Whittaker function

In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker (1904) to make the formulas involving the solutions more symmetric. More generally, Jacquet (1966, 1967) introduced Whittaker functions of reductive groups over local fields, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL₂(R).

Whittaker's equation is

{\frac {d^{2}w}{dz^{2}}}+\left(-{\frac {1}{4}}+{\frac {\kappa }{z}}+{\frac {1/4-\mu ^{2}}{z^{2}}}\right)w=0.

It has a regular singular point at 0 and an irregular singular point at ∞. Two solutions are given by the Whittaker functions M_κ,μ(z), W_κ,μ(z), defined in terms of Kummer's confluent hypergeometric functions M and U by

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)

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).

The Whittaker functions $M_{\kappa ,\mu }(z)$ and $W_{\kappa ,\mu }(z)$ are the same as those with opposite values of $μ$ , in other words considered as a function of $μ$ at fixed $κ$ and $z$ they are even functions. When $κ$ and $z$ are real, the functions give real values for real and imaginary values of $μ$ . These functions of $μ$ play a role in so-called Kummer spaces.^[1]

Whittaker functions appear as coefficients of certain representations of the group SL₂(R), called Whittaker models.

References

↑ Louis de Branges (1968). Hilbert spaces of entire functions. Prentice-Hall. ASIN B0006BUXNM. Sections 55-57.

Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 13". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 504, 537. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. See also chapter 14.
Bateman, Harry (1953), Higher transcendental functions (PDF), 1, McGraw-Hill .
Brychkov, Yu.A.; Prudnikov, A.P. (2001) [1994], "Whittaker function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 .
Daalhuis, Adri B. Olde (2010), "Whittaker function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
- Jacquet, Hervé (1966), "Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples", Comptes Rendus de l'Académie des Sciences, Série A et B, 262: A943--A945, ISSN 0151-0509, MR 0200390
Jacquet, Hervé (1967), "Fonctions de Whittaker associées aux groupes de Chevalley", Bulletin de la Société Mathématique de France, 95: 243–309, ISSN 0037-9484, MR 0271275
Rozov, N.Kh. (2001) [1994], "Whittaker equation", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 .
Slater, Lucy Joan (1960), Confluent hypergeometric functions, Cambridge University Press, MR 0107026 .
Whittaker, Edmund T. (1904), "An expression of certain known functions as generalized hypergeometric functions", Bulletin of the A.M.S., Providence, R.I.: American Mathematical Society, 10: 125–134

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[1] Louis de Branges (1968). Hilbert spaces of entire functions. Prentice-Hall. ASIN B0006BUXNM. Sections 55-57.