Martin Huxley - Revision history

121.241.173.50 at 07:27, 1 December 2014

2014-12-01T07:27:21Z

← Older revision		Revision as of 09:27, 1 December 2014
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	~~[[Image:Complex Riemann Xi.jpg\|right\|thumb\|300px\|Riemann xi function <math> \xi(s) </math> in the [[complex plane]].~~ The ~~color of a point <math> s </math> encodes the value~~ of the ~~function~~. ~~Darker colors denote values closer to zero and hue encodes the value~~'s ~~[[complex number\|argument]].]]~~		The title of the author is Jayson. For years she's been operating as a travel agent. To climb is some thing she would by no means give up. Ohio is exactly where his home is and his family enjoys it.<br><br>my weblog online psychic reading ([http://modenpeople.co.kr/modn/qna/292291 simply click the up coming document])

	~~In [[mathematics]], the '''Riemann Xi function''' is a variant of the [[Riemann zeta function]], and is defined so~~ as ~~to have~~ a ~~particularly simple [[functional equation]]~~. ~~The function~~ is ~~named in honour of [[Bernhard Riemann]].~~

	~~==Definition==~~
	~~Riemann's original lower-case xi-function, ξ, has been renamed with an upper-case Xi, Ξ,~~ by ~~[[Edmund Landau]] (see below)~~. ~~Landau's lower-case xi, ξ,~~ is ~~defined as:<ref>[[Edmund Landau]]~~. ~~Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig 1909. Third edition Chelsea, New York, 1974, §70.</ref>~~
	~~:<math>\xi(s) = \tfrac{1}{2} s(s-1) \pi^{-s/2} \Gamma\left(\tfrac{1}{2} s\right) \zeta(s)</math>~~
	~~for~~ <~~math~~>~~s\in\Bbb{C}~~<~~/math~~>~~. Here ζ~~(~~''s'') denotes the~~ [~~[Riemann zeta function]] and Γ(s) is the [[Gamma function]]. The functional equation (or [[reflection formula]]) for xi is~~
	:~~<math>\xi(1-s) = \xi(s).<~~/~~math>~~
	~~The upper-case Xi, Ξ, is defined by Landau (loc. cit., §71) as~~
	~~:<math>\Xi(z) = \xi(\frac12+zi)</math>~~
	~~and obeys the functional equation~~
	~~:<math>\Xi(-z) =\Xi(z).<~~/~~math>~~
	~~As reported by Landau (loc~~. ~~cit~~.~~, p. 894) this function Ξ is the function Riemann originally denoted by ξ.~~

	~~==Values==~~
	~~The general form for even integers is~~

	~~:<math>\xi(2n) = (-1)^{n+1}\frac{1}{(2n)!}B_{2n}2^{2n-1}\pi^{n}(2n^2-n)(n-1)!<~~/~~math>~~

	~~where ''B<sub>n<~~/~~sub>'' denotes the ''n''-th [[Bernoulli number]]. For example:~~

	~~:<math>\xi(2) = {\pi \over 6} </math>~~

	~~==Series representations==~~
	~~The xi function has the series expansion~~

	~~:<math>\frac{d}{dz} \ln \xi \left(\frac{-z}{1-z}\right) =~~
	~~\sum_{n=0}^\infty \lambda_{n+1} z^n<~~/~~math>~~

	~~This expansion plays a particularly important role in [[Li's criterion]], which states that~~ the ~~[[Riemann hypothesis]] is equivalent to having λ<sub>''n''</sub> > 0 for all positive ''n''.~~

	~~==Hadamard product==~~
	~~A simple [[infinite product~~]~~] expansion is~~

	~~:<math>\Xi(s~~) ~~= \frac{1}{2}\prod_\rho \left(1 - \frac{s}{\rho} \right).\!</math>~~

	~~To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeroes, i.e., the factors for a pair of zeroes of the form ρ and 1−ρ should be combined.~~

	~~==References==~~
	~~{{reflist}}~~

	~~==Further references==~~
	* {{mathworld\|urlname=Xi-Function\|title=Xi-Function}}
	* {{cite journal
	~~\|first1=J.B.~~
	~~\|last1=Keiper~~
	~~\|journal=Mathematics of Computation~~
	~~\|year=1992~~
	~~\|volume=58~~
	~~\|issue=198~~
	~~\|pages=765–773~~
	~~\|title=Power series expansions of Riemann's xi function~~
	~~\|doi=10.1090/S0025-5718-1992-1122072-5~~
	~~\|bibcode=1992MaCom..58..765K~~
	}}

