Bessel Functions Research Articles

Study of Dependence of the Energy of the Field in Weakly Conductive Fiberlight-Guide with Gradient Refractive Index Profile Without Regard to Polarization

A round-in-cross-section weakly guiding fiber optic light guide is considered. The system of Maxwellian equations for electrically conductive transparent media without due regard to polarization with an index of refraction in the case under consideration is reduced to the homogeneous Helmholtz equation, a particular solution of which is found using the Green's function. For a unimodalmode a common decision was obtained for the field and energy inside the fiber in the general case of a gradient refractive index profile depending on the radial coordinate. The concept of normalized energy is introduced as the ratio of the energy inside a fiber with gradient refractive index profile to the energy inside a fiber with a step-index profile. Since in a single-mode regime the zero-order Bessel function, which describes the field inside a step-index profile fiber, can be replaced by a Gaussoid, then, when extending this replacement to the case of a single-mode gradient fiber, as a first approximation we shall obtain a finite expression for the normalized energy for a general gradient profile. The dependences of the normalized energy on the waveguide number were obtained for each of the degrees from the first to the fourth inclusive. It is shown that in the considered approximation, the energy increases up to a certain parameter values (for the first degree – a triangle profile) or slowly decreases (other profiles), after this value the energy increases for all profiles. The results obtained will help to select a suitable single-mode fiber for practical application.

On the Containment of the Unit Disc Image by Analytical Functions in the Lemniscate and Nephroid Domains

Suppose that A1 is a class of analytic functions f:D={z∈C:|z|<1}→C with normalization f(0)=1. Consider two functions Pl(z)=1+z and ΦNe(z)=1+z−z3/3, which map the boundary of D to a cusp of lemniscate and to a twi-cusped kidney-shaped nephroid curve in the right half plane, respectively. In this article, we aim to construct functions f∈A0 for which (i) f(D)⊂Pl(D)∩ΦNe(D) (ii) f(D)⊂Pl(D), but f(D)⊄ΦNe(D) (iii) f(D)⊂ΦNe(D), but f(D)⊄Pl(D). We validate the results graphically and analytically. To prove the results analytically, we use the concept of subordination. In this process, we establish the connection lemniscate (and nephroid) domain and functions, including gα(z):=1+αz2, |α|≤1, the polynomial gα,β(z):=1+αz+βz3, α,β∈R, as well as Lerch’s transcendent function, Incomplete gamma function, Bessel and Modified Bessel functions, and confluent and generalized hypergeometric functions.

Time-dependent behaviour of an M/M/3 heterogeneous server queueing system with multiple vacations subject to system disaster

This research paper investigates the dynamic behaviour of a heterogeneous three-server queuing model with multiple vacation. The model also accounts for the possibility that a disaster could occur during busy periods or on vacation, in which case the restoration process would begin immediately. The time-dependent probabilities of system size are presented in the work with explicit equations computed in terms of modified Bessel functions, employing generating functions. To support the theoretical conclusions, numerical examples are given.

Absolute moments of the variance-gamma distribution

We obtain exact formulas for the absolute raw and central moments of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. When the skewness parameter is equal to zero (the symmetric variance-gamma distribution), the infinite series reduces to a single term. Moreover, for the case that the shape parameter is a half-integer (in our parameterisation of the variance-gamma distribution), we obtain a closed-form expression for the absolute moments in terms of confluent hypergeometric functions. As a consequence, we deduce new exact formulas for the absolute raw and central moments of the asymmetric Laplace distribution and the product of two correlated zero mean normal random variables, and more generally the sum of independent copies of such random variables.

Some formulas for Bessel functions related to diag(1, −1, −1)-matrices and an intertwining operator

In this paper, we have developed novel proofs of two known formulas for the Meijer integral transform and obtained non-trivial extensions of these formulas (in one of them, the Bessel function is replaced by the Coulomb wave function). Also we have derived a new formula for a series involving product of Whittaker functions and modified Bessel functions of the second kind. Our proofs are based on a group theoretical approach and involves the use of an intertwining operator for two representations of one matrix group.

Noncontact Measurements of Resonance of Small Round Liquid Surfaces Using Sound Wave to Determine Surface Tension.

