|
|
| Line 1: |
Line 1: |
| 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]]
| |
Hello, my title is Andrew and online psychic (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 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.
my page: accurate psychic predictions (https://www.machlitim.org.il/subdomain/megila/end/node/12300)