scholarly journals Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders

1959 ◽  
Vol 63B (2) ◽  
pp. 131 ◽  
Author(s):  
F.W.J. Olver
Keyword(s):  
Asymptotic Expansions ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Uniform Asymptotic Expansions

Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations III. The confluent hypergeometric function U ( a , c , z ) for large | c |

1965 ◽  
Vol 284 (1399) ◽  
pp. 531-539 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Confluent Hypergeometric Function ◽  
Parabolic Cylinder ◽  
Exponential Integral ◽  
Parabolic Cylinder Function ◽  
Uniform Expansions ◽  
Kummer's Equation ◽  
Uniform Expansion ◽  
Special Case ◽  
Uniform Asymptotic Expansions

The methods introduced by Jorna (1964 a , b ) are applied to Kummer’s equation, and Green-type, transitional and uniform expansions derived for solutions of the type denoted in Slater (1960) by U ( a , c , z ) which are valid for large | c |. The subsidiary function in the uniform expansion is essentially a parabolic cylinder function of order ½ a . The general exponential integral is studied as a special case. Here the uniform expansion involves as subsidiary function the extensively tabulated error integral.

Derivation of Green-type, transitional, and uniform asymptotic expansions from differential equations II. Whittaker functions W k,m for large k , and for large | K 2 – m 2 |

1964 ◽  
Vol 281 (1384) ◽  
pp. 111-129 ◽  
Keyword(s):  
Differential Equations ◽  
Asymptotic Expansions ◽  
Mellin Transform ◽  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Whittaker Functions ◽  
Transform Method ◽  
Special Cases ◽  
Uniform Expansions ◽  
Uniform Asymptotic Expansions

The method for deriving Green-type asymptotic expansions from differential equations, introduced in I and illustrated therein by detailed calculations on modified Bessel functions, is applied to Whittaker functions W k,m , first for large k , and then for large |k 2 —m 2 |. Following the general theory of I, combination of this procedure with the Mellin transform method yields asymptotic expansions valid in transitional regions, and general uniform expansions. Weber parabolic cylinder and Poiseuille functions are examined as important special cases.

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.

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 Parabolic Cylinder Functions: Examples of Error Bounds for Asymptotic Expansions

2003 ◽  
Vol 01 (03) ◽  
pp. 265-288
Author(s):  
Nico M. Temme ◽  
Raimundas Vidunas
Keyword(s):  
Differential Equation ◽  
Integral Representation ◽  
Computer Algebra ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Parabolic Cylinder ◽  
Numerical Computations ◽  
Parabolic Cylinder Functions ◽  
Uniform Expansions ◽  
Special Case

Several asymptotic expansions of parabolic cylinder functions are discussed and error bounds for remainders in the expansions are presented. In particular, Poincaré-type expansions for large values of the argument z and uniform expansions for large values of the parameter are considered. It is shown how expansions can be derived by using the differential equation, and, for a special case, how an integral representation can be used. The expansions are based on those given in Olver (1959) and on modifications of these expansions given in Temme (2000). Computer algebra techniques are used for obtaining representations of the bounds and for numerical computations.

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