Various methods have been developed for measuring the surface tension of liquids, and the present study proposes a new method for measuring the surface tension of liquids. A small (9 mm diameter) round liquid surface was excited by sound waves with a common speaker. Nine liquids possessing various physical properties and functional groups were employed, and three resonance frequencies (frs) were observed for each liquid in a frequency range of 20-180 Hz. The resonances were analyzed with a forced oscillation model. In addition, the amplitudes and phases of the resonance oscillations at various positions of the liquid surface were measured, and the total deformations of the liquid surface were determined. The deformation was compared with the Bessel functions, and the oscillation modes and boundary conditions were decided. Finally, proportional relationships between frs and σ0.5ρ-0.5 (σ: surface tension, ρ: density) with high correlations were obtained, which were supported by a new theoretical equation using hydrodynamics.

Asymptotics of the partition function of the perturbed Gross–Witten–Wadia unitary matrix model

AbstractWe consider the asymptotics of the partition function of the extended Gross–Witten–Wadia unitary matrix model by introducing an extra logarithmic term in the potential. The partition function can be written as a Toeplitz determinant with entries expressed in terms of the modified Bessel functions of the first kind and furnishes a ‐function sequence of the Painlevé equation. We derive the asymptotic expansions of the Toeplitz determinant up to and including the constant terms as the size of the determinant tends to infinity. The constant terms therein are expressed in terms of the Riemann zeta‐function and the Barnes ‐function. A third‐order phase transition in the leading terms of the asymptotic expansions is also observed.

Computation of integral transforms in terms of double hypergeometric series ψ2 with applications

The principal aim of the present paper is to evaluate a new integral transform involving the product of two Bessel Functions of the first kind expressed in terms of the Humbert function ψ2. We also derive some interesting special cases from the main result of this paper. The presented result will be applied to generate some new laser beams called as the Humbert beams of type-II and study there propagation behaviour. From graphical simulations this type of wave seems interesting and useful in optical trapping which is used in some applications for pushing and moving particles such as: biological cells, neutral atoms, fluorescent, DNA molecules and trapping productivity on nano-sized spheres.

On new identities involving zeros of Bessel functions

We derive new identities involving zeros of the Bessel function Jν and some related functions. These are special cases of more general identities obtained in this note, which might also be of interest.

Developments of the I-Function Several Complex Variables

The purpose of this study is to present the I-function of r-variables, a natural extension of thepopular, intriguing, and practical Fox H-function. The results are quite generalized when r=1, at which pointwe get the I-function. Second, we showed that an integral is formed by multiplying the I-function of a singlevariable by the Bessel function and the General polynomial class. A two-variable I-function may be obtainedby evaluating the Sine transform of the product of the Hermite polynomial and Fox's H-function, and itsspecial case, the Cosine transform of the product of the Legendre polynomial and Fox's H-function. Wehave also provided exceptional examples, basic features, and circumstances under which convergenceoccurs. Results for H-functions in r variables are generalized in this study.

A new modified Bessel‐type radial basis function for meshless methods: Utilized in the analysis of free vibration in 2D functionally graded Euler–Bernoulli beams

AbstractRadial basis functions (RBFs) are extensively employed in mesh‐free methods owing to their distinct properties. This study presents a novel RBF formulation based on a modified first‐kind Bessel function, introduced for the first time. The efficacy and precision of the proposed function are assessed through an examination of the free vibrations of Euler–Bernoulli beams composed of two‐directional functionally graded materials. Longitudinal and thickness property variations are modeled in polynomial and exponential forms, respectively. The performance of the novel RBF is scrutinized under various boundary conditions (clamped, simply supported, and free), and comparative analyses are conducted against similar investigations and an RBF based on the first‐kind Bessel function. Convergence analysis of the proposed modified first‐kind Bessel function‐based RBF reveals superior convergence rates compared to the first‐kind Bessel function‐based RBF. Moreover, a comparison between results obtained from modeling using the proposed RBF and exact solutions underscores the adequacy of this approach, with a maximum discrepancy of 4.933% observed under clamped‐free boundary conditions. In essence, the findings suggest that the proposed modified first‐kind Bessel function‐based RBF holds promise for analyzing the free vibrations of functionally graded Euler–Bernoulli beams. The primary aim of this research is to introduce and validate a new RBF based on a modified first‐kind Bessel function for the analysis of free vibrations in Euler–Bernoulli beams made of two‐directional functionally graded materials. The study focuses on evaluating the performance and accuracy of this novel RBF in comparison with existing RBFs and exact solutions. By addressing the limitations of conventional RBFs and proposing an innovative approach, this research aims to enhance the accuracy and efficiency of meshless methods in structural vibration analysis.

