Unification of Bessel Functions of Different Orders

: The report contains tables of the first five roots of the following transcendental equations: (a) J0(alpha) Y0(alpha) = Y0(alpha) J0(alpha); (b) J1(alpha) Y1(alpha) = Y1(alpha) J1(alpha); (c) J0(alpha) Y1(alpha) = Y0(alpha) J1(alpha) where J0(alpha), Y0(alpha), J1(alpha), Y1(alpha) are Bessel functions of order 0 and 1 respectively. In these equations, alpha is the unknown and k is a parameter which may assume any positive value, other than 0 or 1. Additional tables are included listing an auxiliary quantity gamma which is better suited to interpolation particularly when k is close to unity. (Author)

CHAPTER 17 - Bessel Functions

The Neumann expansion of Bessel functions (of integer order) of a function \(g:\mathbb {C}\rightarrow \mathbb {C}\) corresponds to representing g as a linear combination of basis functions φ0, φ1, …, i.e., \(g(s)=\sum _{\ell = 0}^\infty w_\ell \varphi _\ell (s)\), where φi(s) = Ji(s), i = 0, …, are the Bessel functions. In this work, we study an expansion for a more general class of basis functions. More precisely, we assume that the basis functions satisfy an infinite dimensional linear ordinary differential equation associated with a Hessenberg matrix, motivated by the fact that these basis functions occur in certain iterative methods. A procedure to compute the basis functions as well as the coefficients is proposed. Theoretical properties of the expansion are studied. We illustrate that non-standard basis functions can give faster convergence than the Bessel functions.

Some Integrals Involving Bessel Functions

Bessel functions have shown to be particularly suitable for representing certain classes of signals, since using these basis functions may results in fewer components than using sinusoids. However, as there are no closed form expressions available for such functions, approximations and numerical methods have been adopted for their computation. In this paper the functions called discrete Bessel functions that are expressed as a finite expansion are defined. It is shown that in a finite interval a finite number of such functions that perfectly match Bessel functions of integer order exist. For finite duration sequences it is proven that the subspace spanned by a set of these functions is able to represent the class of finite duration decaying sequences.

Analytic solutions of the fundamental equation of the reference spectrum method for nuclear matter are obtained in the cases of the square well and exponential potentials with hard cores. In the former case analytic wavefunctions are obtained for every partial wave L. These wavefunctions are expressed in terms of spherical Bessel and Hankel functions or in terms of spherical Bessel, Hankel, and Newmann functions of order L. In the latter case an analytic expression is obtained for the s wavefunction only. This is given in terms of the transcendental functions G±v(r) and M±v(r) which are defined as series of well-known functions. Finally an application is made of our results to the calculation of the well depth of a Λ particle in its ground state in nuclear matter.

Some results concerning the non-existence of complex and purely imaginary zeros of the transcendental equation F(z)J_v(z)+G(z)J'_v(z) = 0 are given, where J_v(z) is the Bessel function of first kind and order v (in general complex), J'_v(z) is the derivative of J_v(z) , and F(z) , G(z) are analytic functions. The obtained results improve and generalize previously known ones.

A theoretical model for the calculation of the quantized spectrum of spin modes frequencies in cylindrical magnetic dots of radius ranging from the nanometric to the submicrometric scale in the vortex ground state at zero applied magnetic field is presented. The effective field includes both the surface and the volume dynamic magnetostatic and exchange fields. We also show that the core energy affects the spin dynamics. The modes at lower frequencies present as radial eigenvectors Bessel functions of high orders $(m\ensuremath{\geqslant}1)$, while the axially symmetric modes at higher frequencies correspond to zero order Bessel functions.

On a conjecture regarding the extrema of Bessel functions and its generalization

Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

Using the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function${K}_{ir} (x)$of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of${K}_{ir} (x)$and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of$r$. Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of${K}_{ir} (x)$.

: An identity, relating a certain infinite continued fraction to the ratio of two contiguous Bessel functions, is exploited to derive the representation of a single Bessel function in terms of a series of products of continued fractions. This representation is made the basis of a new method for computing numerically Bessel functions of the first kind. The method is seen to be especially valuable in computing Bessel functions of order much larger tan unity.

Applications of a simple approximation of Bessel functions of integer order, in terms of trigonometric functions, are discussed for several examples from electromagnetism and optics. The method may be applied in the intermediate regime, bridging the ‘small values regime’ and the ‘asymptotic’ one, and covering, in this way, an area of great physical interest. The examples that illustrate our approach are accessible to the undergraduate student.

This paper aims to shape the identification of thin inhomogeneities with different dielectric/magnetic properties from a two-dimensional homogeneous background. The shapes are identified through subspace migration without requiring the diagonal elements of the collected multi-static response matrix. To understand why subspace migration without diagonal elements can retrieve the shape of a thin inhomogeneity, we carefully investigate the relations between the imaging function and Bessel functions of orders 0 and 1. This analysis exploits the fact that when a thin homogeneity exists, the measured far-field pattern can be represented as an asymptotic expansion formula. The investigated relation is supported in numerical experiments with noisy data.

TRAITEMENT AUTOMATIQUE DES SONDAGES ELECTRIQUES AUTOMATIC PROCESSING OF ELECTRICAL SOUNDINGS*