Whittaker function

Short description: In mathematics, a solution to a modified form of the confluent hypergeometric equation

Plot of the Whittaker function M k,m(z) with k=2 and m=½ in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker (1903) 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

[math]\displaystyle{ \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_κ,μ(z), W_κ,μ(z), defined in terms of Kummer's confluent hypergeometric functions M and U by

[math]\displaystyle{ M_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}M\left(\mu-\kappa+\tfrac{1}{2}, 1+2\mu, z\right) }[/math]

[math]\displaystyle{ W_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}U\left(\mu-\kappa+\tfrac{1}{2}, 1+2\mu, z\right). }[/math]

The Whittaker function [math]\displaystyle{ W_{\kappa,\mu}(z) }[/math] is the same as those with opposite values of μ, in other words considered as a function of μ at fixed κ and z it is 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.

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

References

• Abramowitz, Milton; Stegun, Irene Ann, eds (1983). "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. LCCN 65-12253. ISBN 978-0-486-61272-0. http://www.math.sfu.ca/~cbm/aands/page_504.htm.  See also chapter 14.
• Bateman, Harry (1953), Higher transcendental functions, 1, McGraw-Hill, http://apps.nrbook.com/bateman/Vol1.pdf, retrieved 2011-07-30 .
• Hazewinkel, Michiel, ed. (2001), "Whittaker function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=W/w097850 .
• Daalhuis, Adri B. Olde (2010), "Whittaker function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/13 
  • 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 
• Jacquet, Hervé (1967), "Fonctions de Whittaker associées aux groupes de Chevalley", Bulletin de la Société Mathématique de France 95: 243–309, doi:10.24033/bsmf.1654, ISSN 0037-9484, http://www.numdam.org/item?id=BSMF_1967__95__243_0 
• Hazewinkel, Michiel, ed. (2001), "Whittaker equation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=W/w097840 .
• Slater, Lucy Joan (1960), Confluent hypergeometric functions, Cambridge University Press .
• Whittaker, Edmund T. (1903), "An expression of certain known functions as generalized hypergeometric functions", Bulletin of the A.M.S. (Providence, R.I.: American Mathematical Society) 10 (3): 125–134, doi:10.1090/S0002-9904-1903-01077-5 

Further reading

• Hatamzadeh-Varmazyar, Saeed; Masouri, Zahra (2012-11-01). "A fast numerical method for analysis of one- and two-dimensional electromagnetic scattering using a set of cardinal functions" (in en). Engineering Analysis with Boundary Elements 36 (11): 1631–1639. doi:10.1016/j.enganabound.2012.04.014. ISSN 0955-7997. http://www.sciencedirect.com/science/article/pii/S0955799712001129. 
• Gerasimov, A. A.; Lebedev, Dmitrii R.; Oblezin, Sergei V. (2012). "New integral representations of Whittaker functions for classical Lie groups" (in en). Russian Mathematical Surveys 67 (1): 1–92. doi:10.1070/RM2012v067n01ABEH004776. ISSN 0036-0279. Bibcode: 2012RuMaS..67....1G. https://iopscience.iop.org/article/10.1070/RM2012v067n01ABEH004776/meta. 
• Baudoin, Fabrice; O’Connell, Neil (2011). "Exponential functionals of brownian motion and class-one Whittaker functions" (in en). Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 47 (4): 1096–1120. doi:10.1214/10-AIHP401. Bibcode: 2011AIHPB..47.1096B. http://www.numdam.org/item/AIHPB_2011__47_4_1096_0/. 
• McKee, Mark (April 2009). "An Infinite Order Whittaker Function" (in en). Canadian Journal of Mathematics 61 (2): 373–381. doi:10.4153/CJM-2009-019-x. ISSN 0008-414X. 
• Mathai, A. M.; Pederzoli, Giorgio (1997-03-01). "Some properties of matrix-variate Laplace transforms and matrix-variate Whittaker functions" (in en). Linear Algebra and Its Applications 253 (1): 209–226. doi:10.1016/0024-3795(95)00705-9. ISSN 0024-3795. 
• Whittaker, J. M. (May 1927). "On the Cardinal Function of Interpolation Theory" (in en). Proceedings of the Edinburgh Mathematical Society 1 (1): 41–46. doi:10.1017/S0013091500007318. ISSN 1464-3839. 
• Cherednik, Ivan (2009). "Whittaker Limits of Difference Spherical Functions". International Mathematics Research Notices 2009 (20): 3793–3842. doi:10.1093/imrn/rnp065. ISSN 1687-0247. https://ieeexplore.ieee.org/document/8189397. 
• Slater, L. J. (October 1954). "Expansions of generalized Whittaker functions" (in en). Mathematical Proceedings of the Cambridge Philosophical Society 50 (4): 628–631. doi:10.1017/S0305004100029765. ISSN 1469-8064. Bibcode: 1954PCPS...50..628S. https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/expansions-of-generalized-whittaker-functions/E011E597828132A26531306D6794C7C3. 
• Etingof, Pavel (1999-01-12). "Whittaker functions on quantum groups and q-deformed Toda operators". arXiv:math/9901053.
• McNamara, Peter J. (2011-01-15). "Metaplectic Whittaker functions and crystal bases" (in EN). Duke Mathematical Journal 156 (1): 1–31. doi:10.1215/00127094-2010-064. ISSN 0012-7094. https://projecteuclid.org/euclid.dmj/1292509116. 
• Mathai, A. M.; Pederzoli, Giorgio (1998-01-15). "A whittaker function of matrix argument" (in en). Linear Algebra and Its Applications 269 (1): 91–103. doi:10.1016/S0024-3795(97)00059-1. ISSN 0024-3795. 
• Frenkel, E.; Gaitsgory, D.; Kazhdan, D.; Vilonen, K. (1998). "Geometric realization of Whittaker functions and the Langlands conjecture" (in en). Journal of the American Mathematical Society 11 (2): 451–484. doi:10.1090/S0894-0347-98-00260-4. ISSN 0894-0347. https://www.ams.org/jams/1998-11-02/S0894-0347-98-00260-4/.