Uniform, Exponentially Improved, Asymptotic Expansions for the Confluent Hypergeometric Function and Other Integral Transforms

1991 ◽  
Vol 22 (5) ◽  
pp. 1475-1489 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Hypergeometric Function ◽  
Asymptotic Expansions ◽  
Integral Transforms ◽  
Confluent Hypergeometric Function

Asymptotic expansions and Stokes multipliers of the confluent hypergeometric function Φ2, I

1993 ◽  
Vol 123 (6) ◽  
pp. 1165-1177
Author(s):  
Shun Shimomura
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Hypergeometric Function ◽  
Asymptotic Expansions ◽  
Confluent Hypergeometric Function ◽  
Connection Formula ◽  
Regular Type ◽  
Linearly Independent ◽  
Singular Loci

SynopsisThe confluent hypergeometric function Φ2(β,β′, γ, x, y) satisfies a system of partial differential equations which possesses the singular loci x = 0, y = 0, x − y = 0 of regular type and x = ∞, y = ∞ of irregular type. Near x = ∞ (|y| is bounded) and near y = ∞ (|x| is bounded), asymptotic expansions and Stokes multipliers of linearly independent solutions of the system are obtained. By a connection formula, the asymptotic behaviour of Φ2(β,β′, γ, x, y) itself is also clarified near these singular loci.

scholarly journals Analytical and asymptotic evaluations of Dawson’s integral and related functions in mathematical physics

2019 ◽  
Vol 25 (2) ◽  
pp. 179-188
Author(s):  
Victor Nijimbere
Keyword(s):  
Asymptotic Expansion ◽  
Hypergeometric Function ◽  
Complex Plane ◽  
Asymptotic Expansions ◽  
Error Function ◽  
Mathematical Physics ◽  
Confluent Hypergeometric Function ◽  
Dispersion Function ◽  
Plasma Dispersion

Abstract Dawson’s integral and related functions in mathematical physics that include the complex error function (Faddeeva’s integral), Fried–Conte (plasma dispersion) function, Jackson function, Fresnel function and Gordeyev’s integral are analytically evaluated in terms of the confluent hypergeometric function. And hence, the asymptotic expansions of these functions on the complex plane {\mathbb{C}} are derived by using the asymptotic expansion of the confluent hypergeometric function.

On the evaluation of the confluent hypergeometric function

1953 ◽  
Vol 49 (4) ◽  
pp. 612-622 ◽  
Author(s):  
L. J. Slater
Keyword(s):  
Hypergeometric Function ◽  
Asymptotic Expansions ◽  
Confluent Hypergeometric Function

ABSTRACTThis paper contains a table of the confluent hypergeometric function over the range a = − 1·0(0·1) + 1·0, b = 0·1(0·1)1·0, x= 1·0(1·0)10·0, and the expansions in converging factors by means of which the accuracy of the asymptotic expansions for higher values of x can be improved.

scholarly journals A new generalization of confluent hypergeometric function and whittaker function

2018 ◽  
Vol 38 (2) ◽  
pp. 9-26
Author(s):  
Nabiullah Khan ◽  
Talha Usman ◽  
Mohd Ghayasuddin
Keyword(s):  
Hypergeometric Function ◽  
Recurrence Relations ◽  
Integral Transforms ◽  
Whittaker Function ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Extra Parameter

In this article, we introduce a further generalizations of the confluent hypergeometric function and Whittaker function by introducing an extra parameter in the extended con uent hypergeometric function dened by Parmar [15]. We also investigate some integral representations, some integral transforms, differential formulas and recurrence relations of these new generalizations

scholarly journals The Bateman Functions Revisited after 90 Years—A Survey of Old and New Results

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1273
Author(s):  
Alexander Apelblat ◽  
Armando Consiglio ◽  
Francesco Mainardi
Keyword(s):  
Hypergeometric Function ◽  
Laplace Transforms ◽  
Confluent Hypergeometric Function ◽  
Integral Function ◽  
Basic Properties ◽  
Differential Relations

The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman function is only mentioned as a particular case of the confluent hypergeometric function. In order to revive our knowledge on these functions, their basic properties (recurrence functional and differential relations, series, integrals and the Laplace transforms) are presented. Some new results are also included. Special attention is directed to the Bateman and Havelock functions with integer orders, to generalizations of these functions and to the Bateman-integral function known in the literature.

