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Parabolic cylinder function

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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=48106
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