Zeros Of Bessel Functions Research Articles

Recent software developments for special functions in the Santander–Amsterdam project

We give an overview of published algorithms by our group and of current activities and future plans. In particular, we give details on methods for computing special functions and discuss in detail two current lines of research. Firstly, we describe the recent developments for the computation of central and non-central χ-square cumulative distributions (also called Marcum Q-functions), and we present a new quadrature method for computing them. Secondly, we describe the fourth-order methods for computing zeros of special functions recently developed, and we provide an explicit example for the computation of complex zeros of Bessel functions. We end with an overview of published software by our group for computing special functions.

The monotone iterative method and zeros of Bessel functions for nonlinear singular derivative dependent BVP in the presence of upper and lower solutions

In this paper we consider a class of nonlinear singular boundary value problems − ( x α y ′ ( x ) ) ′ + x α f ( x , y ( x ) , x α y ′ ( x ) ) = 0 , 0 < x < 1 , y ′ ( 0 ) = y ′ ( 1 ) = 0 , for α ≥ 1 . We assume that the source function f ( x , y , x α y ′ ) is Lipschitz in x α y ′ and one-sided Lipschitz in y . The initial approximations are an upper solution u 0 ( x ) and a lower solution v 0 ( x ) which can be ordered in one way, v 0 ( x ) ≤ u 0 ( x ) , or the other, u 0 ( x ) ≤ v 0 ( x ) . We propose an iterative scheme and establish the existence of solutions bounded by v 0 and u 0 , and allow ∂ f / ∂ y to take both positive and negative values. The method is constructive in nature and can be used to generate solutions of the nonlinear singular boundary value problems.

Interlacing of positive real zeros of Bessel functions

We unify the three distinct inequality sequences (Abramowitz and Stegun (1972) [1, 9.5.2]) of positive real zeros of Bessel functions into a single one.

On nonsingular potentials of Cox–Thompson inversion scheme

We establish a condition for obtaining nonsingular potentials using the Cox–Thompson inverse scattering method with one phase shift. The anomalous singularities of the potentials are avoided by maintaining unique solutions of the underlying Regge–Newton integral equation for the transformation kernel. As a by-product, new inequality sequences of zeros of Bessel functions are discovered.

A Global Approach with Cutoff Exponential Function, Mathematically Well Defined at the Outset, for Calculating the Casimir Energy: The Example of Scalar Field

A global approach with cutoff exponential functions is used to obtain the Casimir energy of a massless scalar field in the presence of a spherical shell. The proposed method, mathematically well defined at the outset, makes use of two regulators, one of them to make the sum of the orders of Bessel functions finite and the other to regularize the integral involving the zeros of Bessel function. This procedure ensures a consistent mathematical handling in the calculations of the Casimir energy and allows a major comprehension on the regularization process when nontrivial symmetries are under consideration. In particular, we determine the Casimir energy of a scalar field, showing all kinds of divergences. We consider separately the contributions of the inner and outer regions of a spherical shell and show that the results obtained are in agreement with those known in the literature, and this gives a confirmation for the consistence of the proposed approach. The choice of the scalar field was due to its simplicity in terms of physical quantity spin.

Weighted Markov‐type inequalities, norms of Volterra operators, and zeros of Bessel functions

AbstractThe first term of the asymptotics of the best constants in Markov‐type inequalities for higher derivatives of polynomials is determined in the two cases where the underlying norm is the L2 norm with Laguerre weight or the L2 norm with Gegenbauer weight. The coefficient in this term is shown to be the norm of a certain Volterra integral operator which depends on the weight and the order of the derivative. For first order derivatives, the norms of the Volterra operators are expressed in terms of the zeros of Bessel functions. The asymptotic behavior of the coefficients is studied and tight bounds for them are given (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

An efficient algorithm of the electromagnetic dyadic Green’s function in a horizontal-layered anisotropic medium

In this paper, we advance an efficient algorithm of electromagnetic spatial dyadic Greens function in a horizontal-layered anisotropic medium through high order window function and so on. First, we use the high order window function to transform the Sommerfeld integrals of dyadic Greens function into integrals along a generalized steep descent path. And we give a new and higher order of power series of expansion expression of Hankel transforms of the window function and an accurate formula of Lommel function so that the dyadic Greens function can be precisely computed either near or far from the transmitter. Then, we divide the integral path into a series of subintervals based on the zeros of Bessel function and compute integral per subinterval using an improved adaptive Gauss quadrature. Furthermore, an algorithm of the finite continued fraction expansion is used to sum up the integrals on each subinterval to greatly accelerate the convergence velocity of the numerical integration along the descent path. Finally, our numerical results validated the efficiency of the algorithm.

