Bessel Equations and Functions

A reliable and portable software package, called ZEBEC (ZEros of BEssel functions Complex) is presented, which localizes and computes simple zeros of Bessel functions of the first, the second or the third kind, or their derivatives. The Bessel functions are of real order and complex argument. ZEBEC calculates with certainty the total number of zeros within a given box whose edges are parallel to the coordinate axes. Cauchy's Theorem is used for this calculation. Then the program isolates each one of the zeros, utilizing the above mentioned theorem, and finally computes them to a given desired accuracy using a generalized method of bisection.

In his classic book on group representations and special functions Vilenkin studied the matrix elements of irreducible representations of the Euclidean and oscillator Lie algebras with respect to countable bases of eigenfunctions of the Cartan subalgebras, and he computed the summation identities for Bessel functions and Laguerre polynomials associated with the addition theorems for these matrix elements. He also studied matrix elements of the pseudo-Euclidean and pseudo-oscillator algebras with respect to the continuum bases of generalized eigenfunctions of the Cartan subalgebras of these Lie algebras and this resulted in realizations of the addition theorems for the matrix elements as integral transform identities for Bessel functions and for confluent hypergeometric functions. Here we work out q analogs of these results in which the usual exponential function mapping from the Lie algebra to the Lie group is replaced by the q-exponential mappings Eq and eq. This study of representations of the Euclidean quantum algebra and the q-oscillator algebra (not a quantum algebra) leads to summation, integral transform, and q-integral transform identities for q analogs of the Bessel and confluent hypergeometric functions, extending the results of Vilenkin for the q=1 case.

The present paper is a review of different kind of Orthogonal
functions with there important properties, main emphasize is given on the
hypergeometric function. In this paper we have discussed orthogonality
properties of orthogonal polynomials like Rodrigues formula, Recurrence
relation, Generating function classification of Orthogonal polynomials using
lemmas and examples. This paper also deals with all these properties of
Laguerre, Hermite polynomials.

We show that inverse square singularities can be treated as boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter with “a negative number of poles”. More precisely, we treat in a unified manner one-dimensional Schrödinger operators with either an inverse square singularity or a boundary condition containing a rational Herglotz–Nevanlinna function of the eigenvalue parameter at each endpoint, and define Darboux-type transformations between such problems. These transformations allow one, in particular, to transfer almost any spectral result from problems with eigenparameter dependent boundary conditions to those with inverse square singularities, and vice versa.

In this paper, infinite integrals involving the product of Bessel functions of different arguments are solved in closed-form. These solutions provide novel expressions in the error rate analysis for wireless amplify and forward (AF) systems with an arbitrary number of variable-gain relays. Here we show that the error probability can be described by generalized hypergeometric functions, namely Lauricella functions. This work represents a significant improvement on previous contributions and extends previous formulas pertaining to dual-hop transmissions over identical Nakagami-m fading channels. Numerical examples show an excellent match between simulation and theoretical results.

AbstractTwo algorithms which provide a statistical treatment to the estimation of the parameters of the cross-correlation analysis of spaced-receiver data are reviewed. Their results are compared using signals: (1) transmitted by a quasi-stationary polar beacon and received at Goose Bay, Labrador (53.3°N, 60.3°W); and (2) transmitted by the orbiting Hilat satellite and received at Tromsø, Norway (69.7°N, 18.9°E). A good general agreement is displayed in this comparison. The former experiment indicates the possibility of extreme daily variations in the anisotropy of the ground diffraction pattern and in the true drift velocity of the in situ irregularities. The latter experiment displays geometrical enhancements in the intensity scintillation index S4, in the rms phase fluctuation σϕ and in the axial ratio of the ellipse which characterizes the anisotropy of the ground diffraction pattern, around the region of local L shell alignment of the ray paths. Increases of these parameters also are observed northward of Tromsø. These observations are thus consistent with a morphological model for anisotropy of high-latitude nighttime F region irregularities proposed in the literature. Next, a possible dependence of the results of the spaced-receiver measurements on the receiver baselines is discussed. It is argued that this mechanism could be responsible for the relatively small values of the anisotropy of the diffraction pattern obtained from the Hilat measurements at Tromsø. Finally, a procedure which combines a simple propagation model of the spaced-receiver experiment with a nonlinear minimization algorithm is proposed to estimate the anisotropy of the in situ irregularities from that of the diffraction pattern.

We propose a new statistical model for SAR images. According to this model, the SAR image amplitude follows a product of Rayleigh and Bessel functions. We derive the maximum likelihood feature detector for extracting terrain features from synthetic aperture radar (SAR) images. The terrain features are classified as ridges and ravines according to their statistical properties and surrounding neighborhood. These salient features are used as constraints for estimating the SAR terrain surface

To detect buried objects at a not well-defined distance of several meters from the radar system it is preferable to maintain the beam confined in a small spot of a few centimeters for a wide range. To this purpose a pseudo-Bessel beam having a propagation distance of several hundreds of wavelengths can be suitably used. This paper shows how this beam can be conveniently obtained in the microwave band by using a Radial Line Slot Array (RLSA). To assure the required aperture field distribution, the antenna design has been achieved by making use of a holographic approach and by employing an ad hoc optimization tool that provides directly the final antenna slot layout. Simulated and measured antenna characteristics show that high efficiency and polarization purity can be obtained by such a flat antenna, also achieving both manufacturing and setup simplicity.

This article is about finding the possible solutions of fractional kinetic equation associated with the generalized multiindex Bessel function using the method of Laplace transform. Numerical results and graphical solutions of the main theorems are also presented.