SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Whittaker function</title>

</head>

Whittaker function

</h1>

</header>

In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the <a href="Confluent_hypergeometric_equation" class="mw-redirect" title="Confluent hypergeometric equation">confluent hypergeometric equation</a> introduced by <a href="E._T._Whittaker" title="E. T. Whittaker">Whittaker</a> (<a href="#CITEREFWhittaker1903">1903</a>) to make the formulas involving the solutions more symmetric. More generally, <a href="Herv%C3%A9_Jacquet" title="Hervé Jacquet">Jacquet</a> (<a href="#CITEREFJacquet1966">1966</a>, <a href="#CITEREFJacquet1967">1967</a>) introduced Whittaker <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> of <a href="Reductive_group" title="Reductive group">reductive groups</a> over <a href="Local_field" title="Local field">local fields</a>, 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

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left(-{\frac {1}{4}}+{\frac {\kappa }{z}}+{\frac {1/4-\mu ^{2}}{z^{2}}}\right)w=0.}</annotation>

</semantics>

</math><img src="./104ed9bcd2df2d881882a8fa4abc48940e8bf67a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.536ex; height:7.509ex;" alt="{\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left(-{\frac {1}{4}}+{\frac {\kappa }{z}}+{\frac {1/4-\mu ^{2}}{z^{2}}}\right)w=0.}" loading="lazy"></dd></dl>

It has a <a href="Regular_singular_point" title="Regular singular point">regular singular point</a> at 0 and an irregular singular point at ∞.

Two solutions are given by the Whittaker functions Mκ,μ(z), Wκ,μ(z), defined in terms of Kummer's <a href="Confluent_hypergeometric_functions" class="mw-redirect" title="Confluent hypergeometric functions">confluent hypergeometric functions</a> M and U by

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<msup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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)}</annotation>

</semantics>

</math><img src="./22b70e3592752abfaa97b52cfc841be84632938c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:51.86ex; height:4.843ex;" alt="{\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)}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<msup>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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).}</annotation>

</semantics>

</math><img src="./168e90f1eaf09716fcd53dc4d50d258f33b32c3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:52.174ex; height:4.843ex;" alt="{\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).}" loading="lazy"></dd></dl>

The Whittaker function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{\kappa ,\mu }(z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{\kappa ,\mu }(z)}</annotation>

</semantics>

</math><img src="./62dbbcfb3ebd6067f413d9390282fa77e8835c40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.719ex; height:3.009ex;" alt="{\displaystyle W_{\kappa ,\mu }(z)}" loading="lazy"> 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 <a href="Even_function" class="mw-redirect" title="Even function">even functions</a>. 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.<a href="#cite_note-1">[1]</a>

Whittaker functions appear as coefficients of certain representations of the group SL2(R), called <a href="Whittaker_model" title="Whittaker model">Whittaker models</a>.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFLouis_de_Branges1968" class="citation book cs1"><a href="Louis_de_Branges" class="mw-redirect" title="Louis de Branges">Louis de Branges</a> (1968). <a rel="nofollow" class="external text" href="https://archive.org/details/hilbertspacesofe0000debr">Hilbert spaces of entire functions</a>. Prentice-Hall. <a href="ASIN_(identifier)" class="mw-redirect" title="ASIN (identifier)">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.com/dp/B0006BUXNM">B0006BUXNM</a>.</cite> Sections 55-57.

</li>

</ol></div></div>

<ul><li><cite id="CITEREFAbramowitzStegun1983" class="citation book cs1"><a href="Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="Irene_Stegun" title="Irene Stegun">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cbm/aands/page_504.htm">"Chapter 13"</a>. <a href="Abramowitz_and_Stegun" title="Abramowitz and Stegun">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>. Applied Mathematics Series. Vol. 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. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-61272-0</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://www.loc.gov/item/65012253">65-12253</a>.</cite> See also <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cbm/aands/page_537.htm">chapter 14</a>.</li>

<li><cite id="CITEREFBateman1953" class="citation cs2">Bateman, Harry (1953), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110811153220/http://apps.nrbook.com/bateman/Vol1.pdf">Higher transcendental functions</a> (PDF), vol. 1, McGraw-Hill, archived from <a rel="nofollow" class="external text" href="http://apps.nrbook.com/bateman/Vol1.pdf">the original</a> (PDF) on 2011-08-11, retrieved 2011-07-30</cite>.</li>

