Asymptotics of modified Bessel functions of high order (original) (raw)

A Nonstandard Asymptotic Approximation of Bessel Function

There are several methods to obtain asymptotic approximations for special functions or functions represented by integral. The known methods involve cases of real values and values tend to infinity, whereas the cases of infinitesimal values or values infinitely close to exact values was not given before , which is our aim in this paper. Throughout this paper, we try to give some new asymptotic approximations of the integral form of the Bessel special function sin sin 0 0 1 1 ( ) 2 2                  iz ip iz ip p J z e d e d , where z £ and 0   , for a nonstandard values of  , by using some nonstandard concepts to obtain a new asymptotic representations of the integral form of the Bessel function. The main difference in this paper is that the obtained results, are exactly represented by trigonometric and hyperbolic functions unlike the classical approximate results, the values of Bessel function of hyperbolic representation are complex’s, and for trigonometric ...

Expansion and asymptotic in terms of basic Bessel functions

Applied Mathematics and Computation, 2007

This work aims to study the expansion and asymptotic for solutions of q-difference equations in terms of the basic Bessel functions, namely J ð2Þ a ðx; qÞ. For this purpose, we will show that the constructive method introduced by Olver [F.

Basics of Bessel Functions

This paper is a deep exploration of the project Bessel Functions by Martin Kreh of Pennsylvania State University. We begin with a derivation of the Bessel functions Ja(x) and Ya(x), which are two solutions to Bessel's differential equation. Next we find the generating function and use it to prove some useful standard results and recurrence relations. We use these recurrence relations to examine the behavior of the Bessel functions at some special values. Then we use contour integration to derive their integral representations, from which we can produce their asymptotic formulae. We also show an alternate method for deriving the first Bessel function using the generating function. Finally, a graph created using Python illustrates the Bessel functions of order 0, 1, 2, 3, and 4.

The k-Bessel function of the first kind

The authors introduce a k-version k of the Bessel function of the first kind and study some basic properties. Then they present a relationship between this function and the k-Mittag-Leffler and k-Wright functions recently introduced by autors themself.

Generalized series of Bessel functions

Journal of Computational and Applied Mathematics, 2002

Known series of Bessel functions, currently available in handbooks, and many of Neumann type, are generalized to arbitrary order. The underlying result is a Poisson formula due to Titchmarsh. This formula gives rise to a Neumann series involving modiÿed Bessel functions of integral order. The latter is the basis of many of the generalized series that follow. Included are examples of generalized trigonometric identities. The paper concludes by indicating the wide range of results that can be obtained.

A note on the theory ofn-variable generalized bessel functions

Il Nuovo Cimento B, 1991

In this note we introduce a further generalization of Bessel-type functions, discussing the case of a multivariables and one-index function. This kind of function can be usefully exploited in problems in which the dipole approximation does not hold and many higher harmonics are simultaneously operating. We analyse the relevant recurrence properties, the modified forms and the generating functions.

A unified point of view on the theory of generalized Bessel functions

Computers & Mathematics with Applications, 1995

Bessel functions have been generalized in a number of ways and many of these generalizations have been proved to be important tools in applications. In this paper we present a unified treatment, thus proving that many of the seemingly different generalizations may be viewed as particular cases of a two-variable function of the type introduced by Miller during the sixties.

The basics of Bessel functions

Maths at Bondi Beach, 2023

This paper gives an historical perspective on the genesis of Bessel functions and sets out detailed proofs for basic properties using different representations of Bessel functions.