Whittaker Function
The Whittaker functions arise as solutions to the Whittaker differential equation. The linearly independent solutions to this equation are
![画像:z^(1/2+m)e^(-z/2)[1+(1/2+m-k)/(1!(2m+1))z+((1/2+m-k)(3/2+m-k))/(2!(2m+1)(2m+2))z^2+...]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/WhittakerFunction/Inline6.svg){kind=link}
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
| W_(k,m)(z)=(e^(-z/2)z^k)/(Gamma(1/2-k+m))×int_0^inftyt^(-k-1/2+m)(1+t/z)^(k-1/2+m)e^(-t)dt |
(5)
|
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,
| [画像: W_(k,m)(z)=(Gamma(-2m))/(Gamma(1/2-m-k))M_(k,m)(z)+(Gamma(2m))/(Gamma(1/2+m-k))M_(k,-m)(z). ] |
(8)
|
When |arg(-z)|<3pi/2 and 2m is not an integer,
| W_(-k,m)(-z)=(Gamma(-2m))/(Gamma(1/2-m-k))M_(-k,m)(-z)+(Gamma(2m))/(Gamma(1/2+m+k))M_(-k,-m)(-z). |
(9)
|
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 FunctionExplore with Wolfram|Alpha
References
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 FunctionCite this as:
Weisstein, Eric W. "Whittaker Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WhittakerFunction.html