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

This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabolic cylindrical coordinates.

The above equation may be brought into two distinct forms (A) and (B) by completing the square and rescaling , called H. F. Weber's equations: and

is a solution of equation ( is a solution of ( are also solutions of ().

Solutions

There are independent even and odd solutions of the form (). These are given by (following the notation of Abramowitz and Stegun (1965)): is the confluent hypergeometric function.

Other pairs of independent solutions may be formed from linear combinations of the above solutions.

The function approaches zero for large values of and , while diverges for large values of positive real

For half-integer values of a, these (that is, U and V) can be re-expressed in terms of Hermite polynomials; alternatively, they can also be expressed in terms of Bessel functions.

The functions U and V can also be related to the functions (a notation dating back to Whittaker (1902)) that are themselves sometimes called parabolic cylinder functions:

Function was introduced by Whittaker and Watson as a solution of eq.~(. It can be expressed in terms of confluent hypergeometric functions as

Power series for this function have been obtained by Abadir (1993).

Parabolic Cylinder U(a,z) function

Integral representation

Integrals along the real line, The fact that these integrals are solutions to equation() can be easily checked by direct substitution.

Derivative

Recurrence relation

Asymptotic expansion

Power series

Values at z=0

Parabolic cylinder Dν(z) function

is the solution to the Weber differential equation-function.

Integral representation

Asymptotic expansion

is a non-negative integer this series terminates and turns into a polynomial, namely the Hermite polynomial

Connection with quantum harmonic oscillator

function appears naturally in the Schrödinger equation for the one-dimensional quantum harmonic oscillator is the reduced Planck constant

References