<li><cite id="CITEREFBrychkovPrudnikov2001" class="citation cs2">Brychkov, Yu.A.; Prudnikov, A.P. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Whittaker_function">"Whittaker function"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite>.</li>

<li><cite id="CITEREFDaalhuis2010" class="citation cs2">Daalhuis, Adri B. Olde (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/13">"Whittaker function"</a>, in <a href="Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19225-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite>.

<ul><li><cite id="CITEREFJacquet1966" class="citation cs2">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, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0151-0509">0151-0509</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0200390">0200390</a></cite></li></ul></li>

<li><cite id="CITEREFJacquet1967" class="citation cs2">Jacquet, Hervé (1967), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=BSMF_1967__95__243_0">"Fonctions de Whittaker associées aux groupes de Chevalley"</a>, Bulletin de la Société Mathématique de France, 95: 243–309, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fbsmf.1654">10.24033/bsmf.1654</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0037-9484">0037-9484</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0271275">0271275</a></cite></li>

<li><cite id="CITEREFRozov2001" class="citation cs2">Rozov, N.Kh. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Whittaker_equation">"Whittaker equation"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite>.</li>

<li><cite id="CITEREFSlater1960" class="citation cs2">Slater, Lucy Joan (1960), Confluent hypergeometric functions, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0107026">0107026</a></cite>.</li>

<li><cite id="CITEREFWhittaker1903" class="citation cs2">Whittaker, Edmund T. (1903), "An expression of certain known functions as generalized hypergeometric functions", Bulletin of the A.M.S., 10 (3), Providence, R.I.: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>: 125–134, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1903-01077-5">10.1090/S0002-9904-1903-01077-5</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFHatamzadeh-VarmazyarMasouri2012" class="citation journal cs1">Hatamzadeh-Varmazyar, Saeed; Masouri, Zahra (2012-11-01). <a rel="nofollow" class="external text" href="http://www.sciencedirect.com/science/article/pii/S0955799712001129">"A fast numerical method for analysis of one- and two-dimensional electromagnetic scattering using a set of cardinal functions"</a>. Engineering Analysis with Boundary Elements. 36 (11): 1631–1639. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.enganabound.2012.04.014">10.1016/j.enganabound.2012.04.014</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0955-7997">0955-7997</a>.</cite></li>

<li><cite id="CITEREFGerasimovLebedevOblezin2012" class="citation journal cs1">Gerasimov, A. A.; Lebedev, Dmitrii R.; Oblezin, Sergei V. (2012). <a rel="nofollow" class="external text" href="https://iopscience.iop.org/article/10.1070/RM2012v067n01ABEH004776/meta">"New integral representations of Whittaker functions for classical Lie groups"</a>. Russian Mathematical Surveys. 67 (1): 1–92. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0705.2886">0705.2886</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012RuMaS..67....1G">2012RuMaS..67....1G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1070%2FRM2012v067n01ABEH004776">10.1070/RM2012v067n01ABEH004776</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0036-0279">0036-0279</a>.</cite></li>

<li><cite id="CITEREFBaudoinO'Connell2011" class="citation journal cs1">Baudoin, Fabrice; O'Connell, Neil (2011). <a rel="nofollow" class="external text" href="http://www.numdam.org/item/AIHPB_2011__47_4_1096_0/">"Exponential functionals of brownian motion and class-one Whittaker functions"</a>. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques. 47 (4): 1096–1120. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0809.2506">0809.2506</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011AIHPB..47.1096B">2011AIHPB..47.1096B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F10-AIHP401">10.1214/10-AIHP401</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:113388">113388</a>.</cite></li>

<li><cite id="CITEREFMcKee2009" class="citation journal cs1">McKee, Mark (April 2009). <a rel="nofollow" class="external text" href="https://doi.org/10.4153%2FCJM-2009-019-x">"An Infinite Order Whittaker Function"</a>. Canadian Journal of Mathematics. 61 (2): 373–381. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4153%2FCJM-2009-019-x">10.4153/CJM-2009-019-x</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0008-414X">0008-414X</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:55587239">55587239</a>.</cite></li>

