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.

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

$M_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}M\left(\mu -\kappa +{\frac {1}{2}},1+2\mu ;z\right)$

$W_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}U\left(\mu -\kappa +{\frac {1}{2}},1+2\mu ;z\right)$

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

References

• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 13". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 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. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. See also chapter 14.
• Bateman, Harry (1953), Higher transcendental functions, vol. 1, McGraw-Hill.
• Brychkov, Yu.A.; Prudnikov, A.P. (2001) [1994], "Whittaker function", Encyclopedia of Mathematics, EMS Press.
• Daalhuis, Adri B. Olde (2010), "Whittaker function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Rozov, N.Kh. (2001) [1994], "Whittaker equation", Encyclopedia of Mathematics, EMS Press.
• Slater, Lucy Joan (1960), Confluent hypergeometric functions, Cambridge University Press, MR0107026.
• Whittaker, Edmund T. (1904), "An expression of certain known functions as generalized hypergeometric functions", Bulletin of the A. M. S., 10, Providence, R.I.: American Mathematical Society: 125–134