Fermi acceleration - Revision history

en>Tom.Reding: Gen fixes (page/s, endash, , et al., unicodify, concising wikilinks, ref cleanup, etc.) using AWB

2014-12-25T22:58:43Z

Gen fixes (page/s, endash, &nbsp, et al., unicodify, concising wikilinks, ref cleanup, etc.) using AWB

← Older revision		Revision as of 00:58, 26 December 2014
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	~~In mathematics~~, ~~a '''Whittaker function'''~~ is ~~a special solution of '''~~[[E. T. ~~Whittaker\|Whittaker]~~]'s ~~equation''~~'~~, a modified form of the~~ [~~[confluent hypergeometric equation]] introduced by {{harvtxt\|Whittaker\|1904}} to make the formulas involving the solutions more symmetric~~. ~~More generally, {{harvs\|txt\|last=Jacquet\|year1=1966\|year2~~=~~1967}} introduced Whittaker functions of [[reductive group]]s over [[local field~~]~~]s, where~~ the ~~functions studied by Whittaker are essentially the case where the local field~~ is the ~~real numbers and the group is SL<sub>2</sub>('''R''')~~.		Hello, my title is Andrew and online psychic ([http://kjhkkb.net/xe/notice/374835 kjhkkb.net]) my wife doesn't like it at all. For years she's been living in Kentucky but her spouse wants them to transfer. My husband doesn't like [http://www.taehyuna.net/xe/?document_srl=78721 tarot card readings] it the way I do but what I really like performing is caving but I don't have the time lately. He is an information officer.<br><br>my page: accurate psychic predictions ([https://www.machlitim.org.il/subdomain/megila/end/node/12300 https://www.machlitim.org.il/subdomain/megila/end/node/12300])

	~~Whittaker's equation~~ is
	~~:<math>\frac{d^2w}{dz^2}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{1/4-\mu^2}{z^2}\right)w=0.</math>~~
	~~It has a regular singular point at 0 and~~ an ~~irregular singular point at ∞~~.
	~~Two solutions are given by the '''Whittaker functions''' ''M''~~<~~sub~~>~~κ,μ~~<~~/sub~~>(~~''z''), ''W''<sub>κ,μ</sub>(''z''), defined in terms of Kummer's~~ [~~[confluent hypergeometric functions]] ''M'' and ''U'' by~~
	:~~<math>M_{\kappa,\mu}\left(z\right) = \exp\left(-z~~/~~2\right)z^{\mu+\tfrac{1}{2}}M\left(\mu-\kappa+\frac{1}{2}, 1+2\mu; z\right)<~~/~~math>~~
	~~:<math>W_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}U\left(\mu-\kappa+\frac{1}{2}, 1+2\mu; z\right)</math>~~

	~~Whittaker functions appear as coefficients of certain representations of the group SL<sub>2</sub>('''R'''), called [[Whittaker model]]s~~.

	~~==References==~~
	*{{AS ref \|13\|504\|14\|537}}
	*{{citation\|first=Harry\|last=Bateman\|title=Higher transcendental functions\|volume=1\|year=1953\|publisher=McGraw-Hill\|url=http://~~apps.nrbook.com~~/~~bateman~~/~~Vol1.pdf}}.~~
	*{{springer\|id=W/~~w097850\|title=Whittaker function\|first=Yu.A. \|last=Brychkov\|first2=A.P.\|last2= Prudnikov}}.~~
	*{{dlmf\|first=Adri B. Olde\|last= Daalhuis\|id=13}}
	**{{Citation \| last1=Jacquet \| first1=Hervé \| title=Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples \| mr=0200390 \| year=1966 \| journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B \| issn=0151-0509 \| volume=262 \| pages=A943--A945}}
	*{{Citation \| last1=Jacquet \| first1=Hervé \| title=Fonctions de Whittaker associées aux groupes de Chevalley \| url=http://www.~~numdam~~.org/~~item?id=BSMF_1967__95__243_0 \| mr=0271275 \| year=1967 \| journal=Bulletin de la Société Mathématique de France \| issn=0037-9484 \| volume=95 \| pages=243–309}}~~
	*{{springer\|id=W/~~w097840\|title=Whittaker equation\|first=N.Kh. \|last=Rozov}}.~~
	*{{Citation \| last1=Slater \| first1=Lucy Joan \| title=Confluent hypergeometric functions \| publisher=[[Cambridge University Press]] \| mr=0107026 \| year=1960}}.
	*{{Citation \| last1=Whittaker \| first1=Edmund T. \| title=An expression of certain known functions as generalized hypergeometric functions \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| year=1904 \| journal=Bulletin of the A. M. S. \| volume=10 \| pages=125–134}}

