scholarly journals On the squares of Weber's parabolic cylinder functions and certain integrals connected with them

1934 ◽  
Vol 4 (1) ◽  
pp. 27-32 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Image Position

The parabolic cylinder functions Dn(x) and D−(n+1) (± ix) are defined byfor all values of n and x.

scholarly journals On certain infinite integrals involving Struve functions and parabolic cylinder functions

1946 ◽  
Vol 7 (4) ◽  
pp. 171-173 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Present Note ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Struve Functions ◽  
Image Position

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved thatFrom (1)follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.

scholarly journals On the Equation of the Parabolic Cylinder Functions

1913 ◽  
Vol 32 ◽  
pp. 2-14 ◽  
Author(s):  
Arch Milne
Keyword(s):  
Differential Equation ◽  
Positive Integer ◽  
Arbitrary Function ◽  
Parabolic Cylinder ◽  
Orthogonal Functions ◽  
Parabolic Cylinder Functions ◽  
Image Position

Hermite, in 1864 (Comptes Rendus, vol. 58) introduced into analysis the polynomials defined by the relationwhere n is a positive integer. He showed that they satisfied the differential equationthat they were orthogonal functions, and that an arbitrary function f(x) could be expanded in the form

Error bounds for the Liouville–Green (or WKB) approximation

1961 ◽  
Vol 57 (4) ◽  
pp. 790-810 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Error Bounds ◽  
Bessel Functions ◽  
Approximate Solutions ◽  
Parabolic Cylinder ◽  
Real Interval ◽  
Positive Parameter ◽  
Modified Bessel Functions ◽  
Elementary Functions ◽  
Parabolic Cylinder Functions ◽  
Image Position

ABSTRACTError bounds are derived and examined for approximate solutions in terms of elementary functions of the differential equationsin which u is a positive parameter, the functions f and p are free from singularities and p does not vanish. Bounds are also obtained for the remainder terms in the asymptotic expansions of the solutions in descending powers of u. The variable x ranges over a real interval, finite or infinite or over a region of the complex plane, bounded or unbounded.Applications are made to parabolic cylinder functions of large orders, and modified Bessel functions of large orders.

A method for the determination of converging factors, applied to the asymptotic expansions for the parabolic cylinder functions

1952 ◽  
Vol 48 (2) ◽  
pp. 243-254 ◽  
Author(s):  
J. C. P. Miller
Keyword(s):  
Infinite Series ◽  
Asymptotic Expansions ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Image Position

1. The method of converging factors, for hastening the convergence of slowly convergentseries and improving the accuracy of asymptotic expansions, was introduced by J. R. Airey and is well known to computers (see Airey(1) and Rosser(2)). The principle is as follows. It is required to compute a quantity which is expressed as an infinite seriesThe series may be either convergent or asymptotic and divergent.

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.

Two inequalities for parabolic cylinder functions

1961 ◽  
Vol 57 (4) ◽  
pp. 811-822 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Complex Variable ◽  
Asymptotic Theory ◽  
Upper Bounds ◽  
Parabolic Cylinder ◽  
Principal Solution ◽  
Second Order Differential Equations ◽  
Parabolic Cylinder Functions ◽  
Image Position

In this paper upper bounds are established for the principal solution of the differential equationand its derivative, for unrestricted values of the complex variable t and the complex parameter μ. The results may have little interest in their own right, but they are of great value in developing the asymptotic theory of linear second-order differential equations in a domain containing two turning points. Equation (1·1) is the simplest example of a differential equation of this type.

An infinite integral involving Bessel functions and parabolic cylinder functions

1937 ◽  
Vol 33 (2) ◽  
pp. 210-211 ◽  
Author(s):  
R. S. Varma
Keyword(s):  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Infinite Integral

The object of this paper is to evaluate an infinite integral involving Bessel functions and parabolic cylinder functions. The following two lemmas are required:Lemma 1. provided that R(m) > 0.

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