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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.
\end{displaymath} (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

\end{displaymath} (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.
\end{displaymath} (5)

Now letting
$\displaystyle u$ $\textstyle =$ $\displaystyle x+{b\over 2a}$ (6)
$\displaystyle du$ $\textstyle =$ $\displaystyle dx$ (7)

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

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

Equation (4) has the two standard forms
$\displaystyle y''-({\textstyle{1\over 4}}x^2+a)y$ $\textstyle =$ $\displaystyle 0$ (10)
$\displaystyle y''+({\textstyle{1\over 4}}x^2-a)y$ $\textstyle =$ $\displaystyle 0.$ (11)

For a general $a$, the Even and Odd solutions to (10) are
$\displaystyle y_1(x)$ $\textstyle =$ $\displaystyle e^{-x^2/4} \,{}_1F_1({\textstyle{1\over 2}}a+{\textstyle{1\over 4}}; {\textstyle{1\over 2}}; {\textstyle{1\over 2}}x^2)$ (12)
$\displaystyle y_2(x)$ $\textstyle =$ $\displaystyle xe^{-x^2/4} \,{}_1F_1({\textstyle{1\over 2}}a+{\textstyle{3\over 4}}; {\textstyle{3\over 2}}; {\textstyle{1\over 2}}x^2),$ (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}).
\end{displaymath} (14)

Abramowitz and Stegun (1972, p. 687) define standard solutions to (10) as
$\displaystyle U(a,x)$ $\textstyle =$ $\displaystyle \cos[\pi ({\textstyle{1\over 4}}+{\textstyle{1\over 2}}a)]Y_1-\sin[\pi({\textstyle{1\over 4}}+{\textstyle{1\over 2}}a)]Y_2$ (15)
$\displaystyle V(a,x)$ $\textstyle =$ $\displaystyle {\sin[\pi({\textstyle{1\over 4}}+{\textstyle{1\over 2}}a)]Y_1+\co...
...1\over 4}}+{\textstyle{1\over 2}}a)]Y_2\over \Gamma({\textstyle{1\over 2}}-a)},$ (16)

$\displaystyle Y_1$ $\textstyle \equiv$ $\displaystyle {1\over\sqrt{\pi}} {\Gamma({\textstyle{1\over 4}}-{\textstyle{1\over 2}}a)\over 2^{a/2+1/4}}y_1$  
  $\textstyle =$ $\displaystyle {1\over\sqrt{\pi}} {\Gamma({\textstyle{1\over 4}}-{\textstyle{1\o...
...2}}a+{\textstyle{1\over 4}}; {\textstyle{1\over 2}}; {\textstyle{1\over 2}}x^2)$ (17)
$\displaystyle Y_2$ $\textstyle \equiv$ $\displaystyle {1\over\sqrt{\pi}} {\Gamma({\textstyle{3\over 4}}-{\textstyle{1\over 2}}a)\over 2^{a/2+1/4}}y_2$  
  $\textstyle =$ $\displaystyle {1\over\sqrt{\pi}} {\Gamma({\textstyle{3\over 4}}-{\textstyle{1\o...
...}}a+{\textstyle{3\over 4}}; {\textstyle{3\over 2}}; {\textstyle{1\over 2}}x^2).$  

In terms of Whittaker and Watson's functions,

$\displaystyle U(a,x)$ $\textstyle =$ $\displaystyle D_{-a-1/2}(x)$ (19)
$\displaystyle V(a,x)$ $\textstyle =$ $\displaystyle {\Gamma({\textstyle{1\over 2}}+a)[\sin(\pi a)D_{-a-1/2}(x)+D_{-a-1/2}(-x)]\over\pi}.$ (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),
\end{displaymath} (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
\end{displaymath} (22)

D_\nu'(z)+{\textstyle{1\over 2}}z D_\nu(z)-\nu D_{\nu-1}(z)=0.
\end{displaymath} (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
\end{displaymath} (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},
\end{displaymath} (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
\end{displaymath} (26)

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


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.

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© 1996-9 Eric W. Weisstein