scholarly journals Generalized Laplace transform with matrix variables

1987 ◽  
Vol 10 (3) ◽  
pp. 503-511
Author(s):  
R. M. Joshi ◽  
J. M. C. Joshi
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Function ◽  
Hankel Transform ◽  
Laplace Transforms ◽  
Confluent Hypergeometric Function ◽  
Symmetric Matrices ◽  
The Real ◽  
Matrix Argument ◽  
Alpha Beta

In the present paper we have extended generalized Laplace transforms of Joshi to the space ofm×msymmetric matrices using the confluent hypergeometric function of matrix argument defined by Herz as kernel. Our extension is given byg(z)=Γm(α)Γm(β)∫∧>01F1(α:β:−∧z) f(∧)d∧The convergence of this integral under various conditions has also been discussed. The real and complex inversion theorems for the transform have been proved and it has also been established that Hankel transform of functions of matrix argument are limiting cases of the generalized Laplace transforms.

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.

The real zeros of the confluent hypergeometric function

1956 ◽  
Vol 52 (4) ◽  
pp. 626-635 ◽  
Author(s):  
L. J. Slater
Keyword(s):  
Hypergeometric Function ◽  
Numerical Evaluation ◽  
Confluent Hypergeometric Function ◽  
The Real ◽  
Real Zeros ◽  
Image Position

This paper contains a discussion of various points which arise in the numerical evaluation of the small real zeros of the confluent hypergeometric functionwhereThere are two distinct problems, first the determination of those values of x for which M(a, b; x) = 0, given a and b, and secondly the study of the curves represented by M (a, b; x) = 0, for fixed values of x. These curves all lie on the surface M(a, b; x) = 0, of course.

A note on the real zeros of the basic confluent hypergeometric function

1968 ◽  
Vol 64 (3) ◽  
pp. 683-686
Author(s):  
Ramadhar Mishra
Keyword(s):  
Hypergeometric Function ◽  
Newton’S Method ◽  
Newton's Method ◽  
Numerical Evaluation ◽  
Confluent Hypergeometric Function ◽  
The Real ◽  
Real Zeros ◽  
In Series

Some years back, Slater (4) discussed the approximations, based on the expansion in series, for the cases 1F1(a; b; x) = 0, when either of b and x or a and x are fixed. These approximations were based essentially on the well-known Newton's method of approximation and were helpful in the numerical evaluation of the small real zeros of the confluent hypergeometric function 1F1(a; b; x;). In this note, we deal with the corresponding problem for the basic confluent hypergeometric function 1Φ1(a; b; x;).

scholarly journals Eigenvalue distributions of beta-Wishart matrices

2014 ◽  
Vol 03 (02) ◽  
pp. 1450009 ◽  
Author(s):  
Alan Edelman ◽  
Plamen Koev
Keyword(s):  
Hypergeometric Function ◽  
Random Matrices ◽  
The Real ◽  
Wishart Matrices ◽  
Extreme Eigenvalues ◽  
Matrix Argument ◽  
Explicit Expressions

We derive explicit expressions for the distributions of the extreme eigenvalues of the beta-Wishart random matrices in terms of the hypergeometric function of a matrix argument. These results generalize the classical results for the real (β = 1), complex (β = 2), and quaternion (β = 4) Wishart matrices to any β > 0.

scholarly journals On several new Laplace transforms of generalized hypergeometric functions 2F2(x)

2021 ◽  
Vol 39 (4) ◽  
pp. 97-109
Author(s):  
Asmaa Orabi Mohammed ◽  
Medhat A. Rakha ◽  
Mohammed M. Awad ◽  
Arjun K. Rathie
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Laplace Transforms ◽  
Theoretical Physics ◽  
Generalized Hypergeometric Function ◽  
Generalized Hypergeometric Functions ◽  
Special Cases ◽  
And Mathematics

By employing generalizations of Gauss's second, Bailey's and Kummer's summation theorems obtained earlier by Rakha and Rathie, we aim to establish unknown Laplace transform of six rather general formulas of generalized hypergeometric function 2F2[a,b;c,d;x]. The results obtained in this paper are simple, interesting, easily established and may be useful in theoretical physics, engineering and mathematics. Results obtained earlier by Kim et al. and Choi and Rathie follow special cases of our main findings.

scholarly journals A study of generalized summation theorems for the series 2F1 with an applications to Laplace transforms of convolution type integrals involving Kummer's functions 1F1

2018 ◽  
Vol 12 (1) ◽  
pp. 257-272
Author(s):  
Gradimir Milovanovic ◽  
Rakesh Parmar ◽  
Arjun Rathie
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Transform ◽  
Laplace Transforms ◽  
Confluent Hypergeometric Function ◽  
Convolution Type ◽  
Product Theorem ◽  
Confluent Hypergeometric Functions

Motivated by recent generalizations of classical theorems for the series 2F1 [Integral Transform. Spec. Funct. 229(11), (2011), 823-840] and interesting Laplace transforms of Kummer's confluent hypergeometric functions obtained by Kim et al. [Math. Comput. Modelling 55 (2012), 1068-1071], first we express generalized summations theorems in explicit forms and then by employing these, we derive various new and useful Laplace transforms of convolution type integrals by using product theorem of the Laplace transforms for a pair of Kummer's confluent hypergeometric function.

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.

scholarly journals A Relation between Laplace and Hankel Transforms

1962 ◽  
Vol 5 (3) ◽  
pp. 114-115 ◽  
Author(s):  
B. R. Bhonsle
Keyword(s):  
Laplace Transform ◽  
Hankel Transform ◽  
Laplace And Hankel Transforms ◽  
Hankel Transforms ◽  
The Laplace Transform ◽  
Image Position

The Laplace transform of a function f(t) ∈ L(0, ∞) is defined by the equationand its Hankel transform of order v is defined by the equationThe object of this note is to obtain a relation between the Laplace transform of tμf(t) and the Hankel transform of f(t), when ℛ(μ) > − 1. The result is stated in the form of a theorem which is then illustrated by an example.

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.

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