Integrals Of Bessel Functions Research Articles

Triple-spherical Bessel function integrals with exponential and Gaussian damping: towards an analytic N-point correlation function covariance model

Spherical Bessel functions (sBFs) appear commonly in many areas of physics wherein there is both translation and rotation invariance, and often integrals over products of several arise. Thus, analytic evaluation of such integrals with different weighting functions (which appear as toy models of a given physical observable, such as the galaxy power spectrum) is useful. Here, we present a generalization of a recursion-based method for evaluating such integrals. It gives relatively simple closed-form results in terms of Legendre functions (for the exponentially damped case) and Gamma, incomplete Gamma, and hypergeometric functions (for the Gaussian-damped case). We also present a new, non-recursive method to evaluate integrals of products of sBFs with Gaussian damping in terms of incomplete Gamma functions and hypergeometric functions.

A novel method for calculating dislocation Green's functions and deformation in a transversely isotropic and layered elastic half-space

A novel and comprehensive method is proposed for calculating the dislocation Love numbers (DLNs), Green's functions (GFs), and the corresponding deformation in a transversely isotropic and layered elastic half-space. It is based on the newly introduced Fourier-Bessel series system of vector functions, along with the dual variable and position method. Two important features associated with this new system are: (1) it is much faster than the conventional cylindrical system of vector functions; (2) we can even pre-calculate the DLNs which are only possible in terms of this new system. This is due to the fact that the variables to be solved in the new system are functions of the simple discrete zero points of the Bessel functions, instead of the numerical integration of continuous Bessel functions between the neighboring zero points as in the conventional system. The introduced dual variable and position method is unconditionally stable as compared to the traditional propagator matrix method in dealing with layering. Exact asymptotic expressions of the DLNs for large wavenumber are further derived, which makes the Kummer's transformation applicable in accelerating the convergence of the corresponding GFs. For the reduced case of homogeneous and isotropic half-space, the present solutions amazingly reduce to the existing exact closed-form solutions. These new features are further seamlessly combined for calculating the deformation due to a finite dislocation (or a finite fault in geophysics) in the layered structure, which are demonstrated to be accurate and efficient.

The 𝑊^{𝑠,𝑝}-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential

In this paper, we investigate the W s , p W^{s,p} -boundedness for stationary wave operators of the Schrödinger operator with inverse-square potential L a = − Δ + a | x | 2 , a ≥ − ( d − 2 ) 2 4 , \begin{equation*} \mathcal L_a=-\Delta +\tfrac {a}{|x|^2}, \quad a\geq -\tfrac {(d-2)^2}{4}, \end{equation*} in dimension d ≥ 2 d\geq 2 . We construct the stationary wave operators in terms of integrals of Bessel functions and spherical harmonics, and prove that they are W s , p W^{s,p} -bounded for certain p p and s s which depend on a a . As corollaries, we solve some open problems associated with the operator L a \mathcal L_a , which include the dispersive estimates and the local smoothing estimates in dimension d ≥ 2 d\geq 2 . We also generalize some known results such as the uniform Sobolev inequalities, the equivalence of Sobolev norms and the Mikhlin multiplier theorem, to a larger range of indices. These results are important in the description of linear and nonlinear dynamics for dispersive equations with inverse-square potential.

Numerical methods for Cauchy principal value integrals of oscillatory Bessel functions

In this paper, we first propose different combination methods to compute the Cauchy principal value integrals of oscillatory Bessel functions. By special transformations, the considered integrals are converted to finite integrals and infinite integrals. Then, the finite integrals can be calculated through the Filon-type method, the Clenshaw–Curtis–Filon method and the Clenshaw–Curtis–Filon-type method, respectively. We compute the infinite integral through the numerical steepest descent method. Moreover, the error analysis with respect to frequency ω is given through theoretical analysis. Eventually, we present several numerical experiments which are in accord with our analysis. Particularly, the accuracy can be improved by either using more nodes or adding more derivatives interpolation at endpoints. The accuracy will increase drastically with the growth of frequency ω if both the number of nodes and interpolated multiplicity are fixed.

A central limit theorem for integrals of random waves

Abstract We derive a central limit theorem for the mean-square of random waves in the high-frequency limit over shrinking sets. Our proof applies to any compact Riemannian manifold of dimension 3 or higher, thanks to the universality of the local Weyl law. The key technical step is an estimate capturing some cancellation in a triple integral of Bessel functions, which we achieve using Gegenbauer’s addition formula.

Fractional Integral Inequalities concerning Extended Bessel Function in the Kernel

The major purpose of this paper is to use the fractional integral operator in terms of extended generalized Bessel function to estimate new fractional integral inequalities for the extended Chebyshev functional in the sense of synchronous functions. We prove a set of inequalities for the fractional integral operator in terms of extended generalized Bessel function integrals with one and two parameters. Also, we discussed some special cases of the obtained result.

