31:
784:. 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.
618:
454:
30:
252:
1359:
460:
299:
115:
663:
912:
Jacquet, Hervé (1966), "Une interprétation géométrique et une généralisation P-adique des fonctions de
Whittaker en théorie des groupes semi-simples",
1570:
885:
613:{\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).}
449:{\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)}
894:
789:
283:
63:
98:, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL
1059:"A fast numerical method for analysis of one- and two-dimensional electromagnetic scattering using a set of cardinal functions"
247:{\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left(-{\frac {1}{4}}+{\frac {\kappa }{z}}+{\frac {1/4-\mu ^{2}}{z^{2}}}\right)w=0.}
1033:
995:
867:
990:
862:
829:
1580:
1008:
1024:
Whittaker, Edmund T. (1903), "An expression of certain known functions as generalized hypergeometric functions",
1575:
257:
It has a regular singular point at 0 and an irregular singular point at ∞. Two solutions are given by the
1091:
87:
626:
985:
857:
779:
50:
in the complex plane from -2-2i to 2+2i with colors created with
Mathematica 13.1 function ComplexPlot3D
1371:
1164:
1113:
1058:
1407:
Etingof, Pavel (1999-01-12). "Whittaker functions on quantum groups and q-deformed Toda operators".
702:
1549:
1521:
1461:
1435:
1408:
1395:
1346:
1320:
1225:
1182:
1154:
1103:
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1129:
1078:
965:
921:
890:
880:
811:
795:
785:
767:
749:
1509:
1239:"Some properties of matrix-variate Laplace transforms and matrix-variate Whittaker functions"
1531:
1486:
1445:
1379:
1330:
1285:
1250:
1207:
1172:
1121:
1070:
1037:
955:
737:
75:
1016:
977:
933:
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807:
1012:
973:
929:
900:
803:
717:
91:
67:
1375:
1168:
1117:
742:
1491:
1474:
1142:
1074:
27:
In mathematics, a solution to a modified form of the confluent hypergeometric equation
1564:
1399:
1255:
1238:
943:
708:
Whittaker functions appear as coefficients of certain representations of the group SL
682:
1553:
1229:
1125:
1042:
1350:
771:
1536:
1465:
1423:
1186:
17:
821:
781:
Handbook of
Mathematical Functions with Formulas, Graphs, and Mathematical Tables
775:
95:
1308:
74:) to make the formulas involving the solutions more symmetric. More generally,
1449:
1383:
1290:
1273:
1143:"Exponential functionals of brownian motion and class-one Whittaker functions"
1092:"New integral representations of Whittaker functions for classical Lie groups"
1545:
1500:
1457:
1391:
1342:
1299:
1264:
1221:
1133:
1082:
969:
925:
1510:"Geometric realization of Whittaker functions and the Langlands conjecture"
1212:
1195:
1334:
838:
693:
are real, the functions give real values for real and imaginary values of
1526:
1177:
960:
753:
1413:
883:; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),
815:
1440:
1325:
1159:
1147:
Annales de l'Institut Henri
Poincaré, Probabilités et Statistiques
1108:
1090:
Gerasimov, A. A.; Lebedev, Dmitrii R.; Oblezin, Sergei V. (2012).
29:
799:
1364:
Mathematical
Proceedings of the Cambridge Philosophical Society
1508:
Frenkel, E.; Gaitsgory, D.; Kazhdan, D.; Vilonen, K. (1998).
944:"Fonctions de Whittaker associées aux groupes de Chevalley"
1057:
Hatamzadeh-Varmazyar, Saeed; Masouri, Zahra (2012-11-01).
876:
914:
Comptes Rendus de l'Académie des
Sciences, Série A et B
34:
Plot of the
Whittaker function M k,m(z) with k=2 and m=
570:
536:
409:
375:
629:
463:
302:
118:
1309:"Whittaker Limits of Difference Spherical Functions"
1424:"Metaplectic Whittaker functions and crystal bases"
1274:"On the Cardinal Function of Interpolation Theory"
741:
657:
612:
448:
246:
1278:Proceedings of the Edinburgh Mathematical Society
1473:Mathai, A. M.; Pederzoli, Giorgio (1998-01-15).
1237:Mathai, A. M.; Pederzoli, Giorgio (1997-03-01).
