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.

Asymptotic expansions and converging factors IV. Confluent hypergeometric, parabolic cylinder, modified Bessel, and ordinary Bessel functions

1959 ◽  
Vol 249 (1257) ◽  
pp. 270-283 ◽  
Keyword(s):  
Exponential Type ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Modified Bessel Functions ◽  
Confluent Hypergeometric Functions ◽  
Special Cases ◽  
Parabolic Cylinder Functions

A theory of confluent hypergeometric functions is developed, based upon the methods described in the first three papers (I, II and III) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘ basic converging factors ’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of exponential-type integrals, parabolic cylinder functions, modified Bessel functions, and ordinary Bessel functions.

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 Integrals involving products of modified Bessel functions of the second kind

1963 ◽  
Vol 6 (2) ◽  
pp. 70-74 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Positive Integer ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Image Position

It is proposed to establish the two following integrals.where n is a positive integer, x is real and positive, μi and ν are complex, and Δ (n; a) represents the set of parameterswhere n is a positive integer and x is real and positive.

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.

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.

An extension of the method of steepest descents

1957 ◽  
Vol 53 (3) ◽  
pp. 599-611 ◽  
Author(s):  
C. Chester ◽  
B. Friedman ◽  
F. Ursell
Keyword(s):  
Complex Variable ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Airy Function ◽  
Saddle Points ◽  
Positive Parameter ◽  
Implicit Relation ◽  
Image Position ◽  
Uniform Expansions ◽  
Uniformly Regular

ABSTRACTIn the integralthe functions g(z), f(z, α) are analytic functions of their arguments, and N is a large positive parameter. When N tends to ∞, asymptotic expansions can usually be found by the method of steepest descents, which shows that the principal contributions arise from the saddle points, i.e. the values of z at which ∂f/∂z = 0. The position of the saddle points varies with α, and if for some a (say α = 0) two saddle points coincide (say at z = 0) the ordinary method of steepest descents gives expansions which are not uniformly valid for small α. In the present paper we consider this case of two nearly coincident saddle points and construct uniform expansions as follows. A new complex variable u is introduced by the implicit relationwhere the parameters ζ(α), A(α) are determined explicitly from the condition that the (u, z) transformation is uniformly regular near z = 0, α = 0 (see § 2 below). We show that with these values of the parameters there is one branch of the transformation which is uniformly regular. By taking u on this branch as a new variable of integration we obtain for the integral uniformly asymptotic expansions of the formwhere Ai and Ai′ are the Airy function and its derivative respectively, and A(α), ζ(α) are the parameters in the transformation. The application to Bessel functions of large order is briefly described.

scholarly journals Liouville-Green expansions of exponential form, with an application to modified Bessel functions

2019 ◽  
Vol 150 (3) ◽  
pp. 1289-1311 ◽  
Author(s):  
T. M. Dunster
Keyword(s):  
Error Bounds ◽  
Asymptotic Expansions ◽  
Turning Point ◽  
Bessel Functions ◽  
Alternative Form ◽  
Modified Bessel Functions ◽  
Exponential Form ◽  
Positive Order ◽  
Second Order Differential Equations ◽  
Computable Error Bounds

AbstractLinear second order differential equations of the form d2w/dz2 − {u2f(u, z) + g(z)}w = 0 are studied, where |u| → ∞ and z lies in a complex bounded or unbounded domain D. If f(u, z) and g(z) are meromorphic in D, and f(u, z) has no zeros, the classical Liouville-Green/WKBJ approximation provides asymptotic expansions involving the exponential function. The coefficients in these expansions either multiply the exponential or in an alternative form appear in the exponent. The latter case has applications to the simplification of turning point expansions as well as certain quantum mechanics problems, and new computable error bounds are derived. It is shown how these bounds can be sharpened to provide realistic error estimates, and this is illustrated by an application to modified Bessel functions of complex argument and large positive order. Explicit computable error bounds are also derived for asymptotic expansions for particular solutions of the nonhomogeneous equations of the form d2w/dz2 − {u2f(z) + g(z)}w = p(z).

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

Deductions from some artificial refractive index profiles based on the elementary functions

1977 ◽  
Vol 81 (1) ◽  
pp. 121-132 ◽  
Author(s):  
J. Heading
Keyword(s):  
Refractive Index ◽  
Bessel Functions ◽  
Approximate Solutions ◽  
Elementary Functions ◽  
Stokes Phenomenon ◽  
New Device ◽  
Second Order Differential Equations ◽  
Single Transition ◽  
Transcendental Functions ◽  
Transition Points

AbstractThe familiar W.K.B.J. method expresses in terms of elementary functions approximate solutions of second order differential equations in normal form that otherwise have no analytical solutions expressible in terms of the standard elementary or transcendental functions. Proofs of the connexion formulae required to trace these approximate solutions across transition points are usually derived by comparison with known functions such as the Airy integral or Bessel functions, or by an appeal to the Stokes phenomenon in the complex plane. A new device is developed for the synthesis of refractive index profiles with transition points, and yet yielding solutions in terms of the elementary functions. These are then used to derive the W.K.B.J. connexion formula by means of a novel limiting process based on solutions considered only along the real z-axis. The method is obviously more complicated than the more usual approach, but contains special features throwing new light on the connexion formula across a single transition point.

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