorics.org/Volume_3/volume3_2.html#R7 A Note on Alternating Sums]", ''[http://www.combinatorics.org The Electronic Journal of Combinatorics]'', Volume '''3''' (1996) Issue 2 Article 7.

	~~{{DEFAULTSORT:Norlund-Rice integral}}~~
	~~[[Category:Factorial and binomial topics]]~~
	~~[[Category:Complex analysis]]~~
	~~[[Category:Integral transforms]]~~
	~~[[Category:Finite differences]~~]

en>Mmitchell10: added link

2012-11-02T11:32:04Z

added link

← Older revision		Revision as of 13:32, 2 November 2012
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	~~I would like~~ to ~~introduce myself~~ to ~~you, I am Andrew~~ and ~~my spouse doesn~~'~~t like it at all~~. ~~She~~ is ~~really fond~~ of ~~caving~~ but ~~she doesn~~'~~t have~~ the ~~time recently~~. He is ~~an information officer~~. ~~I've always loved living~~ in ~~Alaska~~.<br><br>~~Have~~ a ~~look at my blog~~: ~~spirit messages~~ - [http://~~brazil~~.~~amor-amore~~.~~com~~/~~irboothe Full Content~~],		In [[mathematics]], the '''Nörlund–Rice integral''', sometimes called '''Rice's method''', relates the ''n''th [[forward difference]] of a function to a [[line integral]] on the [[complex plane]]. As such, it commonly appears in the theory of [[finite differences]], and also has been applied in [[computer science]] and [[graph theory]] to estimate [[binary tree]] lengths. It is named in honour of [[Niels Erik Nørlund]] and [[Stephen O. Rice]]. Nørlund's contribution was to define the integral; Rice's contribution was to demonstrate its utility by applying [[Method of steepest descent\|saddle-point technique]]s to its evaluation.

			==Definition==
			The ''n''th [[forward difference]] of a function ''f''(''x'') is given by

			:<math>\Delta^n[f](x)= \sum_{k=0}^n {n \choose k} (-1)^{n-k} f(x+k)</math>

			where <math>{n \choose k}</math> is the [[binomial coefficient]].

			The Nörlund–Rice integral is given by

			:<math>\sum_{k=\alpha}^n {n \choose k} (-1)^{n-k} f(k) =
			\frac{n!}{2\pi i}
			\oint_\gamma \frac{f(z)}{z(z-1)(z-2)\cdots(z-n)}\, \mathrm{d}z</math>

			where ''f'' is understood to be [[meromorphic]], α is an integer, <math>0\leq \alpha \leq n</math>, and the contour of integration is understood to circle the [[pole (complex analysis)\|poles]] located at the integers α, ..., ''n'', but none of the poles of ''f''. The integral may also be written as

			:<math>\sum_{k=\alpha}^n {n \choose k} (-1)^{k} f(k) =
			-\frac{1}{2\pi i}
			\oint_\gamma B(n+1, -z) f(z)\, \mathrm{d}z</math>

			where ''B''(''a'',''b'') is the Euler [[beta function]]. If the function <math>f(z)</math> is [[polynomially bounded]] on the right hand side of the complex plane, then the contour may be extended to infinity on the right hand side, allowing the transform to be written as

			:<math>\sum_{k=\alpha}^n {n \choose k} (-1)^{n-k} f(k) =
			\frac{-n!}{2\pi i}
			\int_{c-i\infty}^{c+i\infty} \frac{f(z)}{z(z-1)(z-2)\cdots(z-n)}\, \mathrm{d}z</math>

			where the constant ''c'' is to the left of α.

			==Poisson–Mellin–Newton cycle==
			The Poisson–Mellin–Newton cycle, noted by Flajolet et al. in 1985, is the observation that the resemblance of the Nørlund–Rice integral to the [[Mellin transform]] is not accidental, but is related by means of the [[binomial transform]] and the [[Newton series]]. In this cycle, let <math>\{f_n\}</math> be a [[sequence]], and let ''g''(''t'') be the corresponding [[Poisson generating function]], that is, let

			:<math>g(t) = e^{-t} \sum_{n=0}^\infty f_n t^n.</math>

			Taking its Mellin transform

			:<math>\phi(s)=\int_0^\infty g(t) t^{s-1}\, \mathrm{d}t,</math>

			one can then regain the original sequence by means of the Nörlund–Rice integral:

			:<math>f_n = \frac{(-1)^n }{2\pi i}
			\int_\gamma
			\frac {\phi(s)}{\Gamma(-s)} \frac{n!}{s(s-1)\cdots (s-n)}\, \mathrm{d}s</math>

			where Γ is the [[gamma function]].

			==Riesz mean==
			A closely related integral frequently occurs in the discussion of [[Riesz mean]]s. Very roughly, it can be said to be related to the Nörlund–Rice integral in the same way that [[Perron's formula]] is related to the Mellin transform: rather than dealing with infinite series, it deals with finite series.

			==Utility==
			The integral representation for these types of series is interesting because the integral can often be evaluated using [[asymptotic expansion]] or [[Method of steepest descent\|saddle-point]] techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large ''n''.

			==See also==
			* [[Table of Newtonian series]]
			* [[List of factorial and binomial topics]]

			==References==
			* Niels Erik Nørlund, ''Vorlesungen uber Differenzenrechnung'', (1954) Chelsea Publishing Company, New York.
			* Donald E. Knuth, ''The Art of Computer Programming'', (1973), Vol. 3 Addison-Wesley.
			* Philippe Flajolet and Robert Sedgewick, "[http://www-rocq.inria.fr/algo/flajolet/Publications/FlSe95.pdf Mellin transforms and asymptotics: Finite differences and Rice's integrals]", ''Theoretical Computer Science'' '''144''' (1995) pp 101–124.
			* Peter Kirschenhofer, "[http://www.combinatorics.org/Volume_3/volume3_2.html#R7 A Note on Alternating Sums]", ''[http://www.combinatorics.org The Electronic Journal of Combinatorics]'', Volume '''3''' (1996) Issue 2 Article 7.

			{{DEFAULTSORT:Norlund-Rice integral}}
			[[Category:Factorial and binomial topics]]
			[[Category:Complex analysis]]
			[[Category:Integral transforms]]
			[[Category:Finite differences]]

en>SmackBot: /* See also */remove Erik9bot category,outdated, tag and general fixes

2009-12-17T02:28:52Z

See also: remove Erik9bot category,outdated, tag and general fixes

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I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. She is really fond of caving but she doesn't have the time recently. He is an information officer. I've always loved living in Alaska.<br><br>Have a look at my blog: spirit messages - [http://brazil.amor-amore.com/irboothe Full Content],