Weber equation

A second-order ordinary linear differential equation:

$$y''+\left(\nu+\frac12-\frac{x^2}{4}\right)y=0,\quad\nu=\text{const},\tag{*}$$

in which the point $x=\infty$ is strongly singular (cf. Singular point). An equation of this type was first studied by H. Weber in potential theory in connection with the parabolic cylinder [1]; it is the result of separation of variables for the Laplace equation in parabolic coordinates. The substitution $y=x^{-1/2}w$, $z=x^2/2$ converts the Weber equation to the Whittaker equation. It is a special case of a confluent hypergeometric equation. The substitution $y=u\exp(-x^2/4)$ converts Weber's equation into

$$u''-xu'+\nu u=0.$$

Solutions of equation (*) are known as parabolic cylinder functions or as Weber–Hermite functions. In particular, if $\nu$ is a non-negative integer, equation (*) is satisfied by the function

$$y=\exp(-x^2/4)H_\nu(x),$$

where $H_\nu(x)$ is the Hermite polynomial (cf. Hermite polynomials) [2], [3], [4].

References

[1]	H.F. Weber, "Ueber die Integration der partiellen Differentialgleichung $\partial^2u/\partial x^2+\partial^2u/\partial y^2+k^2 u = 0$" Math. Ann. , 1 (1869) pp. 1–36
[2]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 2
[3]	H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[4]	E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)