Recursion formulas for multivariable hypergeometric functions

2015 ◽  
Vol 08 (04) ◽  
pp. 1550082 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Radiation Field ◽  
Hypergeometric Functions ◽  
Integral Transforms ◽  
Recursion Formulas ◽  
Appl Math ◽  
Appell Functions

Recently, Opps, Saad and Srivastava [Recursion formulas for Appell’s hypergeometric function [Formula: see text] with some applications to radiation field problems, Appl. Math. Comput. 207 (2009) 545–558] presented the recursion formulas for Appell’s function [Formula: see text] and also gave its applications to radiation field problems. Then Wang [Recursion formulas for Appell functions, Integral Transforms Spec. Funct. 23(6) (2012) 421–433] obtained the recursion formulas for Appell functions [Formula: see text] and [Formula: see text]. In our investigation here, we derive the recursion formulas for 14 three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions.

Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit-circle singularity, I

1984 ◽  
Vol 99 (1-2) ◽  
pp. 51-70 ◽  
Author(s):  
F. V. Atkinson ◽  
C. T. Fulton
Keyword(s):  
Asymptotic Expansion ◽  
Eigenvalue Problem ◽  
Hypergeometric Function ◽  
Finite Interval ◽  
Confluent Hypergeometric Function ◽  
Limit Circle ◽  
Asymptotic Formulae ◽  
Sturm Liouville

SynopsisAsymptotic formulae for the positive eigenvalues of a limit-circle eigenvalue problem for –y” + qy = λy on the finite interval (0, b] are obtained for potentials q which are limit circle and non-oscillatory at x = 0, under the assumption xq(x)∈L1(0,6). Potentials of the form q(x) = C/xk, 0<fc<2, are included. In the case where k = 1, an independent check based on the limit-circle theory of Fulton and an asymptotic expansion of the confluent hypergeometric function, M(a, b; z), verifies the main result.

INTEGRAL TRANSFORM OF K4-FUNCTION

2021 ◽  
Vol 21 (2) ◽  
pp. 429-436
Author(s):  
SEEMA KABRA ◽  
HARISH NAGAR
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Function ◽  
Integral Transform ◽  
Integral Transforms ◽  
Gauss Hypergeometric Function ◽  
Wright Function ◽  
Generalized Wright Function ◽  
Special Cases ◽  
Euler Transform

In this present work we derived integral transforms such as Euler transform, Laplace transform, and Whittaker transform of K4-function. The results are given in generalized Wright function. Some special cases of the main result are also presented here with new and interesting results. We further extended integral transforms derived here in terms of Gauss Hypergeometric function.

A Generalised Kummer-Type Transformation for the pFp(x) Hypergeometric Function

2012 ◽  
Vol 55 (3) ◽  
pp. 571-578
Author(s):  
A. R. Miller ◽  
R. B. Paris
Keyword(s):  
Hypergeometric Function ◽  
Confluent Hypergeometric Function ◽  
Reduction Formula ◽  
Type Transformation

AbstractIn a recent paper, Miller derived a Kummer-type transformation for the generalised hypergeometric function pFp(x) when pairs of parameters differ by unity, by means of a reduction formula for a certain Kampé de Fériet function. An alternative and simpler derivation of this transformation is obtained here by application of the well-known Kummer transformation for the confluent hypergeometric function corresponding to p = 1.

Sign in / Sign up

Export Citation Format

Share Document