User:PAR/Test

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In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation

\frac{d^2f}{dz^2} + \left(az^2+bz+c\right)f=0

This equation is found, for example, when the technique of separation of variables is used on differential equations which are expressed in parabolic cylindrical coordinates.

The above equation may be brought into two distinct forms (A) and (B) by performing the following sequence of operations: complete the square in the term in the brackets to obtain,

\frac{d^2f}{dz^2} + \left(a{\left(z-\frac{b}{2a}\right)}^2 - \frac{b^2}{4a}+c\right)f=0

And perform the change of variable,

z' = \sqrt{a}\left(z-\frac{b}{2a}\right)\,\,\,\,\,\,\,\,\,z' = i\sqrt{a}\left(z-\frac{b}{2a}\right)

Then, retaining the original symbol z for the independent variable and lumping the constant terms into a for simplicity, the two equations are written as:

\frac{d^2f}{dz^2} - \left(\frac{z^2}{4}+a\right)f=0 (A)

and

\frac{d^2f}{dz^2} + \left(\frac{z^2}{4}-a\right)f=0 (B)

If

f(a,z)\,

is a solution, then so are

f(a,-z), f(-a,iz)\, and f(-a,-iz)\,.

If

f(a,z)\,

is a solution of equation (A), then

f(-ia,ze^{i\pi/4})\,

is a solution of (B), and, by symmetry,

f(-ia,-ze^{i\pi/4}), f(ia,-ze^{-i\pi/4})\, and f(ia,ze^{-i\pi/4})\,

are also solutions of (B).

[edit] Solutions

There are independent even and odd solutions of the form (A). These are given by (following the notation of Abramowitz and Stegun):

y_1(a;z) = \exp(-z^2/4) \;_1F_1 \left(\frac{a}{2}+\frac{1}{4}; \; \frac{1}{2}\; ; \; \frac{z^2}{2}\right)\,\,\,\,\,\, (\mathrm{even})

and

y_2(a;z) = z\exp(-z^2/4) \;_1F_1 \left(\frac{a}{2}+\frac{3}{4}; \; \frac{3}{2}\; ; \; \frac{z^2}{2}\right)\,\,\,\,\,\, (\mathrm{odd})

where \;_1F_1 (a;b;z)=M(a;b;z) is the confluent hypergeometric function.

Other pairs of independent solutions may be formed from linear combinations of the above solutions (see Abramowitz and Stegun). One such pair is based upon their behavior at infinity:

U(a,z)=\frac{1}{2^\xi\sqrt{\pi}} \left[ \cos(\xi\pi)\Gamma(1/2-\xi)\,y_1(a,z) -\sqrt{2}\sin(\xi\pi)\Gamma(1-\xi)\,y_2(a,z) \right]
V(a,z)=\frac{1}{2^\xi\sqrt{\pi}\Gamma[1/2-a]} \left[ \sin(\xi\pi)\Gamma(1/2-\xi)\,y_1(a,z) +\sqrt{2}\cos(\xi\pi)\Gamma(1-\xi)\,y_2(a,z) \right]

where

\xi=\frac{1}{4}+\frac{a}{2}

U(a,z) approaches zero for large values of |z|  and |arg(z)|<π/2, while V(a,z) diverges for large values of positive real z .

\lim_{|z|\rightarrow\infty}U(a,z)=e^{-x^2/2}x^{-a-1/2}\,\,\,\,(\mathrm{for}\,|\arg(z)|<\pi/2)

and

\lim_{|z|\rightarrow\infty}V(a,z)=\sqrt{\frac{2}{\pi}}e^{x^2/2}x^{a-1/2}\,\,\,\,(\mathrm{for}\,\arg(z)=0)

For half-integer values of a, these can be re-expressed in terms of Hermite polynomials; alternately, they can also be expressed in terms of Bessel functions.

U is essentially Whittaker's function (Dn)(z)), which is closely related to the Hermite function (ψn(z)) which, for non-negative integer values of n , are related to the (physicist's) Hermite polynomials (Hn)(z)):

U(-n-1/2,z)=D_n(z) = \psi_n(z/\sqrt{2})=\frac{1}{\sqrt{n!2^n\sqrt{\pi}}}\,e^{-z^2/2}H_n(z)\,

Orthogonality for non-negative integer n, expressed in terms of Whittaker's function

\int_0^\infty D_m(z)D_n(z) dz = n!\sqrt{2\pi}\,\,\delta_{mn}

[edit] References

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