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Whittaker Differential Equation

(d^2u)/(dz^2)+(du)/(dz)+(k/z+(1/4-m^2)/(z^2))u=0.
(1)

Let u=e^(-z/2)W_(k,m)(z), where W_(k,m)(z) denotes a Whittaker function. Then (1) becomes

d/(dz)(-1/2e^(-z/2)W+e^(-z/2)W^')+(-1/2e^(-z/2)W+e^(-z/2)W^') +(k/z+(1/4-m^2)/(z^2))e^(-z/2)W=0.
(2)

Rearranging,

(1/4e^(-z/2)W-1/2e^(-z/2)W^'-1/2e^(-z/2)W^'+e^(-z/2)W^(''))p +(-1/2e^(-z/2)W+e^(-z/2)W^')+(k/z+(1/4-m^2)/(z^2))e^(-z/2)W=0
(3)
-1/4e^(-z/2)W+e^(-z/2)W^('')+(k/z+(1/4-m^2)/(z^2))e^(-z/2)W=0,
(4)

so

W^('')+(-1/4+k/z+(1/4-m^2)/(z^2))W=0,
(5)

where W^'=dW/dz (Abramowitz and Stegun 1972, p. 505; Zwillinger 1997, p. 128). The solutions are known as Whittaker functions. Replacing W(z) by y(x), the solutions can also be written in the form

y=e^(-x/2)x^(m+1/2)[C_1U(1/2-k+m,2m+1,x)+C_2L_(-1/2+k-m)^(2m)(x)],
(6)

where U(a,b,z) is a confluent hypergeometric function of the second kind and L_n^a(x) is a generalized Laguerre polynomial.

See also

Whittaker Function

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 505, 1972.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 128, 1997.

Referenced on Wolfram|Alpha

Whittaker Differential Equation

Cite this as:

Weisstein, Eric W. "Whittaker Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WhittakerDifferentialEquation.html

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