scholarly journals Uniform asymptotic expansions for solutions of the parabolic cylinder and Weber equations

2021 ◽  
pp. 69-107
Author(s):  
T. M. Dunster
Keyword(s):  
Asymptotic Expansions ◽  
Parabolic Cylinder ◽  
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.

VACUUM ENERGY IN SPHERICAL DOMAINS: WKB AND UNIFORM ASYMPTOTIC EXPANSIONS

1996 ◽  
Vol 11 (22) ◽  
pp. 4129-4146 ◽  
Author(s):  
AUGUST ROMEO
Keyword(s):  
Zeta Function ◽  
Asymptotic Expansions ◽  
Zero Point ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Point Energy ◽  
Dimensional Dependence ◽  
Spherical Domains ◽  
Spectral Zeta Function ◽  
Uniform Asymptotic Expansions

We evaluate the finite part of the regularized zero-point energy for a massless scalar field confined in the interior of a D-dimensional spherical region. While some insight is offered into the dimensional dependence of the WKB approximations by examining the residues of the spectral-zeta-function poles, a mode-sum technique based on an integral representation of the Bessel spectral zeta function is applied with the help of uniform asymptotic expansions (u.a.e.’s).

Uniform Asymptotic Expansions of Certain Integrals

1967 ◽  
Vol 15 (6) ◽  
pp. 1422-1433 ◽  
Author(s):  
Norman Bleistein ◽  
Richard A. Handelsman
Keyword(s):  
Asymptotic Expansions ◽  
Uniform Asymptotic Expansions

Uniform asymptotic expansions for oblate spheroidal functions I: positive separation parameter λ

1992 ◽  
Vol 121 (3-4) ◽  
pp. 303-320 ◽  
Author(s):  
T. M. Dunster
Keyword(s):  
Wave Equation ◽  
Half Plane ◽  
Positive Constant ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Small Positive Constant ◽  
Separation Parameter ◽  
Oblate Spheroidal ◽  
Uniform Asymptotic Expansions

SynopsisUniform asymptotic expansions are derived for solutions of the spheroidal wave equation, in the oblate case where the parameter µ is real and nonnegative, the separation parameter λ is real and positive, and γ is purely imaginary (γ = iu). As u →∞, three types of expansions are derived for oblate spheroidal functions, which involve elementary, Airy and Bessel functions. Let δ be an arbitrary small positive constant. The expansions are uniformly valid for λ/u2 fixed and lying in the interval (0,2), and for λ / u2when 0<λ/u2 < 1, and when 1 = 1≦λ/u2 < 2. The union of the domains of validity of the various expansions cover the half- plane arg (z)≦ = π/2.

scholarly journals Uniform asymptotic expansions for the Whittaker functions M κ , μ ( z ) and W κ , μ ( z ) with μ large

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210360
Author(s):  
T. M. Dunster
Keyword(s):  
Analytic Continuation ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Whittaker Functions ◽  
Airy Functions ◽  
Elementary Functions ◽  
Confluent Hypergeometric Functions ◽  
Uniform Asymptotic Expansions

Uniform asymptotic expansions are derived for Whittaker’s confluent hypergeometric functions M κ , μ ( z ) and W κ , μ ( z ) , as well as the numerically satisfactory companion function W − κ , μ ( z   e − π i ) . The expansions are uniformly valid for μ → ∞ , 0 ≤ κ / μ ≤ 1 − δ < 1 and 0 ≤ arg ⁡ ( z ) ≤ π . By using appropriate connection and analytic continuation formulae, these expansions can be extended to all unbounded non-zero complex z . The approximations come from recent asymptotic expansions involving elementary functions and Airy functions, and explicit error bounds are either provided or available.

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