Asymptotic expansions and converging factors V. Lommel, Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions

1959 ◽  
Vol 249 (1257) ◽  
pp. 284-292 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Lommel Functions ◽  
Special Cases

A theory of Lommel functions is developed, based upon the methods described in the first four papers (I to IV) 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 Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions.

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.

Transient Analysis of a Two-Heterogeneous Severs Queue with Impatient Behaviour and Multiple Vacations

2018 ◽  
Vol 6 (1) ◽  
pp. 69-84
Author(s):  
Jia Xu ◽  
Liwei Liu ◽  
Taozeng Zhu
Keyword(s):  
Generating Functions ◽  
Transient Analysis ◽  
Bessel Functions ◽  
Queueing System ◽  
System Size ◽  
Heterogeneous Servers ◽  
Modified Bessel Functions ◽  
Multiple Vacations ◽  
Special Cases ◽  
Mean And Variance

AbstractWe consider anM/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities, mean and variance of the system size at timetby employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.

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.

scholarly journals Liouville-Green expansions of exponential form, with an application to modified Bessel functions

2019 ◽  
Vol 150 (3) ◽  
pp. 1289-1311 ◽  
Author(s):  
T. M. Dunster
Keyword(s):  
Error Bounds ◽  
Asymptotic Expansions ◽  
Turning Point ◽  
Bessel Functions ◽  
Alternative Form ◽  
Modified Bessel Functions ◽  
Exponential Form ◽  
Positive Order ◽  
Second Order Differential Equations ◽  
Computable Error Bounds

AbstractLinear second order differential equations of the form d2w/dz2 − {u2f(u, z) + g(z)}w = 0 are studied, where |u| → ∞ and z lies in a complex bounded or unbounded domain D. If f(u, z) and g(z) are meromorphic in D, and f(u, z) has no zeros, the classical Liouville-Green/WKBJ approximation provides asymptotic expansions involving the exponential function. The coefficients in these expansions either multiply the exponential or in an alternative form appear in the exponent. The latter case has applications to the simplification of turning point expansions as well as certain quantum mechanics problems, and new computable error bounds are derived. It is shown how these bounds can be sharpened to provide realistic error estimates, and this is illustrated by an application to modified Bessel functions of complex argument and large positive order. Explicit computable error bounds are also derived for asymptotic expansions for particular solutions of the nonhomogeneous equations of the form d2w/dz2 − {u2f(z) + g(z)}w = p(z).

scholarly journals Some Novel Fractional Integral Inequalities over a New Class of Generalized Convex Function

2022 ◽  
Vol 6 (1) ◽  
pp. 42
Author(s):  
Soubhagya Kumar Sahoo ◽  
Muhammad Tariq ◽  
Hijaz Ahmad ◽  
Bibhakar Kodamasingh ◽  
Asif Ali Shaikh ◽  
...  
Keyword(s):  
Applied Sciences ◽  
Fractional Integral ◽  
Bessel Functions ◽  
Fractional Operators ◽  
Modified Bessel Functions ◽  
New Class ◽  
Special Means ◽  
Special Cases ◽  
Related Integral ◽  
Increasing Function

The comprehension of inequalities in convexity is very important for fractional calculus and its effectiveness in many applied sciences. In this article, we handle a novel investigation that depends on the Hermite–Hadamard-type inequalities concerning a monotonic increasing function. The proposed methodology deals with a new class of convexity and related integral and fractional inequalities. There exists a solid connection between fractional operators and convexity because of its fascinating nature in the numerical sciences. Some special cases have also been discussed, and several already-known inequalities have been recaptured to behave well. Some applications related to special means, q-digamma, modified Bessel functions, and matrices are discussed as well. The aftereffects of the plan show that the methodology can be applied directly and is computationally easy to understand and exact. We believe our findings generalise some well-known results in the literature on s-convexity.

Derivation of Green-type, transitional, and uniform asymptotic expansions from differential equations I. General theory, and application to modified Bessel functions of large order

1964 ◽  
Vol 281 (1384) ◽  
pp. 99-110 ◽  
Keyword(s):  
General Theory ◽  
Differential Equations ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Asymptotic Solutions ◽  
Modified Bessel Functions ◽  
Mellin Transforms ◽  
Convenient Procedure ◽  
Uniform Expansions ◽  
Uniform Asymptotic Expansions

The problem of deriving Green-type asymptotic solutions from differential equations of general form d 2 y /dz 2 = X(a 2 >, z)y , for large values of a 2 , is reformulated. Combination of this formulation with the method of Mellin transforms leads further to a particularly convenient procedure for finding asymptotic expansions valid in transitional regions, and general uniform expansions. The methods are illustrated by detailed calculations for modified Bessel functions.

(p, q)-Extended Bessel and Modified Bessel Functions of the First Kind

2017 ◽  
Vol 72 (1-2) ◽  
pp. 617-632 ◽  
Author(s):  
Dragana Jankov Maširević ◽  
Rakesh K. Parmar ◽  
Tibor K. Pogány
Keyword(s):  
Bessel Functions ◽  
Modified Bessel Functions
Sign in / Sign up

Export Citation Format

Share Document