Monotonic sequences related to zeros of Bessel functions

In the course of their work on Salem numbers and uniform distribution modulo 1, A. Akiyama and Y. Tanigawa proved some inequalities concerning the values of the Bessel function J0 at multiples of π, i.e., at the zeros of J1/2. This raises the question of inequalities and monotonicity properties for the sequences of values of one cylinder function at the zeros of another such function. Here we derive such results by differential equations methods.

Intrinsic cylindrical and spherical waves

Intrinsic waveforms associated with cylindrical and spherical Bessel functions are obtained by eliminating the factors responsible for the inverse radius and inverse square radius laws of wave power per unit area of wavefront. The resulting expressions are Riccati–Bessel functions for both cases and these can be written in terms of amplitude and phase functions of order v and wave variable z. When z is real, it is shown that a spatial phase angle of the intrinsic wave can be defined and this, together with its amplitude function, is systematically investigated for a range of fixed orders and varying z. The derivatives of Riccati–Bessel functions are also examined. All the component functions exhibit different behaviour in the near field depending on the order being less than, equal to or greater than 1/2. Plots of the phase angle can be used to display the locations of the zeros of the general Riccati–Bessel functions and lead to new relations concerning the ordering of the real zeros of Bessel functions and the occurrence of multiple zeros when the argument of the Bessel function is fixed.

Glueball spectrum of pure Yang-Mills theory in2+1dimensions

We present details of the analytic computation of the spectrum of lowest spin glueballs in pure Yang-Mills theory in $2+1$ dimensions. The new ingredient is provided by the conjectured new nontrivial expression for the (quasi)Gaussian part of the ground state wave functional. We show that this wave functional can be derived by solving the Schr\odinger equation under certain assumptions. The mass spectrum of the theory is determined by the zeros of Bessel functions, and the agreement with available lattice data is excellent.

Transition from Vibrational to Rotational Regimes in the Pairing Phase

An approximate solution at the critical point of the pairing transition from harmonic vibration (normal nuclear behavior) to deformed rotation (superconducting behavior) in gauge space is found by analytic solution of a collective pairing Hamiltonian. The eigenvalues are expressed in terms of the zeros of Bessel functions of integer order. Comparison of the results to the pairing bands based on the Pb isotopes suggests that this description may provide a simple approach to explaining the observed anharmonicities of the pairing vibrational structure around 208 Pb. The validity of the model is reinforced by comparing the results to an exact treatment of the pairing-phase transition in a two-level model.

Advances in light-front quantization and new perspectives for QCD from AdS/CFT

The light-front quantization of gauge theories in light-cone gauge provides a frame-independent wavefunction representation of relativistic bound states, simple forms for current matrix elements, explicit unitarity, and a Fock space built on a trivial vacuum. The AdS/CFT correspondence has led to important insights into the properties of quantum chromodynamics even though QCD is a broken conformal theory. We have recently shown how a model based on a truncated AdS space can be used to obtain the hadronic spectrum of q q ¯ , qqq and gg bound states, as well as their respective light-front wavefunctions. Specific hadrons are identified by the correspondence of string modes with the dimension of the interpolating operator of the hadron's valence Fock state, including orbital angular momentum excitations. The predicted mass spectrum is linear M ∝ L at high orbital angular momentum, in contrast to the quadratic dependence M 2 ∝ L found in the description of spinning strings. Since only one parameter, the QCD scale Λ Q C D , is introduced, the agreement with the pattern of physical states is remarkable. In particular, the ratio of Δ to nucleon trajectories is determined by the ratio of zeros of Bessel functions. As a specific application of QCD dynamics from AdS/CFT duality, we describe a computation of the proton magnetic form factor in both the space-like and time-like regions. The extended AdS/CFT space-time theory also provides an analytic model for hadronic light-front wavefunctions, thus providing a relativistic description of hadrons in QCD at the amplitude level. The model wavefunctions display confinement at large inter-quark separation and conformal symmetry at short distances. In particular, the scaling and conformal properties of the LFWFs at high relative momenta agree with perturbative QCD. These AdS/CFT model wavefunctions could be used as an initial ansatz for a variational treatment of the light-front QCD Hamiltonian.

