Zeros Of Bessel Functions Research Articles

Uniform Asymptotic Expansions for the Zeros of Bessel Functions

Uniform Asymptotic Expansions for the Zeros of Bessel Functions

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.

Calibration complex of radio measuring equipment

Modern trends in the development of radio measuring equipment require improvement of the principles of metrological support of these devices and corresponding reference base. The new approach is based on the digitalisation of equipment, programming of measurement procedures, and the transition from single measurement standards to measurement complexes. The paper describes the complex developed at the NSC “Institute of Metrology” and designed for verification and calibration of a wide range of radio measuring equipment, where all these approaches are realised, presents methodological, software and hardware features of its construction, as well as the advantages and possibilities of this approach in metrological practice. On the example of the method of “Bessel Function Zeros”, which was somewhat forgotten due to the cumbersome nature of its implementation by analogue methods and equipment, it is shown that with the new approach this method gets a “second birth”, moreover, at a higher metrological level. It is shown that the use of direct digital synthesis oscillators can significantly improve the “electronic counting frequency meter” method, in particular, prevent the phenomenon of “frequency capture”, which limited the capabilities of the method and introduced additional measurement uncertainty. Moreover, the additional properties of the complex associated with the use of the R&S FSL spectrum analyser with the K7 analogue signal demodulator option are explained in more detail. A list of other methods realised in the complex is given, among which the “combination frequencies” method, deserves a special attention, providing the highest resolution in measuring nonlinear distortions of both quasi-harmonic and modulated signals. It is shown that the complex is a flexible apparatus, the work programme of which can be adjusted and supplemented. Information on the structure of the complex, hardware solution of a number of nodes, programming language, etc. is given.

Pilot-wave dynamics in a rotating frame: the onset of orbital instability

We report the results of a theoretical investigation of the stability of a hydrodynamic analogue of Landau levels, specifically circular orbits arising when a millimetric droplet self-propels along the surface of a vibrating, rotating liquid bath. Our study elucidates the form of the stability diagram characterising the critical memory at which circular orbits destabilise, and the form of instability. Particular attention is given to rationalising observations reported in prior experimental works, including the prevalence of resonant wobbling instabilities, in which the instability frequency is approximately twice the orbital frequency. We also explore the physical mechanism responsible for the onset of instability. Specifically, we compare the efficacy of different heuristic arguments proposed in prior studies, including propositions that the most unstable orbits arise when their radii correspond to the zeros of Bessel functions or when their associated wave intensity is extremised. We establish a new relation between orbital stability and the mean wave field, which supersedes existing heuristic arguments and suggests a rationale for the alternate wobbling and monotonic instabilities arising at onset as the orbital radius is increased progressively.

Two formulae with nodes related to zeros of Bessel functions for semi-infinite integrals: extending Gauss–Jacobi-type rules

For approximating integrals $$\varvec{\int _0^{\infty }\!\!} \varvec{x}^{\varvec{\alpha }}\varvec{f(x)dx}$$ ( $$\varvec{\alpha >-1}$$ ) over a semi-infinite interval $$\varvec{[0,\infty )}$$ with a given function $$\varvec{f(x)}$$ , two formulae, one of them new and another associated with an existing formula, are presented. They are constructed in a limiting process to a semi-infinite interval $$\varvec{[0,\infty ]}$$ with a linear transformation for a well-known approximation method, the Gauss–Jacobi (GJ) rule and its family rules: the Gauss–Jacobi–Radau (GR) and Gauss–Jacobi–Lobatto (GL) rules on a finite interval. This procedure was used in constructing our previous limit Clenshaw–Curtis-type formulae. The limit GJ formula (LGJ) constructed in this way uses as nodes zeros of the Bessel function $$\varvec{\varvec{J_{\alpha }(x)}}$$ squared after multiplied by a positive constant a and the limit GR (LGR) (and limit GL (LGL)) formula those with zeros of $$\varvec{J_{\alpha +1}(x)}$$ . The LGJ formula is also shown to be derived from the formula developed by Frappier and Olivier (Math. Comp. 60:303–316, 1993) for an integral on $$\varvec{[0,\infty )}$$ . The LGR and LGL formulae give the same and new formula. We show that for a function f(z) analytic on a domain in the complex plane z and satisfying some appropriate conditions, there exists a constant $${d>0}$$ such that the errors of both formulae are $$\varvec{O(e^{-2d/a})}$$ as $$\varvec{a\rightarrow +0}$$ . The average of the LGJ and LGR formulae gives smaller quadrature errors than each formula. Numerical examples confirm these behaviors and show that the LGJ and LGR formulae give asymptotically the same quadrature errors of opposite sign. Consequently, the LGR formula behaves like an anti-LGJ formula in the same way as the Lobatto rule for integrals on $$\varvec{[-1,1]}$$ behaving like the anti-Gauss rule.