1360:"Expansions of generalized Whittaker functions"
948:Bulletin de la Société Mathématique de France
669:, in other words considered as a function of
665:is the same as those with opposite values of
8:
1514:Journal of the American Mathematical Society
1063:Engineering Analysis with Boundary Elements
1313:International Mathematics Research Notices
1141:Baudoin, Fabrice; O'Connell, Neil (2011).
856:Brychkov, Yu.A.; Prudnikov, A.P. (2001) ,
837:, vol. 1, McGraw-Hill, archived from
1535:
1525:
1490:
1475:"A whittaker function of matrix argument"
1439:
1412:
1324:
1289:
1254:
1211:
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1158:
1107:
1041:
959:
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468:
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367:
350:
307:
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222:
211:
196:
190:
177:
164:
144:
126:
119:
117:
71:
886:NIST Handbook of Mathematical Functions
729:
83:
79:
1196:"An Infinite Order Whittaker Function"
7:
1479:Linear Algebra and Its Applications
1243:Linear Algebra and Its Applications
658:{\displaystyle W_{\kappa ,\mu }(z)}
1005:Confluent hypergeometric functions
744:Hilbert spaces of entire functions
284:confluent hypergeometric functions
25:
1422:McNamara, Peter J. (2011-01-15).
1075:10.1016/j.enganabound.2012.04.014
64:confluent hypergeometric equation
1571:Special hypergeometric functions
282:), defined in terms of Kummer's
1200:Canadian Journal of Mathematics
1126:10.1070/RM2012v067n01ABEH004776
1043:10.1090/S0002-9904-1903-01077-5
875:Daalhuis, Adri B. Olde (2010),
831:Higher transcendental functions
1358:Slater, L. J. (October 1954).
889:, Cambridge University Press,
652:
646:
1:
1537:10.1090/S0894-0347-98-00260-4
1492:10.1016/S0024-3795(97)00059-1
1272:Whittaker, J. M. (May 1927).
1034:American Mathematical Society
1256:10.1016/0024-3795(95)00705-9
1096:Russian Mathematical Surveys
991:Encyclopedia of Mathematics
863:Encyclopedia of Mathematics
1597:
1194:McKee, Mark (April 2009).
1009:Cambridge University Press
1003:Slater, Lucy Joan (1960),
1450:10.1215/00127094-2010-064
1428:Duke Mathematical Journal
1384:10.1017/S0305004100029765
1291:10.1017/S0013091500007318
701:play a role in so-called
62:, a modified form of the
58:is a special solution of
1307:Cherednik, Ivan (2009).
109:Whittaker's equation is
1032:(3), Providence, R.I.:
942:Jacquet, Hervé (1967),
828:Bateman, Harry (1953),
623:The Whittaker function
86:) introduced Whittaker
1213:10.4153/CJM-2009-019-x
1026:Bulletin of the A.M.S.
984:Rozov, N.Kh. (2001) ,
659:
614:
450:
248:
51:
697:. These functions of
660:
615:
451:
249:
33:
986:"Whittaker equation"
877:"Whittaker function"
858:"Whittaker function"
627:
461:
300:
116:
60:Whittaker's equation
1376:1954PCPS...50..628S
1335:10.1093/imrn/rnp065
1169:2011AIHPB..47.1096B
1118:2012RuMaS..67....1G
259:Whittaker functions
1178:10.1214/10-AIHP401
961:10.24033/bsmf.1654
881:Olver, Frank W. J.
768:Abramowitz, Milton
655:
610:
579:
545:
446:
418:
384:
244:
56:Whittaker function
54:In mathematics, a
52:
18:Whittaker equation
1581:Special functions
1319:(20): 3793–3842.
1069:(11): 1631–1639.
896:978-0-521-19225-5
791:978-0-486-61272-0
772:Stegun, Irene Ann
748:. Prentice-Hall.
578:
544:
417:
383:
228:
185:
172:
151:
16:(Redirected from
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1539:
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1527:alg-geom/9703022
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1153:(4): 1096–1120.
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998:
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963:
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907:
870:
851:
850:
849:
843:
836:
820: See also
819:
774:, eds. (1983) .
759:
757:
747:
738:Louis de Branges
734:
718:Whittaker models
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92:reductive groups
49:
47:
46:
43:
40:
21:
1596:
1595:
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1590:
1589:
1587:
1586:
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1576:E. T. Whittaker
1561:
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1472:
1421:
1406:
1357:
1306:
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1140:
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1053:
1051:Further reading
1023:
1002:
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911:
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855:
847:
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841:
834:
827:
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766:
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758:Sections 55-57.
