An extension of a theorem of Mehler's on Hermite polynomials

1945 ◽  
Vol 41 (1) ◽  
pp. 12-15 ◽  
Author(s):  
W. F. Kibble
Keyword(s):  
Hermite Polynomial ◽  
Hermite Polynomials ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Function ◽  
Cylinder Function ◽  
Image Position

It was shown by Mehler (1866) thatwhere Hk(x) denotes the Hermite polynomial(Hermite, 1864a, b), which can be expressed in terms of Weber's parabolic cylinder function (Whittaker, 1903). The series is convergent if | ρ | < 1, and divergent if | ρ | > 1. If ρ = 1 and x = y = 0 the series is divergent, and Hille's work (1938) shows that it will therefore be divergent for all real or complex values, except possibly real positive values, of x and y.

scholarly journals Expansions in terms of Parabolic Cylinder Functions

1948 ◽  
Vol 8 (2) ◽  
pp. 50-65 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Wave Equation ◽  
Positive Integer ◽  
Plane Wave ◽  
Arbitrary Function ◽  
Hermite Polynomials ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Image Position

When the plane wave equation is expressed in terms of parabolic co-ordinates x, y, the variables are separable, and the elementary solutions have the formwhere x, y, μ are real. In this context, therefore, the functions Dν (z) which are directly significant are those where amp z = ± π/4 and ν + ½ is purely imaginary, rather than those where z is real and ν is a positive integer. The expansion of an arbitrary function in terms of the latter sort of D-function (substantially, in terms of Hermite polynomials) is well known. This paper is concerned with the expansion in terms of the former sort of D-function.

The asymptotic behaviour of Pearcey’s integral for complex variables

1991 ◽  
Vol 432 (1886) ◽  
pp. 391-426 ◽  
Keyword(s):  
Asymptotic Behaviour ◽  
Oscillatory Integral ◽  
Complex Variables ◽  
Stationary Points ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Function ◽  
Cylinder Function ◽  
Arbitrary Complex ◽  
The Rich ◽  
Asymptotic Character

A deeper understanding of the rich structure of the canonical form of the oscillatory integral describing the cusp diffraction catastrophe, generally known as Pearcey’s integral P '( X , Y ), can be obtained by considering its analytic continuation to arbitrary complex variables X and Y . A new integral representation for P '( X , Y ) is given in the form of a contour integral involving a Weber parabolic cylinder function whose order is the variable of integration. It is shown how the asymptotics of P '( X , Y ) may be obtained from this representation for complex X and Y when either | X | or | Y | → ∞, without reference to the usual stationary points of the integrand. For the case | X | → ∞, Y finite the full asymptotic expansion of P '( X , Y ) is derived and its asymptotic character is found to be either exponentially large or algebraic in certain sectors of the X - plane. The case | Y | → ∞ , X finite is complicated by the presence of exponentially small subdominant terms in certain sectors of the Y - plane, and only the first terms in the expansion are given. The asymptotic behaviour of P '( X , Y ) on the caustic Y 2 + (2/3 X ) 3 = 0 is also obtained from the new representation and is shown to agree with recent results of D. Kaminski. The various properties of the Weber parabolic cylinder function required in this paper are collected together in the Appendix.

The Parabolic Cylinder Function D v (x)

2008 ◽  
pp. 471-484
Author(s):  
Keith B. Oldham ◽  
Jan C. Myland ◽  
Jerome Spanier
Keyword(s):  
Parabolic Cylinder ◽  
Parabolic Cylinder Function ◽  
Cylinder Function
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