Weber function, Weber–Hermite function
A solution of the differential equation
 | (*) |
which is obtained as a result of separating the variables in the wave equation 
in parabolic cylindrical coordinates (cf. Parabolic coordinates). Ordinarily one uses the solution
where 
is the confluent hypergeometric function. The functions 
and 
also satisfy equation (*). The functions 
and 
are linearly independent for arbitrary 
, and 
and 
are linearly independent for 
. The parabolic cylinder functions are entire functions of 
. The function 
is real for real 
and 
.
The differentiation formulas are, 
:
The recurrence formulas are:
Asymptotically, for fixed 
and 
, as 
, one has
and, for bounded 
and 
, as 
, one has
Parabolic cylinder functions are related to other functions as follows (
): To the Hermite polynomials by
to the probability integral (error function) by
and to the Bessel functions by
References
| [1] | H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953) |
| [2] | J.C.P. Miller, "Giving solutions of the differential equation  , tables of Weber parabolic cylinder functions" , H.M. Stationary Office (1955) |
How to Cite This Entry:
Parabolic cylinder function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_cylinder_function&oldid=13870
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