On the spectrum of a mixed-type operator with applications to rotating wave solutions

We study rotating wave solutions of the nonlinear wave equation ∂t2v-Δv+mv=|v|p-2vinR×Bv=0onR×∂B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} \\partial _{t}^2 v - \\Delta v + m v = |v|^{p-2} v &{} \ ext {in }{\\mathbb {R} \ imes {\ extbf{B}}} \\\\ v = 0 &{} \\hbox {on }\\mathbb {R} \ imes \\partial {\ extbf{B}} \\end{array} \\right. \\end{aligned}$$\\end{document}where 2<p<infty , m in mathbb {R} and {textbf{B}} subset mathbb {R}^2 denotes the unit disk. If the angular velocity alpha of the rotation is larger than 1, this leads to a semilinear boundary value problem on textbf{B} involving a mixed-type operator, whose spectrum is related to the zeros of Bessel functions and could generally be badly behaved. Based on new estimates for these zeros, we find values of alpha such that the spectrum only consists of eigenvalues with finite multiplicity and has no accumulation point. Combined with suitable spectral estimates, this allows us to formulate an appropriate indefinite variational setting and find ground state solutions of the reduced equation for p in (2,4). Using a minimax characterization of the ground state energy, we ultimately show that these ground states are nonradial and thus yield nontrivial rotating waves, provided m is sufficiently large.

About Inverse Laplace Transform of a Dynamic Viscosity Function.

A dynamic viscosity function plays an important role in water hammer modeling. It is responsible for dispersion and decay of pressure and velocity histories. In this paper, a novel method for inverse Laplace transform of this complicated function being the square root of the ratio of Bessel functions of zero and second order is presented. The obtained time domain solutions are dependent on infinite exponential series and Calogero–Ahmed summation formulas. Both of these functions are based on zeros of Bessel functions. An analytical inverse will help in the near future to derive a complete analytical solution of this unsolved mathematical problem concerning the water hammer phenomenon. One can next present a simplified approximate form of this solution. It will allow us to correctly simulate water hammer events in large ranges of water hammer number, e.g., in oil–hydraulic systems. A complete analytical solution is essential to prevent pipeline failures while still designing the pipe network, as well as to monitor sensitive sections of hydraulic systems on a continuous basis (e.g., against possible overpressures, cavitation, and leaks that may occur). The presented solution has a high mathematical value because the inverse Laplace transforms of square roots from the ratios of other Bessel functions can be found in a similar way.

Weighted Uniform Convergence of Entire Grünwald Operators on the Real Line

We consider weighted uniform convergence of entire analogues of the Grünwald operator on the real line. The main result deals with convergence of entire interpolations of exponential type \(\tau >0\) at zeros of Bessel functions in spaces with homogeneous weights. We discuss extensions to Grünwald operators from de Branges spaces.

Azimuthal jittered sampling of bandlimited functions in the two-dimensional Fourier transform and the Hankel transform domains

Stark’s sampling theorems in polar coordinates have important significance in the field of medical imaging. However, the direct use of Stark’s results may mislead as in most real-world scenarios the jitter occurs in azimuth due to uncertainty of sampling at the medical imaging system transmitter end. On the basis of the classical Stark’s interpolation formulae, this paper addresses the image reconstruction problem when the sample data are collected on nonuniform polar radius and polar azimuth lattices, where the sample points are normalized zeros of Bessel functions in radius while random jittered ones in azimuth that are clustered around uniform points with a given probability distribution. By using the existing basis functions found in Stark’s formulae, it follows a biased estimate of the original angularly periodic images that are bandlimited both to the highest frequency and in the Fourier transform or the Hankel transform domain. With the newly designed angular generating functions, there are a kind of unbiased estimators whose variance approaches zero as the jitter becomes less and less pronounced.

Sound source localization using multiple circular microphone arrays based on harmonic analysis.

This paper proposes a multiple signal classification (MUSIC) framework for direction-of-arrival estimation by combining multiple circular arrays in the circular-harmonics domain. We jointly transform the received signals of all sub-arrays into the circular harmonics domain to generate a set of sound field coefficients containing the direction information of the sound sources. These coefficients are then formulated in a form in which MUSIC algorithm can be applied. Compared with the conventional circular-harmonics-domain localization methods, which are based on a single circular array, the proposed method can provide sufficient spatial resolution over different frequency ranges by adjusting the distribution of sub-arrays. Furthermore, the mean square error of the estimated sound field coefficients is derived for guiding this adjustment. Numerical simulation results indicate that a reasonable distribution of sub-arrays can effectively avoid the performance degradation caused by the zeros of Bessel functions, which is an inherent problem of the modal array signal processing. Simulation and experimental results with different configuration parameters demonstrate that the proposed method provides a better localization performance compared to the state-of-the-art methods.

