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Whittaker Function

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

M_(k,m)(z)=z^(m+1/2)e^(-z/2)sum_(n=0)^(infty)((m-k+1/2)_n)/(n!(2m+1)_n)z^n
(1)
=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+...]
(2)

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

M_(k,m)(z)=e^(-z/2)z^(m+1/2)_1F_1(1/2+m-k,1+2m;z)
(3)
W_(k,m)(z)=e^(-z/2)z^(m+1/2)U(1/2+m-k,1+2m;z)
(4)

(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

erfc(x)=(e^(-x^2/2))/(sqrt(pix))W_(-1/4,1/4)(x^2)
(6)

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

D_n(z)=1/(sqrt(z))2^(n/2+1/4)W_(n/2+1/4,-1/4)(1/2z^2).
(7)

When |argz|<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).
(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

W_(k,m)(z)=z^(1/2)W_(k-1/2,m-1/2)(z)+(1/2-k+m)W_(k-1,m)(z)
(10)
W_(k,m)(z)=z^(1/2)W_(k-1/2,m+1/2)(z)+(1/2-k-m)W_(k-1,m)(z)
(11)
zW_(k,m)^'(z)=(k-1/2z)W_(k,m)(z)-[m^2-(k-1/2)^2]W_(k-1,m)(z).
(12)

See also

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

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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.

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Whittaker Function

Cite this as:

Weisstein, Eric W. "Whittaker Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WhittakerFunction.html

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