Fuzzy Differential Subordination of the Atangana–Baleanu Fractional Integral

The present paper continues the study on the relatively new concept of fuzzy differential subordination conducted in some recently published cited papers. In this article, certain fuzzy subordination results for analytical functions involving the Atangana–Baleanu fractional integral of Bessel functions are presented. Theorems giving the best dominants for some fuzzy differential subordinations are proved, and interesting corollaries are provided with the use of particular functions as fuzzy best dominants.

An Application of the Principle of Differential Subordination to Analytic Functions Involving Atangana–Baleanu Fractional Integral of Bessel Functions

The aim of this paper is to establish certain subordination results for analytic functions involving Atangana–Baleanu fractional integral of Bessel functions. Studying subordination properties by using various types of operators is a technique that is widely used.

On the spherical expansion for calculating the sound radiated by a baffled circular piston

An efficient and accurate method for calculating the sound radiated by a baffled circular rigid piston is using spherical harmonics, and the solution is a series containing the integral of spherical Bessel functions. The integral is usually calculated with the generalized hypergeometric functions in existing literatures, which shows poor convergence at middle and high frequencies due to the overflow and the loss of significant figures. A rigorous and closed form solution of the integral is derived in this paper based on the recurrence method, which is accurate in the whole frequency range and thousands of times faster than the existing methods. It is shown that the proposed method can be extended for the calculation of the sound radiated by a baffled piston and an unbaffled resilient disk with axisymmetric velocity and pressure profiles, respectively, and some baffled rotating sources where the velocity profile is asymmetric.

On the fully anisotropic honeycomb lattice Green function, Bessel function integrals and Pearson random walks

The analytical properties of the lattice Green function for the fully anisotropic honeycomb lattice are studied, where (α 1, α 2, α 3) are anisotropy parameters with {α j ∈ (0, ∞): j = 1, 2, 3}, and w = u + iv is a complex variable in a (u, v) plane. This integral defines a single-valued analytic function G H(α 1, α 2, α 3; w) provided that a cut is made along the real axis from u = −(α 1 + α 2 + α 3) to u = (α 1 + α 2 + α 3). We show that G H(α 1, α 2, α 3; w) is a solution of a second-order linear differential equation with ten ordinary regular singular points and four apparent singular points. The apparent singularities are removed by constructing a particular differential equation of fourth order. Next, the series solution where |w| > (α 1 + α 2 + α 3), and is introduced. It is proved that, in general, satisfies a five-term linear recurrence relation. The asymptotic behaviour of as n → ∞ is also established. In order to determine the behaviour of G H(α 1, α 2, α 3; w) along the edges of the cut we define the limit function where u ∈ [−(α 1 + α 2 + α 3), (α 1 + α 2 + α 3)]. Integral representations are established for and . In particular, it is found that where J 0(z) and Y 0(z) denote Bessel functions of the first and second kind, respectively, and u ∈ (0, α 1 + α 2 + α 3). It is also demonstrated that the piecewise functions and can be sectionally evaluated exactly for all u ∈ (0, α 1 + α 2 + α 3), in terms of complete elliptic integrals of the first kind K(k), where k 2 ≡ k 2(α 1, α 2, α 3, u) is a rational function of (α 1, α 2, α 3) and u. Finally, applications of the results are made to the lattice Green function for the fully anisotropic simple cubic lattice, and to the theory of Pearson random walks in a plane. In particular, various Bessel function integrals are evaluated in order to derive a new exact formula for the mean end-to-end distance of a general three-step random walk.

Interfacial debonding of an orthotropic half-plane bonded to a rigid foundation

A partial debonding of the interface of an orthotropic elastic half-plane bonded to a rigid foundation is studied when shear and compressive loads are applied. A debonding/crack face smoothly contacts with a rigid base under applied compressive load, but relatively slides. The Fourier transform method is used to reduce a mixed boundary value problem to dual integral equations. Full elastic fields at any position are obtained in terms of preliminary functions with the aid of the well-known integrals of Bessel functions for odd and even shear loads. Asymptotic elastic stresses near the debonding tips together with stress intensity factors are determined explicitly. Only inverse square-root singularity near the debonding tips appears. The influence of material orthotropy parameters on energy release rate and stress distribution is discussed.

Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

AbstractExplicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for some smoothing estimates as well as for the L2 restrictions of Fourier transforms onto spheres in which are independent of the radius of the sphere. For more special weights these restrictions are shown to be Hölder continuous with a Hölder constant having this independence as well. To illustrate the use of these results a uniform resolvent estimate of the free Dirac operator with mass in dimensions is derived.

Magnetic Field Distortion From Conductive Layers for High-Frequency Speed Sensor Applications

In previous works, the known scalar approach for the vector potential was extended to determine the influences of eddy currents in metallic layers in a closed form, based on a connection between infinite integrals of Bessel functions and the complete elliptic integrals. While the original idea treats only thin layers with respect to the penetration depth, in this paper, the formalism is extended to the opposite limit of thick layers. It was shown that the developed formalism can be used as a framework to analyze and classify the development of eddy currents in speed sensor systems and even to advocate on chip layout. This extension to thick layers is of great interest in this context as it describes the regime where eddy currents strongly influence the sensor output signal.