	~~[[Category:Special hypergeometric functions]~~]

en>Solomonfromfinland: Category:Acceleration; Acceleration

2014-01-29T03:31:31Z

Category:Acceleration; Acceleration

← Older revision		Revision as of 05:31, 29 January 2014
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	~~The individual who wrote~~ the ~~article~~ is ~~called Jayson Hirano~~ and ~~he completely digs that title~~. ~~Ohio is where his house~~ is and ~~his family members enjoys it~~. ~~To play lacross is~~ the ~~factor I love most~~ of ~~all. I am an invoicing officer~~ and I'~~ll be promoted quickly.~~<br><br>~~My web~~-~~site~~; [http://m-~~card~~.co.kr/xe/~~mcard_2013_promote01/29877 certified psychics~~]		In mathematics, a '''Whittaker function''' is a special solution of '''[[E. T. Whittaker\|Whittaker]]'s equation''', a modified form of the [[confluent hypergeometric equation]] introduced by {{harvtxt\|Whittaker\|1904}} to make the formulas involving the solutions more symmetric. More generally, {{harvs\|txt\|last=Jacquet\|year1=1966\|year2=1967}} introduced Whittaker functions of [[reductive group]]s over [[local field]]s, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL<sub>2</sub>('''R''').

			Whittaker's equation is
			:<math>\frac{d^2w}{dz^2}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{1/4-\mu^2}{z^2}\right)w=0.</math>
			It has a regular singular point at 0 and an irregular singular point at ∞.
			Two solutions are given by the '''Whittaker functions''' ''M''<sub>κ,μ</sub>(''z''), ''W''<sub>κ,μ</sub>(''z''), defined in terms of Kummer's [[confluent hypergeometric functions]] ''M'' and ''U'' by
			:<math>M_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}M\left(\mu-\kappa+\frac{1}{2}, 1+2\mu; z\right)</math>
			:<math>W_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}U\left(\mu-\kappa+\frac{1}{2}, 1+2\mu; z\right)</math>

			Whittaker functions appear as coefficients of certain representations of the group SL<sub>2</sub>('''R'''), called [[Whittaker model]]s.

			==References==
			*{{AS ref \|13\|504\|14\|537}}
			*{{citation\|first=Harry\|last=Bateman\|title=Higher transcendental functions\|volume=1\|year=1953\|publisher=McGraw-Hill\|url=http://apps.nrbook.com/bateman/Vol1.pdf}}.
			*{{springer\|id=W/w097850\|title=Whittaker function\|first=Yu.A. \|last=Brychkov\|first2=A.P.\|last2= Prudnikov}}.
			*{{dlmf\|first=Adri B. Olde\|last= Daalhuis\|id=13}}
			**{{Citation \| last1=Jacquet \| first1=Hervé \| title=Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples \| mr=0200390 \| year=1966 \| journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B \| issn=0151-0509 \| volume=262 \| pages=A943--A945}}
			*{{Citation \| last1=Jacquet \| first1=Hervé \| title=Fonctions de Whittaker associées aux groupes de Chevalley \| url=http://www.numdam.org/item?id=BSMF_1967__95__243_0 \| mr=0271275 \| year=1967 \| journal=Bulletin de la Société Mathématique de France \| issn=0037-9484 \| volume=95 \| pages=243–309}}
			*{{springer\|id=W/w097840\|title=Whittaker equation\|first=N.Kh. \|last=Rozov}}.
			*{{Citation \| last1=Slater \| first1=Lucy Joan \| title=Confluent hypergeometric functions \| publisher=[[Cambridge University Press]] \| mr=0107026 \| year=1960}}.
			*{{Citation \| last1=Whittaker \| first1=Edmund T. \| title=An expression of certain known functions as generalized hypergeometric functions \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| year=1904 \| journal=Bulletin of the A. M. S. \| volume=10 \| pages=125–134}}

			[[Category:Special hypergeometric functions]]

en>GoingBatty: changed {{No footnotes}} to {{More footnotes}} & general fixes using AWB (8097)

2012-07-18T03:08:24Z

changed {{No footnotes}} to {{More footnotes}} & general fixes using AWB (8097)

New page

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