Uniform Asymptotic Expansions for Whittaker’s Confluent Hypergeometric Functions

1989 ◽  
Vol 20 (3) ◽  
pp. 744-760 ◽  
Author(s):  
T. M. Dunster
Keyword(s):  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Uniform Asymptotic Expansions

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.

UNIFORM ASYMPTOTIC EXPANSIONS FOR HYPERGEOMETRIC FUNCTIONS WITH LARGE PARAMETERS I

2003 ◽  
Vol 01 (01) ◽  
pp. 111-120 ◽  
Author(s):  
A. B. OLDE DAALHUIS
Keyword(s):  
Asymptotic Expansion ◽  
Hypergeometric Function ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Gauss Hypergeometric Function ◽  
Uniform Asymptotic Expansions

In this paper, we obtain an asymptotic expansion for the Gauss hypergeometric function 2F1(a, b - λ; c + λ; -z), as |λ| → ∞. The expansion holds for fixed values of a, b, and c, and is uniformly valid for z in the domain | ph z| < π.

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.

Uniform asymptotic expansions for hypergeometric functions with large parameters IV

2014 ◽  
Vol 12 (06) ◽  
pp. 667-710 ◽  
Author(s):  
S. Farid Khwaja ◽  
A. B. Olde Daalhuis
Keyword(s):  
Hypergeometric Function ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Critical Values ◽  
Technical Detail ◽  
Gauss Hypergeometric Function ◽  
Uniform Asymptotic Expansions

In this paper, we obtain asymptotic expansions for the Gauss hypergeometric function [Formula: see text], where ej = 0, ± 1, j = 1, 2, 3, as |λ| → ∞. We complete the results of three previous publications [Uniform asymptotic expansions for hypergeometric functions with large parameters I, Anal. Appl. (Singap.) 1 (2003) 111–120; Uniform asymptotic expansions for hypergeometric functions with large parameters II, Anal. Appl. (Singap.) 1 (2003) 121–128; Uniform asymptotic expansions for hypergeometric functions with large parameters III, Anal. Appl. (Singap.) 8 (2010) 199–210], discuss all cases and, what is new, we consider now all critical values of z. For one case, the full details of the well-known Bleistein method are given, since a new technical detail is observed.

UNIFORM ASYMPTOTIC EXPANSIONS FOR HYPERGEOMETRIC FUNCTIONS WITH LARGE PARAMETERS II

2003 ◽  
Vol 01 (01) ◽  
pp. 121-128 ◽  
Author(s):  
A. B. OLDE DAALHUIS
Keyword(s):  
Asymptotic Expansion ◽  
Hypergeometric Function ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Gauss Hypergeometric Function ◽  
Uniform Asymptotic Expansions

In this paper, we obtain an asymptotic expansion for the Gauss hypergeometric function 2F1(a + λ, b + 2λ; c; -z), as |λ| → ∞. The expansion holds for fixed values of a, b, and c, and is uniformly valid for z in the domain | ph z| < π.

UNIFORM ASYMPTOTIC EXPANSIONS FOR HYPERGEOMETRIC FUNCTIONS WITH LARGE PARAMETERS III

2010 ◽  
Vol 08 (02) ◽  
pp. 199-210 ◽  
Author(s):  
A. B. OLDE DAALHUIS
Keyword(s):  
Hypergeometric Function ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Gauss Hypergeometric Function ◽  
Uniform Asymptotic Expansions

In this paper, we discuss asymptotic expansions for the Gauss hypergeometric function 2F1(a + e1λ, b + e2λ; c + e3λ; z), where ej = 0, ±1, as |λ| → ∞. We complete the results of two previous publications, extend the sectors of validity, and give more details on the computation of the coefficients.

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