Parabolic Cylinder Function

These functions are sometimes called Weber Functions. Whittaker and Watson (1990, p. 347) define the parabolic cylinder functions as solutions to the Weber Differential Equation

$${d^2D_n(z)\over dz^2}+(n+{\textstyle{1\over 2}}-{\textstyle{1\over 4}}z^2)D_n(z)=0.$$ (1)

The two independent solutions are given by $D_n(z)$ and $D_{-n-1}(ze^{i\pi/2})$, where

$D_n(z)=2^{n/2+1/4}z^{-1/2}W_{n/2+1/4,-1/4}({\textstyle{1\over 2}}z^2)$	(2)
$={\Gamma({\textstyle{1\over 2}})2^{n/2+1/4}z^{-1/2}\over \Gamma({\textstyle{1\o... ... \Gamma(-{\textstyle{1\over 2}}n)} M_{n/2+1/4,1/4} ({\textstyle{1\over 2}}z^2).$	(3)

Here, $W_{a,b}(z)$ is a Whittaker Function and $M_{a,b}(z)={}_1F_1(a;b;z)$ are Confluent Hypergeometric Functions.

Abramowitz and Stegun (1972, p. 686) define the parabolic cylinder functions as solutions to

$$y''+(ax^2+bx+c)=0.$$ (4)

This can be rewritten by Completing the Square,

$$y''+\left[{a\left({x+{b\over 2a}}\right)^2-{b^2\over 4a}+c}\right]y=0.$$ (5)

Now letting

$$u = x+{b\over 2a}$$ (6)

$$du = dx$$ (7)

gives

$${d^2y\over du^2}+(au^2+d)y=0$$ (8)

where

$$d\equiv {b^2\over 4a}+c.$$ (9)

Equation (4) has the two standard forms

$$y''-({\textstyle{1\over 4}}x^2+a)y = 0$$ (10)

$$y''+({\textstyle{1\over 4}}x^2-a)y = 0.$$ (11)

For a general $a$, the Even and Odd solutions to (10) are

$$y_1(x) = e^{-x^2/4} \,{}_1F_1({\textstyle{1\over 2}}a+{\textstyle{1\over 4}}; {\textstyle{1\over 2}}; {\textstyle{1\over 2}}x^2)$$ (12)

$$y_2(x) = xe^{-x^2/4} \,{}_1F_1({\textstyle{1\over 2}}a+{\textstyle{3\over 4}}; {\textstyle{3\over 2}}; {\textstyle{1\over 2}}x^2),$$ (13)

where ${}_1F_1(a;b;z)$ is a Confluent Hypergeometric Function. If $y(a,x)$ is a solution to (10), then (11) has solutions

$$y(\pm ia, xe^{\mp i\pi/4}), y(\pm ia, -xe^{\mp i\pi/4}).$$ (14)

Abramowitz and Stegun (1972, p. 687) define standard solutions to (10) as

$$U(a,x) = \cos[\pi ({\textstyle{1\over 4}}+{\textstyle{1\over 2}}a)]Y_1-\sin[\pi({\textstyle{1\over 4}}+{\textstyle{1\over 2}}a)]Y_2$$ (15)

$$V(a,x) = {\sin[\pi({\textstyle{1\over 4}}+{\textstyle{1\over 2}}a)]Y_1+\co... ...1\over 4}}+{\textstyle{1\over 2}}a)]Y_2\over \Gamma({\textstyle{1\over 2}}-a)},$$ (16)

where

$$Y_1 \equiv {1\over\sqrt{\pi}} {\Gamma({\textstyle{1\over 4}}-{\textstyle{1\over 2}}a)\over 2^{a/2+1/4}}y_1$$

$$= {1\over\sqrt{\pi}} {\Gamma({\textstyle{1\over 4}}-{\textstyle{1\o... ...2}}a+{\textstyle{1\over 4}}; {\textstyle{1\over 2}}; {\textstyle{1\over 2}}x^2)$$ (17)

$$Y_2 \equiv {1\over\sqrt{\pi}} {\Gamma({\textstyle{3\over 4}}-{\textstyle{1\over 2}}a)\over 2^{a/2+1/4}}y_2$$

$$= {1\over\sqrt{\pi}} {\Gamma({\textstyle{3\over 4}}-{\textstyle{1\o... ...}}a+{\textstyle{3\over 4}}; {\textstyle{3\over 2}}; {\textstyle{1\over 2}}x^2).$$ (18)

In terms of Whittaker and Watson's functions,

$$U(a,x) = D_{-a-1/2}(x)$$ (19)

$$V(a,x) = {\Gamma({\textstyle{1\over 2}}+a)[\sin(\pi a)D_{-a-1/2}(x)+D_{-a-1/2}(-x)]\over\pi}.$$ (20)

For Nonnegative Integer $n$, the solution $D_n$ reduces to

$$D_n(x)=2^{-n/2}e^{-x^2/4} H_n\left({x\over\sqrt{2}}\right)= e^{-x^2/4} {\rm He}_n(x),$$ (21)

where $H_n(x)$ is a Hermite Polynomial and He${}_n$ is a modified Hermite Polynomial.

The parabolic cylinder functions $D_\nu$ satisfy the Recurrence Relations

$$D_{\nu+1}(z)-zD_\nu(z)+\nu D_{\nu-1}(z)=0$$ (22)

$$D_\nu'(z)+{\textstyle{1\over 2}}z D_\nu(z)-\nu D_{\nu-1}(z)=0.$$ (23)

The parabolic cylinder function for integral $n$ can be defined in terms of an integral by

$$D_n(z)={1\over \pi} \int_0^\pi \sin(n\theta -z\sin\theta)\,d\theta$$ (24)

(Watson 1966, p. 308), which is similar to the Anger Function. The result

$$\int_{-\infty}^\infty D_m(x)D_n(x)\,dx =\delta_{mn} n!\sqrt{2\pi},$$ (25)

where $\delta_{ij}$ is the Kronecker Delta, can also be used to determine the Coefficients in the expansion

$$f(z)=\sum_{n=0}^\infty a_nD_n$$ (26)

as

$$a_n={1\over n!\sqrt{2\pi}} \int_{-\infty}^\infty D_n(t)f(t)\,dt.$$ (27)

For $\nu$ real,

$$\int_0^\infty [D_\nu(t)]^2\,dt =\pi^{1/2}2^{-3/2} {\phi_0({\... ... 2}}\nu)-\phi_0(-{\textstyle{1\over 2}}\nu)\over \Gamma(-\nu)}$$ (28)

(Gradshteyn and Ryzhik 1980, p. 885, 7.711.3), where $\Gamma(z)$ is the Gamma Function and $\phi_0(z)$ is the Polygamma Function of order 0.

See also Anger Function, Bessel Function, Darwin's Expansions, Hh Function, Struve Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Parabolic Cylinder Function.'' Ch. 19 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 685-700, 1972.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1979.

Iyanaga, S. and Kawada, Y. (Eds.). ``Parabolic Cylinder Functions (Weber Functions).'' Appendix A, Table 20.III in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1479, 1980.

Spanier, J. and Oldham, K. B. ``The Parabolic Cylinder Function $D_\nu(x)$.'' Ch. 46 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 445-457, 1987.

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.