math lessons - Parabolic cylinder functions

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 can be brought into two distinct forms (A) and (B) by change of variable, to

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

and

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

f(a,z)

is a solution, then so are

f(a,−z), f(−a,iz) and f(−a,−iz).

f(a,z)

is a solution of equation (A), then

f(−ia,ze^iπ/4)

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

f(−ia,−ze^iπ/4), f(ia,−ze^−iπ/4) and f(ia,ze^−iπ/4)

are also solutions of (B).

Solutions

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

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

and

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

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

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.

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables 1972, Dover: New York. (See chapter 19.)