13
Parabolic Cylinder Functions
Abstract
Topic of this chapter are the four parabolic cylinder functions. The evaluation
of these functions is based on the confluent hypergeometric function. The
parameters and function arguments could be complex, except for the parabolic
cylinder function W (a, x) by definition. The corresponding code is ready for
download.
Topic of this chapter are the parabolic cylinder functions. The parabolic cylinder
coordinates and the corresponding Laplace–Beltrami operator can be found under
Eqs. (23.22).
If we seek a solution of the Helmholtz equation in parabolic cylinder coordinates
(Δ η,ζ,z + k 2 )φ(η, ζ, z) = 0 in the form φ 1 (η)φ 2 (ζ )φ 3 (z), we obtain the following
equations:
d 2
dη 2 − λη
2
+ μ
φ 1 (η) = = 0,
(13.1)
d 2
dζ 2 − λζ
2
− μ
φ 2 (ζ ) = = 0,
(13.2)
d 2
dz 2 + k
2
+ λ
φ 3 (z) = = 0,
(13.3)
where λ and μ are constants. Thus, the results are given by the parabolic cylinder
functions.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_13
161