Whittaker Function

The Whittaker functions arise as solutions to the Whittaker differential equation. The linearly independent solutions to this equation are


and M_(k,-m)(z), where is a confluent hypergeometric function of the second kind and (z)_n is a Pochhammer symbol. In terms of confluent hypergeometric functions of the first and second kinds, these solutions are


(Abramowitz and Stegun 1972, p. 505; Whittaker and Watson 1990, pp. 339-351).

These functions are implemented in the Wolfram Language as WhittakerM[k, m, z] and WhittakerW[k, m, z], respectively.

Whittaker and Watson (1990, p. 340) define


whenever R[k-1/2-m]<=0 and k-1/2-m is not an integer.

A particular case is given by


for x>0 (Whittaker and Watson 1990, p. 341, adjusting the normalization of erfc(z) to conform to the modern convention).

The Whittaker functions are related to the parabolic cylinder functions through


When |argz|<3pi/2 and 2m is not an integer,


When |arg(-z)|<3pi/2 and 2m is not an integer,


Whittaker functions satisfy the recurrence relations


See also

Associated Laguerre Polynomial, Confluent Hypergeometric Function of the Second Kind, Cunningham Function, Kummer's Formulas, Schlömilch's Function

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Abramowitz, M. and Stegun, I. A. (Eds.). "Confluent Hypergeometric Functions." Ch. 13 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 503-515, 1972.Becker, P. A. "On the Integration of Products of Whittaker Functions with Respect to the Second Index." J. Math. Phys. 45, 761-773, 2004.Iyanaga, S. and Kawada, Y. (Eds.). "Whittaker Functions." Appendix A, Table 19.II in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1469-1471, 1980.Meijer, C. S. "Über die Integraldarstellungen der Whittakerschen Funktion W_(k,m)(z) und der Hankelschen und Besselschen Funktionen." Nieuw Arch. Wisk. 18, 35-57, 1936.Whittaker, E. T. "An Expression of Certain Known Functions as Generalised Hypergeometric Functions." Bull. Amer. Math. Soc. 10, 125-134, 1904.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

Referenced on Wolfram|Alpha

Whittaker Function

Cite this as:

Weisstein, Eric W. "Whittaker Function." From MathWorld--A Wolfram Web Resource.

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