Bessel functions on GL(n), I

In the context of the Kuznetsov trace formula, we outline the theory of the Bessel functions on GL(n) as a series of conjectures designed as a blueprint for the construction of Kuznetsov-type formulas with given ramification at infinity. We are able to prove one of the conjectures at full generality on GL(n) and most of the conjectures in the particular case of the long Weyl element; as with previous papers, we give some unconditional results on Archimedean Whittaker functions, now on GL(n) with arbitrary weight. We expect the heuristics here to apply at the level of real reductive groups. A forthcoming paper will address the initial conjectures up to Mellin-Barnes integral representations in the case of GL(4) Bessel functions.

Nonlinear Vlasov and Fokker-Planck Dynamics in Confined Systems with Drag: A Numerical Study

Abstract We explore numerically the behavior of a one-dimensional many-body system consisting of particles that interact through short range repulsive forces, and are also under the effects of drag forces, and of an external confining potential. The statistical dynamics of systems of this kind exhibits interesting links with the thermostatistical formalism based on the Sq non-additive entropies. In the regime of overdamped motion, these systems admit an effective description in terms of a non-linear Fokker-Planck equation. When the overdamped condition is relaxed, and inertial effects are explicitly taken into account, the system can be described by a Vlasov-like effective mean field dynamics. The Vlasov-like description of this type of systems has been recently investigated in the literature from an analytical point of view. In the present contribution we explore the behaviour of these system numerically, through direct molecular dynamics simulations. We consider examples of systems with four different short range repulsive forces, with a dependence on distance given by exponential, Heaviside, Lorentzian, and Bessel functions. The results of our numerical simulations are fully consistent with the predictions derived from the Vlasov-like mean field description. In particular, we verify that, in the asymptotic limit of large times, the system evolves towards a state exhibiting a spatial distribution of particles that coincides with the stationary solution of an appropriate nonlinear Fokker-Planck equation. This limit spatial distribution has the form of a q-Gaussian.

A theoretical framework for robust implementation of in situ measurements of ocean currents and waves in dynamics of mooring systems

We develop an approximated method to solve analytically the equations of motion that describe mooring line dynamics in a one-dimensional model. For the first time, we derive integral closed-form expressions to compute dynamic properties of mooring lines subject to ocean currents and waves of arbitrary time and spatial dependence, in terms of modified Bessel functions. This is done by decomposing the mooring line in three regions where different approximations and mathematical techniques of solution are carried out. Our analytical results provide a robust framework to simulate and analyze extreme realistic oceanic events when data from in situ ocean observation systems are available, regardless of the resolution or coarseness of subsurface measurements and even for long acquisition times. In order to prove the advantages of this approach, we have processed data from two stations in the National Data Buoy Center of the National Oceanic and Atmospheric Administration. From simulations with ocean currents data, we have gained insights into the coupling of the spatial modulation of ocean currents with the characteristic wavelengths of elastic lines. From simulations with ocean waves data, we have defined a scheme to analyze wave data and identify the contribution of each subset of frequency peaks to the net fluctuation of mooring line tension. This could be useful for classification of irregular waves based on their impact on mooring line tension. The development of better tools that integrate theoretical and experimental findings is necessary for the assessment of marine structures under the environmental conditions associated with climate change.

Extreme eigenvalues of random matrices from Jacobi ensembles

Two-term asymptotic formulæ for the probability distribution functions for the smallest eigenvalue of the Jacobi β-Ensembles are derived for matrices of large size in the régime where β > 0 is arbitrary and one of the model parameters α1 is an integer. By a straightforward transformation this leads to corresponding results for the distribution of the largest eigenvalue. The explicit expressions are given in terms of multi-variable hypergeometric functions, and it is found that the first-order corrections are proportional to the derivative of the leading order limiting distribution function. In some special cases β = 2 and/or small values of α1, explicit formulæ involving more familiar functions, such as the modified Bessel function of the first kind, are presented.

