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{{Short description|In mathematics, a solution to a modified form of the confluent hypergeometric equation}}
[[File: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.svg|alt=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|thumb|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 {{harvs|txt|last=Whittaker|year=1903}} to make the formulas involving the solutions more symmetric. More generally, {{harvs|txt|authorlink=Hervé Jacquet|last=Jacquet|year1=1966|year2=1967}} introduced Whittaker [[Function (mathematics)|functions]] of [[Reductive group|reductive group]]s over [[Local field|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+\tfrac{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+\tfrac{1}{2}, 1+2\mu, z\right).</math>

The Whittaker function <math>W_{\kappa,\mu}(z)</math> is the same as those with opposite values of {{mvar|μ}}, in other words considered as a function of {{mvar|μ}} at fixed {{mvar|κ}} and {{mvar|z}} it is even functions. When {{mvar|κ}} and {{mvar|z}} are real, the functions give real values for real and imaginary values of {{mvar|μ}}. These functions of {{mvar|μ}} play a role in so-called Kummer spaces.<ref>{{cite book|title=Hilbert spaces of entire functions|url=https://archive.org/details/hilbertspacesofe0000debr|url-access=registration|publisher=Prentice-Hall|author=Louis de Branges|asin=B0006BUXNM|date=1968}} Sections 55-57.</ref>

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

==References==
{{Reflist}}
*{{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|access-date=2011-07-30|archive-date=2011-08-11|archive-url=https://web.archive.org/web/20110811153220/http://apps.nrbook.com/bateman/Vol1.pdf|url-status=dead}}.
*{{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 de l'Académie des Sciences, Série 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| doi=10.24033/bsmf.1654 | doi-access=free }}
*{{springer|id=W/w097840|title=Whittaker equation|first=N.Kh. |last=Rozov}}.
*{{Citation | last1=Slater | first1=Lucy Joan | title=Confluent hypergeometric functions | publisher={{wipe|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=1903 | journal=Bulletin of the A.M.S. | volume=10 | issue=3 | pages=125–134| doi=10.1090/S0002-9904-1903-01077-5 | doi-access=free }}

==Further reading==
* {{Cite journal|last1=Hatamzadeh-Varmazyar|first1=Saeed|last2=Masouri|first2=Zahra|date=2012-11-01|title=A fast numerical method for analysis of one- and two-dimensional electromagnetic scattering using a set of cardinal functions|url=http://www.sciencedirect.com/science/article/pii/S0955799712001129|journal=Engineering Analysis with Boundary Elements|language=en|volume=36|issue=11|pages=1631–1639|doi=10.1016/j.enganabound.2012.04.014|issn=0955-7997}}
* {{Cite journal|last1=Gerasimov|first1=A. A.|last2=Lebedev|first2=Dmitrii R.|last3=Oblezin|first3=Sergei V.|date=2012|title=New integral representations of Whittaker functions for classical Lie groups|url=https://iopscience.iop.org/article/10.1070/RM2012v067n01ABEH004776/meta|journal=Russian Mathematical Surveys|language=en|volume=67|issue=1|pages=1–92|doi=10.1070/RM2012v067n01ABEH004776|arxiv=0705.2886|bibcode=2012RuMaS..67....1G|issn=0036-0279}}
* {{Cite journal|last1=Baudoin|first1=Fabrice|last2=O’Connell|first2=Neil|date=2011|title=Exponential functionals of brownian motion and class-one Whittaker functions|url=http://www.numdam.org/item/AIHPB_2011__47_4_1096_0/|journal= Annales de l'Institut Henri Poincaré, Probabilités et Statistiques|language=en|volume=47|issue=4|pages=1096–1120|doi=10.1214/10-AIHP401|bibcode=2011AIHPB..47.1096B|s2cid=113388|doi-access=free|arxiv=0809.2506}}
* {{Cite journal|last=McKee|first=Mark|date=April 2009|title=An Infinite Order Whittaker Function|journal=Canadian Journal of Mathematics|language=en|volume=61|issue=2|pages=373–381|doi=10.4153/CJM-2009-019-x|s2cid=55587239|issn=0008-414X|doi-access=free}}
* {{Cite journal|last1=Mathai|first1=A. M.|last2=Pederzoli|first2=Giorgio|date=1997-03-01|title=Some properties of matrix-variate Laplace transforms and matrix-variate Whittaker functions|journal=Linear Algebra and Its Applications|language=en|volume=253|issue=1|pages=209–226|doi=10.1016/0024-3795(95)00705-9|issn=0024-3795|doi-access=free}}
* {{Cite journal|last=Whittaker|first=J. M.|date=May 1927|title=On the Cardinal Function of Interpolation Theory|journal=Proceedings of the Edinburgh Mathematical Society|language=en|volume=1|issue=1|pages=41–46|doi=10.1017/S0013091500007318|issn=1464-3839|doi-access=free}}
* {{Cite journal|last=Cherednik|first=Ivan|date=2009|title=Whittaker Limits of Difference Spherical Functions|url=https://ieeexplore.ieee.org/document/8189397|journal=International Mathematics Research Notices|volume=2009|issue=20|pages=3793–3842|doi=10.1093/imrn/rnp065|arxiv=0807.2155|s2cid=6253357|issn=1687-0247}}
* {{Cite journal|last=Slater|first=L. J.|date=October 1954|title=Expansions of generalized Whittaker functions|url=https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/expansions-of-generalized-whittaker-functions/E011E597828132A26531306D6794C7C3|journal=Mathematical Proceedings of the Cambridge Philosophical Society|language=en|volume=50|issue=4|pages=628–631|doi=10.1017/S0305004100029765|bibcode=1954PCPS...50..628S|s2cid=122348447 |issn=1469-8064}}
* {{cite arXiv|last=Etingof|first=Pavel|date=1999-01-12|title=Whittaker functions on quantum groups and q-deformed Toda operators|eprint=math/9901053}}
* {{Cite journal|last=McNamara|first=Peter J.|date=2011-01-15|title=Metaplectic Whittaker functions and crystal bases|url=https://projecteuclid.org/euclid.dmj/1292509116|journal=Duke Mathematical Journal|language=EN|volume=156|issue=1|pages=1–31|doi=10.1215/00127094-2010-064|arxiv=0907.2675|s2cid=979197|issn=0012-7094}}
* {{Cite journal|last1=Mathai|first1=A. M.|last2=Pederzoli|first2=Giorgio|date=1998-01-15|title=A whittaker function of matrix argument|journal=Linear Algebra and Its Applications|language=en|volume=269|issue=1|pages=91–103|doi=10.1016/S0024-3795(97)00059-1|issn=0024-3795|doi-access=free}}
* {{Cite journal|last1=Frenkel|first1=E.|last2=Gaitsgory|first2=D.|last3=Kazhdan|first3=D.|last4=Vilonen|first4=K.|date=1998|title=Geometric realization of Whittaker functions and the Langlands conjecture|url=https://www.ams.org/jams/1998-11-02/S0894-0347-98-00260-4/|journal=Journal of the American Mathematical Society|language=en|volume=11|issue=2|pages=451–484|doi=10.1090/S0894-0347-98-00260-4|s2cid=13221400|issn=0894-0347|doi-access=free|arxiv=alg-geom/9703022}}

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

{{Sourceattribution|Whittaker function}}