<li><cite id="CITEREFMathaiPederzoli1997" class="citation journal cs1">Mathai, A. M.; Pederzoli, Giorgio (1997-03-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0024-3795%2895%2900705-9">"Some properties of matrix-variate Laplace transforms and matrix-variate Whittaker functions"</a>. Linear Algebra and Its Applications. 253 (1): 209–226. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0024-3795%2895%2900705-9">10.1016/0024-3795(95)00705-9</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0024-3795">0024-3795</a>.</cite></li>

<li><cite id="CITEREFWhittaker1927" class="citation journal cs1">Whittaker, J. M. (May 1927). <a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0013091500007318">"On the Cardinal Function of Interpolation Theory"</a>. Proceedings of the Edinburgh Mathematical Society. 1 (1): 41–46. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0013091500007318">10.1017/S0013091500007318</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1464-3839">1464-3839</a>.</cite></li>

<li><cite id="CITEREFCherednik2009" class="citation journal cs1">Cherednik, Ivan (2009). <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/8189397">"Whittaker Limits of Difference Spherical Functions"</a>. International Mathematics Research Notices. 2009 (20): 3793–3842. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0807.2155">0807.2155</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fimrn%2Frnp065">10.1093/imrn/rnp065</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1687-0247">1687-0247</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6253357">6253357</a>.</cite></li>

<li><cite id="CITEREFSlater1954" class="citation journal cs1">Slater, L. J. (October 1954). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/expansions-of-generalized-whittaker-functions/E011E597828132A26531306D6794C7C3">"Expansions of generalized Whittaker functions"</a>. <a href="Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society" title="Mathematical Proceedings of the Cambridge Philosophical Society">Mathematical Proceedings of the Cambridge Philosophical Society</a>. 50 (4): 628–631. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1954PCPS...50..628S">1954PCPS...50..628S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0305004100029765">10.1017/S0305004100029765</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1469-8064">1469-8064</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122348447">122348447</a>.</cite></li>

<li><cite id="CITEREFEtingof1999" class="citation arxiv cs1">Etingof, Pavel (1999-01-12). "Whittaker functions on quantum groups and q-deformed Toda operators". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9901053">math/9901053</a>.</cite></li>

<li><cite id="CITEREFMcNamara2011" class="citation journal cs1">McNamara, Peter J. (2011-01-15). <a rel="nofollow" class="external text" href="https://projecteuclid.org/euclid.dmj/1292509116">"Metaplectic Whittaker functions and crystal bases"</a>. Duke Mathematical Journal. 156 (1): 1–31. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0907.2675">0907.2675</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2F00127094-2010-064">10.1215/00127094-2010-064</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0012-7094">0012-7094</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:979197">979197</a>.</cite></li>

<li><cite id="CITEREFMathaiPederzoli1998" class="citation journal cs1">Mathai, A. M.; Pederzoli, Giorgio (1998-01-15). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0024-3795%2897%2900059-1">"A whittaker function of matrix argument"</a>. Linear Algebra and Its Applications. 269 (1): 91–103. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0024-3795%2897%2900059-1">10.1016/S0024-3795(97)00059-1</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0024-3795">0024-3795</a>.</cite></li>

<li><cite id="CITEREFFrenkelGaitsgoryKazhdanVilonen1998" class="citation journal cs1">Frenkel, E.; Gaitsgory, D.; Kazhdan, D.; Vilonen, K. (1998). <a rel="nofollow" class="external text" href="https://www.ams.org/jams/1998-11-02/S0894-0347-98-00260-4/">"Geometric realization of Whittaker functions and the Langlands conjecture"</a>. Journal of the American Mathematical Society. 11 (2): 451–484. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/alg-geom/9703022">alg-geom/9703022</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0894-0347-98-00260-4">10.1090/S0894-0347-98-00260-4</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0894-0347">0894-0347</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13221400">13221400</a>.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-07" href="https://en.wikipedia.org/wiki/?title=Whittaker_function&oldid=1299301371">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>