Feedback of Eddy Currents in Layered Materials for Magnetic Speed Sensing

In this paper, we present a new analytical approach for studying the development and the influence of eddy currents in layered systems of cylindrical symmetry. It is based on the development of a scalar model and exploits a connection between the complete elliptic integral and infinite integrals of Bessel functions to find the solution in an analytic form. The method is specifically designed to help identify the critical model parameters and to understand their functional dependencies. In the context of magnetic speed sensor systems, connections between the feedback of eddy currents and critical parameters, such as penetration depth, lead frame thickness, and the position of the sensitive elements, are investigated, aiming to advocate on sensor layouts to improve signal stability. The developed formalism also bridges the gap between the scalar approach and the well-known off-axis magnetic field of a current loop. In addition, a collection of solutions for infinite integrals of Bessel functions is included of which only a few special cases are found in the literature.

Hydro-acoustic frequencies of the weakly compressible mild-slope equation

We present a novel analytical solution for hydro-acoustic waves in a weakly compressible fluid flow over a slowly varying bottom. Application of a multiple-scale perturbation technique and matched asymptotic analysis leads to a uniform analytical solution of the depth-averaged governing equations in three dimensions. We show that the slow depth variation has a leading-order effect on the evolution of the normal mode amplitude and direction. This dynamics is much richer than the two-dimensional limit analysed in previous studies. For tsunamigenic disturbances, we show that the hydro-acoustic wave field is made up by longshore trapped and offshore propagating components, which were not explicated in previous work. For a plane beach, we find an exact analytical solution of the model equation in terms of integrals of Bessel functions. Our model offers a physical insight into the evolution of hydro-acoustic waves of interest for the design of tsunami early warning systems.

WITHDRAWN: Evaluation of a family of Bessel function integrals

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Numerical Simulation of Unsteady-State Flow in Dual Porous Coalbed Methane Horizontal Wells with Complex Boundary Conditions

The bottom-hole pressure response which can reflect the gas flow characteristics is important to study. A mathematical model for description of gas from porous coalbed methane (CBM) reservoirs with complex boundary conditions flowing into horizontal wells has been developed. Meanwhile, basic solution of boundary elements has been acquired by combination of Lord Kelvin point source solution, the integral of Bessel function, and Poisson superimpose formula for CBM horizontal wells with complex boundary conditions. Using this model, type curves of dimensionless pressure and pressure derivative are obtained, and flow characteristics of horizontal wells in complex boundary reservoirs and relevant factors are accordingly analyzed.

On a Generalized Multipole Expansion with Application to Propeller Design Synthesis

The radiation of sound by surfaces in motion in a non-uniform flow, including the effects of reflections from obstacles on noise, is specified by an extension of the Kirchhoff integral that leads to a generalized multipole expansion that extends the classical series of spherical harmonics to account for the effects of ( i) the moving medium and ( ii) the rotating sound sources. The corresponding radiation integrals are evaluated analytically for an arbitrary source distribution along the blades of a propeller at an angular inflow. The effects of the incident flow and propeller rotation on the amplitude and phase of sound, e.g. through the retarded time, lead to an extension of the generating function for Legendre polynomials; this provides the representation of the acoustic field as the generalized multipolar series. Each generalized multipole sound field is shown to consist of a fundamental blade passng frequency (BPF) plus all harmonics, with directivities specified by integrals of Bessel functions, that are evaluated analytically. The theory is validated by comparison with noise measurements using two different model propellers in two distinct aeroacoustic wind tunnels. The experimental results are used to illustrate the principle of propeller noise synthesis, relating the radiated sound field to the sound source distribution.

The modal noise covariance matrix for an array of vector sensors

A modal noise covariance matrix for an array of vector sensors is presented. It is assumed that the sensors measure pressure and gradients or velocities on three axes. The noise covariance matrix is obtained as a discrete modal sum. The derivation relies on the differentiation of the complex pressure field and the application of a set of Bessel function integrals. The modal representation is restricted to a horizontally stratified environment and assumes that the noise sources form a layer of uncorrelated monopoles. The resulting noise field is horizontally isotropic, but vertically non-isotropic. Particular attention is paid to the effect of the noise source intensity on the normalization of the covariance matrix and, consequently, to the effect of noise on the output of the array of vector sensors.

Time-Domain Absorbing Boundary Terminations for Waveguide Ports Based on State-Space Models

Absorbing boundary conditions for waveguide ports in time domain are important elements of transient approaches to treat RF structures. A successful way to implement these termination conditions is the decomposition of the transient fields in the absorbing plane in terms of modal field patterns. The absorbing condition is then accomplished by transferring the wave impedances (or admittances) of the modes to time domain, which leads to convolution operations involving Bessel functions and integrals of Bessel functions. This paper presents a new alternative approach: the convolution operations are approximated by appropriate state-space models whose system responses can be conveniently computed by standard integration schemes. These schemes are indispensable for transient simulations anyhow. Sufficiently far away from the cutoff frequency, a wideband match is achieved.