# Bessel Functions And Their Applications ((TOP))

It is the aim of this paper to discover the roles played by Bessel functions in a variety of mathematical fields. It was found that Bessel functions attribute to the theories of spherical harmonics, transformations, as well as partial differential equations in relation to quantum mechanics, electrostatics, and classical mechanics in cylindrical and spherical coordinates. In particular, this paper uses Sturm's theorems to prove that Bessel functions have an infinite number of zeros, which has important applications in the study of Laplace's equation.

## Bessel functions and their applications

The importance of Bessel's Function in Mathematical Physics is indicated by their application to the modern solutions of problems in Wave-Theory, Elasticity, Hydrodynamics and Optics, The designation of Bessel's Functions as cylindrical function has its source in the use of these functions to express solutions of such physical problems as flow of heat or electricity in solid circular cylinder. However, the Bessel's Function first appeared in connection with the problem of the vibration of a heavy string suspended from one end. No method was given for determining the coefficients of the series. It was in problems connected with astronomy that the first completely successful application of Bessel's Functions occurred.

Bessel functions are associated with a wide range of problems in important areas of mathematical physics. Bessel function theory is applied to problems of acoustics, radio physics, hydrodynamics, and atomic and nuclear physics. Bessel Functions and Their Applications consists of two parts. In Part One, the author presents a clear and rigorous introduction to the theory of Bessel functions. Part Two is devoted to the application of Bessel functions to physical problems, particularly in the mechanics of solids and heat transfer. This volume was designed for engineers and researchers interested in the applications of the theory, and as such, it provides an indispensable source of reference.

Abstract:The theory of Generalized Bessel Functions is reviewed and their application to various problems in the study of electro-magnetic processes is presented. We consider the cases of emission of bremsstrahlung radiation by ultra-relativistic electrons in linearly polarized undulators, including also exotic configurations, aimed at enhancing the harmonic content of the emitted radiation. The analysis is eventually extended to the generalization of the FEL pendulum equation to treat Free Electron Laser operating with multi-harmonic undulators. The paper aims at picking out those elements supporting the usefulness of the Generalized Bessel Functions in the elaboration of the theory underlying the study of the spectral properties of the bremsstrahlung radiation emitted by relativistic charges, along with the relevant flexibility in accounting for a large variety of apparently uncorrelated phenomenolgies, like multi-photon processes, including non linear Compton scattering.Keywords: free electron laser; synchrotron radiation; Bessel functions; generating functions; integral representation

The theory of Bessel functions is a rich subject due to its essential role in providing solutions for differential equations associated with many applications. As fractional calculus has become an efficient and successful tool for analyzing various mathematical and physical problems, the so-called fractional Bessel functions were introduced and studied from different viewpoints. This paper is primarily devoted to the study of developing two aspects. The starting point is to present a fractional Laplace transform via conformable fractional-order Bessel functions (CFBFs). We establish several important formulas of the fractional Laplace Integral operator acting on the CFBFs of the first kind. With this in hand, we discuss the solutions of a generalized class of fractional kinetic equations associated with the CFBFs in view of our proposed fractional Laplace transform. Next, we derive an orthogonality relation of the CFBFs, which enables us to study an expansion of any analytic functions by means of CFBFs and to propose truncated CFBFs. A new approximate formula of conformable fractional derivative based on CFBFs is provided. Furthermore, we describe a useful scheme involving the collocation method to solve some conformable fractional linear (nonlinear) multiorder differential equations. Accordingly, several practical test problems are treated to illustrate the validity and utility of the proposed techniques and examine their approximate and exact solutions. The obtained solutions of some fractional differential equations improve the analog ones provided by various authors using different techniques. The provided algorithm may be beneficial to enrich the Bessel function theory via fractional calculus.

The present work proposes an approach to approximating the solution for some important linear and nonlinear FDEs in the conformable sense. The paper is designed with two objectives. The first is to establish some interesting properties of fractional Laplace-type integrals of functions via CFBFs. Then, we use the obtained results to establish the possible solutions of conformable fractional kinetic equations through CFBFs. The second objective is concerned with developing applications of the fundamental process of the proposed approach in terms of CFBFs. To achieve that, we derive an orthogonality relation, expand functions in terms of the truncated CFBFs, and effectively formulate a scheme involving the collocation method which employed to provide solutions of certain types of linear and nonlinear CFBEs.

Some new inequalities for quotients of modified Bessel functions of the first and second kinds are deduced. Moreover, some developments on bounds for modified Bessel functions of the first and second kinds, higher-order monotonicity properties of these functions and applications to a special function that arises in finite elasticity, are summarized. The key tool in our proofs is a frequently used criterion for the monotonicity of the quotient of two Maclaurin series.

Many applications can benefit from the use of pupil filters for controlling the light intensity distribution near the focus of an optical system. Most of the design methods for such filters are based on a second-order expansion of the Point Spread Function (PSF). Here, we present a new procedure for designing radially-symmetric pupil filters. It is more precise than previous procedures as it considers the exact expression of the PSF, expanded as a function of first-order Bessel functions. Furthermore, this new method presents other advantages: the height of the side lobes can be easily controlled, it allows the design of amplitude-only, phase-only or hybrid filters, and the coefficients of the PSF expansion can be directly related to filter parameters. Finally, our procedure allows the design of filters with very different behaviours and optimal performance.

Annotation:The class of G-functions and the class of H-functions were designed by Meijer and Fox as a generalization of a broad class of mathematical functions which with different parameters cover almost all known elementary and special functions (e.g. exponential function, gamma function, hypergeometric function, and Bessel functions). In addition, these functions have many good mathematical properties that can be advantageously used for their manipulation (e.g., the Mellin transform, the Laplace transform, and the Fourier transform). It turns out that density (PDF) respectively the cumulative distribution function (CDF) of many probability distributions can be expressed using these special functions (as e.g. gamma distribution, beta distribution, chi-square distribution, F-distribution). It is a very broad class of probability distributions that find applications in various fields of natural, technical and biomedical sciences. It follows from the above properties of these functions that the distribution of a combination of independent random variables with this type of distribution can also be expressed by simple manipulation of these functions. On the other hand, there are obviously many probability distributions that can be expressed by G-functions or H-functions that have not yet been identified, but may be useful for modeling complex physical or biological processes. The PhD thesis has several important objectives. The main objective is focused on research of advanced methods for measurement evaluation and development of methods and algorithms for computing probability distributions based on combinations of Meier G-functions and Fox H-functions and their applications. The specific objective of the project is the characterization of probability distributions expressible by G-functions and H-functions and development of efficient numerical methods and algorithms for computing the distributions of algebraic functions of independent random variables which can be made by using these distributions. 041b061a72