	~~{{PlanetMath attribution\|id=3943\|title=Riemann Ξ function}}~~

	~~[[Category:Zeta and L-functions]]~~

en>Waacstats: Add persondata short description using AWB

2013-12-31T14:48:33Z

Add persondata short description using AWB

← Older revision		Revision as of 16:48, 31 December 2013
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	The ~~author~~'s ~~title~~ is ~~Christy~~. ~~For years she~~'s been ~~operating~~ as ~~a journey agent~~. I'~~ve usually loved living in Kentucky but now I~~'~~m considering other choices~~. The ~~preferred pastime~~ for ~~him~~ and ~~his kids~~ is ~~to play lacross and he~~'~~ll be beginning some thing else along with it~~.<br><br>~~my web~~-~~site~~ [~~http~~:/~~/www~~.~~prograd~~.~~uff~~.~~br/novo~~/~~facts~~-~~about~~-~~growing~~-~~greater~~-~~organic~~-~~garden free online tarot card readings~~]		[[Image:Complex Riemann Xi.jpg\|right\|thumb\|300px\|Riemann xi function <math> \xi(s) </math> in the [[complex plane]]. The color of a point <math> s </math> encodes the value of the function. Darker colors denote values closer to zero and hue encodes the value's [[complex number\|argument]].]]

			In [[mathematics]], the '''Riemann Xi function''' is a variant of the [[Riemann zeta function]], and is defined so as to have a particularly simple [[functional equation]]. The function is named in honour of [[Bernhard Riemann]].

			==Definition==
			Riemann's original lower-case xi-function, ξ, has been renamed with an upper-case Xi, Ξ, by [[Edmund Landau]] (see below). Landau's lower-case xi, ξ, is defined as:<ref>[[Edmund Landau]]. Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig 1909. Third edition Chelsea, New York, 1974, §70.</ref>
			:<math>\xi(s) = \tfrac{1}{2} s(s-1) \pi^{-s/2} \Gamma\left(\tfrac{1}{2} s\right) \zeta(s)</math>
			for <math>s\in\Bbb{C}</math>. Here ζ(''s'') denotes the [[Riemann zeta function]] and Γ(s) is the [[Gamma function]]. The functional equation (or [[reflection formula]]) for xi is
			:<math>\xi(1-s) = \xi(s).</math>
			The upper-case Xi, Ξ, is defined by Landau (loc. cit., §71) as
			:<math>\Xi(z) = \xi(\frac12+zi)</math>
			and obeys the functional equation
			:<math>\Xi(-z) =\Xi(z).</math>
			As reported by Landau (loc. cit., p. 894) this function Ξ is the function Riemann originally denoted by ξ.

			==Values==
			The general form for even integers is

			:<math>\xi(2n) = (-1)^{n+1}\frac{1}{(2n)!}B_{2n}2^{2n-1}\pi^{n}(2n^2-n)(n-1)!</math>

			where ''B<sub>n</sub>'' denotes the ''n''-th [[Bernoulli number]]. For example:

			:<math>\xi(2) = {\pi \over 6} </math>

			==Series representations==
			The xi function has the series expansion

			:<math>\frac{d}{dz} \ln \xi \left(\frac{-z}{1-z}\right) =
			\sum_{n=0}^\infty \lambda_{n+1} z^n</math>

			This expansion plays a particularly important role in [[Li's criterion]], which states that the [[Riemann hypothesis]] is equivalent to having λ<sub>''n''</sub> > 0 for all positive ''n''.

			==Hadamard product==
			A simple [[infinite product]] expansion is

			:<math>\Xi(s) = \frac{1}{2}\prod_\rho \left(1 - \frac{s}{\rho} \right).\!</math>

			To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeroes, i.e., the factors for a pair of zeroes of the form ρ and 1−ρ should be combined.

			==References==
			{{reflist}}

			==Further references==
			* {{mathworld\|urlname=Xi-Function\|title=Xi-Function}}
			* {{cite journal
			\|first1=J.B.
			\|last1=Keiper
			\|journal=Mathematics of Computation
			\|year=1992
			\|volume=58
			\|issue=198
			\|pages=765–773
			\|title=Power series expansions of Riemann's xi function
			\|doi=10.1090/S0025-5718-1992-1122072-5
			\|bibcode=1992MaCom..58..765K
			}}

			{{PlanetMath attribution\|id=3943\|title=Riemann Ξ function}}

			[[Category:Zeta and L-functions]]

en>Rjwilmsi: Journal cites (journal names):, using AWB (8060)

2012-05-20T12:01:00Z

Journal cites (journal names):, using AWB (8060)

New page

The author's title is Christy. For years she's been operating as a journey agent. I've usually loved living in Kentucky but now I'm considering other choices. The preferred pastime for him and his kids is to play lacross and he'll be beginning some thing else along with it.<br><br>my web-site [http://www.prograd.uff.br/novo/facts-about-growing-greater-organic-garden free online tarot card readings]