Collective interlacing and ranges of the positive zeros of Bessel functions

As a generalization of the classical interlacing property for pairs of Bessel functions, it is shown that the positive zeros of ν-parameter families of Bessel functions Jν or Yν are simultaneously interlaced under certain conditions. Our method of proof is based on the range description of each ν-variation map of positive zeros of the same rank.

Parametrization of algebraic differentiators for disturbance annihilation with an application to the differentiation of quantized signals

A method for the systematic parametrization of algebraic differentiators is introduced. It allows the annihilation of disturbances having transfer functions exhibiting repetitive peaks. Using McMahon's expansion for large zeros of Bessel functions, an approach reducing the number of free parameters of the differentiators from five to one is derived. The choice of the parameters is discussed in detail, and the existence of the parametrization for any such disturbance is proven. Error bounds for the annihilation of the repetitive peaks are also derived. The practical applicability of the approach is demonstrated in an experimental case study where the derivative of a quantized signal is numerically estimated using only an algebraic differentiator. A deterministic model for the quantization error which shows repetitive peaks in its transfer function is also proposed.

A physicist’s guide to explicit summation formulas involving zeros of Bessel functions and related spectral sums

In this pedagogical review, we summarize the mathematical basis and practical hints for the explicit analytical computation of spectral sums that involve the eigenvalues of the Laplace operator in simple domains such as [Formula: see text]-dimensional balls (with [Formula: see text]), an annulus, a spherical shell, right circular cylinders, rectangles and rectangular cuboids. Such sums appear as spectral expansions of heat kernels, survival probabilities, first-passage time densities, and reaction rates in many diffusion-oriented applications. As the eigenvalues are determined by zeros of an appropriate linear combination of a Bessel function and its derivative, there are powerful analytical tools for computing such spectral sums. We discuss three main strategies: representations of meromorphic functions as sums of partial fractions, Fourier–Bessel and Dini series, and direct evaluation of the Laplace-transformed heat kernels. The major emphasis is put on a pedagogic introduction, the practical aspects of these strategies, their advantages and limitations. The review gathers many summation formulas for spectral sums that are dispersed in the literature.

Quantum mechanical analogue of optical microdisk resonators

Abstract The complex wave number plane furnishes us with a natural way to seek resonant modes of quantum potentials and optical cavities. In this work we present the solution of the quantum mechanical analogue of an optical microdisk resonator. The rich dynamical behavior of the states is shown for a circular potential barrier in response to continuous variations of the parameters of the potential. In particular, the flows of the resonances go to limit points which are zeros of Hankel functions in complex wave number coordinates and zeros of Bessel functions in an alternative complex potentiodynamic coordinate system.

A generalized spectral theory for continuous metrics on compact Riemann surfaces

We extend the spectral theory of generalized Laplacians to continuous metrics on compact Riemann surfaces. We define a holomorphic analytic torsion for any continuous metric. As an application of this theory, we partly recover some results of the theory of Bessel functions, for instance, Lommel's theorem on the reality of the zeros of Bessel functions of order exceeding −1.