Solving Pure Yang-Mills Theory in2+1Dimensions

We analytically compute the spectrum of the spin zero glueballs in the planar limit of pure Yang-Mills theory in $2+1$ dimensions. The new ingredient is provided by our computation of a new nontrivial form of the ground state wave functional. The mass spectrum of the theory is determined by the zeroes of Bessel functions, and the agreement with large $N$ lattice data is excellent.

Electromagnetic interpretation of the quarter-wave stack condition by means of the phase calculation in Bragg fibers

This paper presents an expression for the eigenvalue equation and fields in a Bragg fiber by calculating the phase change that is based on geometrical optics. The quarter-wave stack condition makes it possible to consider that the Bragg fiber has approximately no cladding from the electromagnetic point of view, despite the fact that the Bragg fiber has a periodic cladding. As a result, its eigenvalue equation can be represented in terms of the zeros of Bessel functions and only the core parameters, for a specific case. The eigenvalue equations for HE and EH modes in the Bragg fiber have a formal equivalence to those for TE and TM modes, respectively, in the circular metallic waveguide. Results obtained are in agreement, under a specific limit, with those derived by an asymptotic expansion method.

Critical-Point Description of the Transition from Vibrational to Rotational Regimes in the Pairing Phase

An approximate solution at the critical point of the pairing transition from harmonic vibration to deformed rotation in gauge space is found by analytic solution of the collective pairing Hamiltonian. The eigenvalues are expressed in terms of the zeros of Bessel functions of integer order. The results are compared to the pairing bands based on the Pb isotopes.

Hadronic Spectrum of a Holographic Dual of QCD

We compute the spectrum of light hadrons in a holographic dual of QCD defined on AdS5 x S5 which has conformal behavior at short distances and confinement at large interquark separation. Specific hadrons are identified by the correspondence of string modes with the dimension of the interpolating operator of the hadron's valence Fock state. Higher orbital excitations are matched quanta to quanta with fluctuations about the AdS background. Since only one parameter, the QCD scale Lambda(QCD), is used, the agreement with the pattern of physical states is remarkable. In particular, the ratio of delta to nucleon trajectories is determined by the ratio of zeros of Bessel functions.

Partial fractions expansions and identities for products of Bessel functions

We derive several partial fractions expansions for products of Bessel functions, and use them to prove algebraic relationships between infinite series involving squares of Bessel functions. We also give formulas for the Taylor series coefficients of the zeros of Bessel functions, when the zeros are regarded as functions of the order x of Jx(y)⁠.

Acoustic measurements and computational results on material specimens with harmonic variation of the cross section

The present work represents both an experimental and theoretical investigation of the behavior of finite cylindrical rods with harmonic variation of the cross section. The matrix method was used to compute the transfer power spectra of elastic rods with uniform circular cross section and of rods with harmonic variation of the cross section with distance. Theoretical and experimental results show that for a rod with periodical variation of the cross section, a new set of supplementary frequencies appear for which the transfer power coefficient has significant values, which are in relation with the space period of the inhomogeneity. Also, due to the radial component of the displacement certain modes are enhanced which satisfy boundary conditions on the surface and are obtained from the zeroes of Bessel functions.

New inequalities from classical Sturm theorems

Inequalities satisfied by the zeros of the solutions of second-order hypergeometric equations are derived through a systematic use of Liouville transformations together with the application of classical Sturm theorems. This systematic study allows us to improve previously known inequalities and to extend their range of validity as well as to discover inequalities which appear to be new. Among other properties obtained, Szegő's bounds on the zeros of Jacobi polynomials P n ( α , β ) ( cos θ ) for | α | < 1 2 , | β | < 1 2 are completed with results for the rest of parameter values, Grosjean's inequality (J. Approx. Theory 50 (1987) 84) on the zeros of Legendre polynomials is shown to be valid for Jacobi polynomials with | β | ⩽ 1 , bounds on ratios of consecutive zeros of Gauss and confluent hypergeometric functions are derived as well as an inequality involving the geometric mean of zeros of Bessel functions.

A new summability property of the zeros of Bessel functions

A new summability property of the zeros of Bessel functions