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Parabolic Cylinder Function

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The parabolic cylinder functions are a class of functions sometimes called Weber functions. There are a number of slightly different definitions in use by various authors.

Whittaker and Watson (1990, p. 347) define the parabolic cylinder functions D_nu(z) as solutions to the Weber differential equation

y^('')(z)+(nu+1/2-1/4z^2)y(z)=0.
(1)

The two independent solutions are given by y=D_nu(z) and y=D_(-nu-1)(iz), where

D_nu(z)=2^(nu/2+1/4)z^(-1/2)W_(nu/2+1/4,-1/4)(1/2z^2)
(2)
=(2^(nu/2)e^(-z^2/4)(-iz)^(1/4)(iz)^(1/4))/(sqrt(z))U(-1/2nu,1/2,1/2z^2),
(3)

which, in the right half-plane R[z]>0, is equivalent to

D_nu(z)=2^(nu/2)e^(-z^2/4)U(-1/2nu,1/2,1/2z^2),
(4)

where W_(k,m)(z) is the Whittaker function (Whittaker and Watson 1990, p. 347; Gradshteyn and Ryzhik 2000, p. 1018) and U(a,b,z) is a confluent hypergeometric function of the first kind.

This function is implemented in the Wolfram Language as ParabolicCylinderD[nu, z].

ParabolicCylinderD

For nu a nonnegative integer n, the solution D_n reduces to

D_n(x)=2^(-n/2)e^(-x^2/4)H_n(x/(sqrt(2)))
(5)
=e^(-x^2/4)He_n(x),
(6)

where H_n(x) is a Hermite polynomial and He_n is a modified Hermite polynomial. Special cases include

D_(-1)(z)=e^(z^2/4)sqrt(pi/2)erfc(z/(sqrt(2)))
(7)
D_(-1/2)(z)=sqrt(z/(2pi))K_(1/4)(1/4z^2)
(8)

for R[z]>0, where K_nu(z) is an modified Bessel function of the second kind.

ParabolicCylinderDReIm
ParabolicCylinderDContours

Plots of the function D_1(z) in the complex plane are shown above.

The parabolic cylinder functions D_nu satisfy the recurrence relations

D_(nu+1)(z)-zD_nu(z)+nuD_(nu-1)(z)=0
(9)
D_nu^'(z)+1/2zD_nu(z)-nuD_(nu-1)(z)=0.
(10)

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

D_n(z)=1/piint_0^pisin(ntheta-zsintheta)dtheta
(11)

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

int_(-infty)^inftyD_m(x)D_n(x)dx=delta_(mn)n!sqrt(2pi),
(12)

where delta_(ij) is the Kronecker delta, can also be used to determine the coefficients in the expansion

f(z)=sum_(n=0)^inftya_nD_n
(13)

as

a_n=1/(n!sqrt(2pi))int_(-infty)^inftyD_n(t)f(t)dt.
(14)

For nu real,

int_0^infty[D_nu(t)]^2dt=pi^(1/2)2^(-3/2)(phi_0(1/2-1/2nu)-phi_0(-1/2nu))/(Gamma(-nu))
(15)

(Gradshteyn and Ryzhik 2000, p. 885, 7.711.3), where Gamma(z) is the gamma function and phi_0(z) is the polygamma function of order 0.

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

y^('')+(ax^2+bx+c)y=0,
(16)

sometimes called the parabolic cylinder differential equation (Zwillinger 1995, p. 414; Zwillinger 1997, p. 126). This can be rewritten by completing the square,

y^('')+[a(x+b/(2a))^2-(b^2)/(4a)+c]y=0.
(17)

Now letting

u=x+b/(2a)
(18)
du=dx
(19)

gives

(d^2y)/(du^2)+(au^2+d)y=0
(20)

where

d=(b^2)/(4a)+c.
(21)

Equation (◇) has the two standard forms

y^('')-(1/4x^2+a)y=0
(22)
y^('')+(1/4x^2-a)y=0.
(23)

For a general a, the even and odd solutions to (◇) are

y_1(x)=e^(-x^2/4)_1F_1(1/2a+1/4;1/2;1/2x^2)
(24)
y_2(x)=xe^(-x^2/4)_1F_1(1/2a+3/4;3/2;1/2x^2),
(25)

where _1F_1(a;b;z) is a confluent hypergeometric function of the first kind. If y(a,x) is a solution to (22), then (23) has solutions

y(+/-ia,xe^(∓ipi/4)),y(+/-ia,-xe^(∓ipi/4)).
(26)

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

U(a,x)=cos[pi(1/4+1/2a)]Y_1-sin[pi(1/4+1/2a)]Y_2
(27)
V(a,x)=(sin[pi(1/4+1/2a)]Y_1+cos[pi(1/4+1/2a)]Y_2)/(Gamma(1/2-a)),
(28)
Y_1=1/(sqrt(pi))(Gamma(1/4-1/2a))/(2^(a/2+1/4))y_1
(29)
=1/(sqrt(pi))(Gamma(1/4-1/2a))/(2^(a/2+1/4))e^(-x^2/4)_1F_1(1/2a+1/4;1/2;1/2x^2)
(30)
Y_2=1/(sqrt(pi))(Gamma(3/4-1/2a))/(2^(a/2-1/4))y_2
(31)
=1/(sqrt(pi))(Gamma(3/4-1/2a))/(2^(a/2-1/4))xe^(-x^2/4)_1F_1(1/2a+3/4;3/2;1/2x^2).
(32)

In terms of Whittaker and Watson's functions,

U(a,x)=D_(-a-1/2)(x)
(33)
V(a,x)=(Gamma(1/2+a)[sin(pia)D_(-a-1/2)(x)+D_(-a-1/2)(-x)])/pi.
(34)

See also

Anger Function, Bessel Function, Darwin's Expansions, Hh Function, Parabolic Cylindrical Coordinates, Parabolic Cylinder Differential Equation, Struve Function, Whittaker Function

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References

Abramowitz, M. and Stegun, I. 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. "Parabolic Cylinder Functions" and "Parabolic Cylinder Functions D_p(z)" §7.7 and 9.24-9.25 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 835-842, 1018-1021, 2000.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.Jeffreys, H. and Jeffreys, B. S. "The Parabolic Cylinder, Hermite, and Hh Functions" et seq. §23.08-23.081 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 620-627, 1988.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. "The Parabolic Cylinder Function." §16.5 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 347-348, 1990.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 414, 1995.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 126, 1997.

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Parabolic Cylinder Function

Cite this as:

Weisstein, Eric W. "Parabolic Cylinder Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParabolicCylinderFunction.html

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