An infinite Integral involving Bessel functions, parabolic cylinder functions, and the confluent hypergeometric functions

1939 ◽  
Vol 44 (1) ◽  
pp. 789-793 ◽  
Author(s):  
N. A. Shastri
Keyword(s):  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Confluent Hypergeometric Functions ◽  
Parabolic Cylinder Functions ◽  
Infinite Integral

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.

An infinite integral involving Bessel functions and parabolic cylinder functions

1937 ◽  
Vol 33 (2) ◽  
pp. 210-211 ◽  
Author(s):  
R. S. Varma
Keyword(s):  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Infinite Integral

The object of this paper is to evaluate an infinite integral involving Bessel functions and parabolic cylinder functions. The following two lemmas are required:Lemma 1. provided that R(m) > 0.

scholarly journals Distributions of the Ratio and Product of Two Independent Weibull and Lindley Random Variables

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
N. J. Hassan ◽  
A. Hawad Nasar ◽  
J. Mahdi Hadad
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Random Variables ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Parabolic Cylinder ◽  
Generalized Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Parabolic Cylinder Functions ◽  
The Moment

In this paper, we derive the cumulative distribution functions (CDF) and probability density functions (PDF) of the ratio and product of two independent Weibull and Lindley random variables. The moment generating functions (MGF) and the k-moment are driven from the ratio and product cases. In these derivations, we use some special functions, for instance, generalized hypergeometric functions, confluent hypergeometric functions, and the parabolic cylinder functions. Finally, we draw the PDF and CDF in many values of the parameters.

scholarly journals On the Roots of Confluent Hypergeometric Functions

1914 ◽  
Vol 33 ◽  
pp. 48-64 ◽  
Author(s):  
Archd Milne
Keyword(s):  
Hypergeometric Functions ◽  
Graphical Form ◽  
Parabolic Cylinder ◽  
Confluent Hypergeometric Functions ◽  
Special Values ◽  
Parabolic Cylinder Functions ◽  
Class Of Functions

In the present paper the disposition of the roots of the confluent hypergeometric functions — denoted by Wk, m(z) — as affected by changing the parameters k and m is investigated. The results are then shewn in a graphical form, and various typical illustrations of the functions are given. By giving special values to k and m it is then exemplified how the roots of other functions expressible in terms of Wk, m(z) may be studied. The zeros of the parabolic cylinder functions are then discussed. Some of the properties of an allied class of functions, denoted by ψn(z), are then given, and finally, it is shewn how the properties of Abel's function φm(z) may be obtained from results already given.

Application of the E-operator to evaluate an infinite integral

1969 ◽  
Vol 65 (3) ◽  
pp. 725-730 ◽  
Author(s):  
F. Singh
Keyword(s):  
Finite Difference ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
General Nature ◽  
Difference Operators ◽  
Generalized Hypergeometric Function ◽  
Generalized Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Infinite Integral

1. The object of this paper is to evaluate an infinite integral, involving the product of H-functions, generalized hypergeometric functions and confluent hypergeometric functions by means of finite difference operators E. As the generalized hypergeometric function and H-function are of a very general nature, the integral, on specializing the parameters, leads to a generalization of many results some of which are known and others are believed to be new.

scholarly journals Integrals of Lommel's Type for Confluent Hypergeometric Functions

1952 ◽  
Vol 1 (1) ◽  
pp. 28-31
Author(s):  
D. Martin
Keyword(s):  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Confluent Hypergeometric Functions

In this note we derive some integrals involving confluent hypergeometric functions and analogous to Lommel's integrals for Bessel functions. Although the method of derivation is straightforward, the integrals do not seem to be mentioned in the literature.

Error bounds for the Liouville–Green (or WKB) approximation

1961 ◽  
Vol 57 (4) ◽  
pp. 790-810 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Error Bounds ◽  
Bessel Functions ◽  
Approximate Solutions ◽  
Parabolic Cylinder ◽  
Real Interval ◽  
Positive Parameter ◽  
Modified Bessel Functions ◽  
Elementary Functions ◽  
Parabolic Cylinder Functions ◽  
Image Position

ABSTRACTError bounds are derived and examined for approximate solutions in terms of elementary functions of the differential equationsin which u is a positive parameter, the functions f and p are free from singularities and p does not vanish. Bounds are also obtained for the remainder terms in the asymptotic expansions of the solutions in descending powers of u. The variable x ranges over a real interval, finite or infinite or over a region of the complex plane, bounded or unbounded.Applications are made to parabolic cylinder functions of large orders, and modified Bessel functions of large orders.

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