On quadrature of highly oscillatory Bessel function via asymptotic analysis of simplex integrals

In this article, two methods for evaluating highly oscillatory Bessel integrals are explored. Firstly, a polynomial is analyzed as an effective approximation of the simplex integral of a highly oscillatory Bessel function based on Laplace transform, and its error rapidly decreases as the frequency increases. Furthermore, the inner product of f and highly oscillatory Bessel function can be approximated by two other forms of inner product by which one depends on a polynomial and the higher derivatives of f, another depends on Bessel function and the interpolation polynomial of f. In addition, three issues related to highly oscillatory Bessel integrals have also been discussed: inequalities for the convergence rate of Filon-type methods, evaluation of Cauchy principal values, and simplified evaluation on infinite intervals. Through some preliminary numerical experiments, our theoretical analysis has been preliminarily confirmed, and the proposed numerical method is accurate and effective.

Efficient computation of Fourier–Bessel transforms for transverse-momentum dependent parton distributions and other functions

We present a method for the numerical computation of Fourier–Bessel transforms on a finite or infinite interval. The function to be transformed needs to be evaluated on a grid of points that is independent of the argument of the Bessel function. We demonstrate the accuracy of the algorithm for a wide range of functions, including those that appear in the context of transverse-momentum dependent parton distributions in Quantum Chromodynamics.

Cutoff Ergodicity Bounds in Wasserstein Distance for a Viscous Energy Shell Model with Lévy Noise

This article establishes explicit non-asymptotic ergodic bounds in the renormalized Wasserstein–Kantorovich–Rubinstein (WKR) distance for a viscous energy shell lattice model of turbulence with random energy injection. The system under consideration is driven either by a Brownian motion, a symmetric α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-stable Lévy process, a stationary Gaussian or α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-stable Ornstein–Uhlenbeck process, or by a general Lévy process with second moments. The obtained non-asymptotic bounds establish asymptotically abrupt thermalization. The analysis is based on the explicit representation of the solution of the system in terms of convolutions of Bessel functions.

Integral inequalities of h‐superquadratic functions and their fractional perspective with applications

The purpose of this article is to provide a number of Hermite–Hadamard and Fejér type integral inequalities for a class of ‐superquadratic functions. We then develop the fractional perspective of inequalities of Hermite–Hadamard and Fejér types by use of the Riemann–Liouville fractional integral operators and bring up with few particular cases. Numerical estimations based on specific relevant cases and graphical representations validate the results. Another motivating component of the study is that it is enriched with applications of modified Bessel function of first type, special means, and moment of random variables by defining some new functions in terms of modified Bessel function and considering uniform probability density function. The results in this paper have not been initiated before in the frame of ‐superquadraticity. We are optimistic that this effort will greatly stimulate and encourage additional research.

Bessel–Bessel–Gaussian vortex laser beams

Abstract We obtain and investigate Bessel–Bessel–Gaussian vortex beams (BBG beams) with the complex amplitude being equal to a product of the Gaussian function with two Bessel functions, whose arguments are expressed as complicated radicals including the cylindrical coordinates and a free parameter that defines the shape of the intensity distribution. If this parameter is small, the intensity has the shape of an inhomogeneous ring. For larger values of this parameter, the intensity has the shape of two arcs or ‘crescents’, oriented by their concave sides to each other. The complex amplitude of such beams is derived in explicit form for an arbitrary distance from the waist. We demonstrate that the BBG beams rotate upon propagation anomalously fast: at a distance much shorter than the Rayleigh length, the intensity distribution is already rotated by almost 45°, whereas typically, the rotation angle of vortex Gaussian beams is equal to the Gouy phase. It is also shown that the parameter of the BBG beam allows controlling its topological charge (TC): when the parameter value is positive and increases, the beam TC also increases stepwise by an even number. Besides, we study two other similar vortex BBG beams: either with four local intensity maxima, lying on the Cartesian coordinates axes, or with one intensity maximum with a crescent shape, whose center is on the horizontal axis. The derived three new families of asymmetric vortex laser beams, whose complex amplitude is described by explicit analytical expressions at an arbitrary distance from the waist, extend the variety of laser beams that can be used for manipulating and rotating microparticles, free space data transmission, and in quantum informatics.