Interview

Jorge Ruiz-Cruz Dr Jorge Ruiz-Cruz from the Autonomous University of Madrid, Spain, talks to Electronics Letters about the paper ‘Analytical Expressions of the Q-Factor for the Complete Resonant Mode Spectrum of the Equilateral Triangular Waveguide Cavity’, page 944. Our field of research is the analysis and design of microwave and millimetre-wave passive devices, especially in waveguide technology. We work on numerical methods in the context of applied electromagnetics for engineering, with both quasi-analytical methods such as mode-matching or more generic techniques such as the finite element method. With these methods, we combine our expertise in the theoretical aspects of electromagnetic fields and microwave circuits with advanced optimisation methods. The result is the design of high-performance microwave devices such as filters, power dividers, multiplexers, polarisers, ortho-mode transducers, etc. We are particularly interested in developing different kinds of networks for radar systems and satellite communications using novel synthesis and manufacturing techniques. The work in our Letter is related to the numerical methods and microwave component research we have performed in recent years, especially with waveguide cavity filters for satellite communications. We have noticed that with the advance of new manufacturing technologies, there is a broad range of problems that were not addressed in the past because they were not practical, and now we can approach them since the fabrication of diverse geometries is not an issue anymore. In this way, a common geometry such as the equilateral triangular cavity can be used for other applications like the design of microwave filters or for checking numerical codes. Although the Q-factor of waveguide cavity resonators is a well-established concept in the microwave and millimetre-wave community (and in other related fields) the range of problems with a closed-form solution is very limited. For microwave and millimetre-wave practitioners, it is common to use the closed-form formulas found in the literature for rectangular or circular waveguide cavities, for diverse applications such as implementing filters or other circuits, as reference cases for checking numerical methods, or for calibration in measurement systems. With this work, we contribute a new closed-form expression not reported previously, since the equilateral triangle cavity had not been treated in detail, as have other canonical geometries, in the literature. The new analytical expressions provided for the equilateral triangular waveguide cavity in the paper are as simple as in the case of the rectangular waveguide cavity (the ones for the circular waveguide cavity are also simple, but they involve zeroes of Bessel functions). We have derived the closed-form expressions with works from other authors, who have presented the complete solution to the Helmholtz equation with homogeneous Neumann and Dirichlet boundary conditions in the specific geometry of an equilateral triangle. Starting with those solutions, we have been able to obtain very simple expressions for the Q-factor after a careful integration process. The analytical closed-form expressions that we have presented complement the classic cases (rectangular and circular waveguide resonators). They have potential use in microwave filters, validation of numerical solvers, or integration with microwave measurement systems. Cavities with larger volumes usually provide resonant modes with greater Q-factor. However, in this case, the spurious-free window is usually reduced. Filter designers must pursue a trade-off between these two parameters, so that a good out-of-band rejection and low insertion losses in the passband are achieved. To that end, resonant mode charts and Q-charts, like the ones provided in our Letter, are typically used to design the physical dimensions of the resonators, which can be used in microwaves up till the terahertz regime. This work has applications in several fields. In the area of filter design, we are developing several prototypes at different frequency bands in order to test the approach, along with a variety of topologies and manufacturing techniques. In the area of electromagnetic analysis, we plan to incorporate the closed-form expressions developed in this work not only for testing but also as part of in-house hybrid analysis techniques we are currently developing. Microwave and millimetre-wave engineering is continuously taking advantage of both new manufacturing facilities and enhanced computational capabilities. Regarding the former, technology is always changing the way we, engineers, think when a design is tackled, as new construction geometries and topologies are enabled with novel manufacturing methods. Therefore, problems that in the past might have been looked at as merely theoretical ones may now become of real practical interest. As for computational capabilities, no matter how high they reach, there will always be a need to develop efficient and accurate analysis methods to be included in microwave design tools. We strongly believe this contribution unites these two approaches and we will work on developing similar derivations and techniques.

Primary Standard of the Unit of Frequency Deviation. New Opportunities and Prospectives

The possibility of expanding working frequency ranges and increasing the precision of State Primary Special Standard of the unit of frequency deviation GET 166–2004 is considered. Methods and means of measuring the frequency deviation that may be incorporated into the improved GET 166–2004 are proposed. Results of studies of the following four independent methods of measuring frequency deviation are presented: method of electronic computer frequency meter; method of direct digital synthesis; method of Bessel function zeros; and method of measuring the extremal values of the period.

Explicit tight bounds on the stably recoverable information for the inverse source problem

For the inverse source problem with the two-dimensional Helmholtz equation, the singular values of the source-to-near-field operator reveal a sharp frequency cut-off in the stably recoverable information on the source. We prove and numerically validate an explicit, tight lower bound for the spectral location of this cut-off. We also conjecture, justify and support numerically a tight upper bound for the cut-off. The bounds are expressed in terms of zeros of Bessel functions of the first and second kind. Finally, we propose our near-field upper bound as an improvement of a commonly used upper bound on the spectral cutoff for the source-to-far-field operator.

On the paper “On an identity for the zeros of Bessel functions” by Baricz et al.

In this note we offer some criticism on the paper “On an identity for zeros of Bessel functions” by Baricz et al. [3]. The paper gives identities of type Stieltjes–Calogero for the sums of reciprocals of differences of fourth powers of zeros of Bessel functions. Although interesting in principle, by containing one too many sums of similar complexity the identities fail to convey the true spirit of the work of Stieltjes and Calogero. We rectify this by providing what we think is the correct type of identity for the above-said sums, in the general setup of entire functions of order <2.

Bernoulli–Dunkl and Apostol–Euler–Dunkl polynomials with applications to series involving zeros of Bessel functions

We introduce Bernoulli–Dunkl and Apostol–Euler–Dunkl polynomials as generalizations of Bernoulli and Apostol–Euler polynomials, where the role of the derivative is now played by the Dunkl operator on the real line. We use them to find the sum of many different series involving the zeros of Bessel functions.