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1562:
1559:
1558:
1520:(2): 451–484.
1505:
1470:
1419:
1404:
1370:(4): 628–631.
1355:
1304:
1269:
1249:(1): 209–226.
1234:
1206:(2): 373–381.
1191:
1138:
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1047:
1021:
1000:
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939:
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937:
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790:
761:
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709:
683:even functions
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66:introduced by
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1485:(1): 91–103.
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953:
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935:
931:
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923:
920:: A943–A945,
919:
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909:
906:
902:
898:
892:
888:
887:
882:
878:
873:
869:
865:
864:
859:
854:
844:on 2011-08-11
840:
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783:
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764:
755:
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746:
745:
739:
733:
730:
723:
721:
719:
715:
706:
704:
703:Kummer spaces
684:
649:
641:
638:
635:
631:
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32:
19:
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1482:
1478:
1431:
1427:
1414:math/9901053
1367:
1363:
1316:
1312:
1284:(1): 41–46.
1281:
1277:
1246:
1242:
1203:
1199:
1150:
1146:
1099:
1095:
1066:
1062:
1029:
1025:
1004:
989:
951:
947:
917:
913:
884:
861:
846:, retrieved
839:the original
830:
780:
776:"Chapter 13"
743:
732:
713:
707:
622:
290:
286:
279:
272:
268:
261:
258:
256:
108:
103:
96:local fields
59:
55:
53:
1434:(1): 1–31.
1102:(1): 1–92.
1036:: 125–134,
954:: 243–309,
1565:Categories
848:2011-07-30
822:chapter 14
754:B0006BUXNM
724:References
716:), called
1546:0894-0347
1501:0024-3795
1458:0012-7094
1441:0907.2675
1400:122348447
1392:1469-8064
1343:1687-0247
1326:0807.2155
1300:1464-3839
1265:0024-3795
1222:0008-414X
1160:0809.2506
1134:0036-0279
1109:0705.2886
1083:0955-7997
996:EMS Press
970:0037-9484
926:0151-0509
868:EMS Press
673:at fixed
642:μ
636:κ
594:μ
564:κ
561:−
558:μ
530:μ
506:−
498:
476:μ
470:κ
433:μ
403:κ
400:−
397:μ
369:μ
345:−
337:
315:μ
309:κ
209:μ
205:−
180:κ
162:−
88:functions
68:Whittaker
1554:13221400
1230:55587239
816:65-12253
800:64-60036
740:(1968).
1372:Bibcode
1351:6253357
1165:Bibcode
1114:Bibcode
1017:0107026
978:0271275
934:0200390
905:2723248
808:0167642
685:. When
78: (
76:Jacquet
70: (
48:
36:
1552:
1544:
1499:
1466:979197
1464:
1456:
1398:
1390:
1349:
1341:
1298:
1263:
1228:
1220:
1187:113388
1185:
1132:
1081:
1015:
976:
968:
932:
924:
903:
893:
814:
806:
798:
788:
752:
681:it is
1550:S2CID
1522:arXiv
1462:S2CID
1436:arXiv
1409:arXiv
1396:S2CID
1347:S2CID
1321:arXiv
1226:S2CID
1183:S2CID
1155:arXiv
1104:arXiv
879:, in
842:(PDF)
835:(PDF)
94:over
1542:ISSN
1497:ISSN
1454:ISSN
1388:ISSN
1339:ISSN
1317:2009
1296:ISSN
1261:ISSN
1218:ISSN
1130:ISSN
1079:ISSN
966:ISSN
922:ISSN
891:ISBN
812:LCCN
796:LCCN
786:ISBN
750:ASIN
689:and
677:and
289:and
84:1967
80:1966
72:1903
1532:doi
1487:doi
1483:269
1446:doi
1432:156
1380:doi
1331:doi
1286:doi
1251:doi
1247:253
1208:doi
1173:doi
1122:doi
1071:doi
1038:doi
956:doi
918:262
495:exp
334:exp
293:by
276:κ,μ
271:),
265:κ,μ
106:).
90:of
1567::
1548:.
1540:.
1530:.
1518:11
1516:.
1512:.
1495:.
1481:.
1477:.
1460:.
1452:.
1444:.
1430:.
1426:.
1394:.
1386:.
1378:.
1368:50
1366:.
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