Bessel Functions Of Order Research Articles

Direct sampling method for imaging small dielectric inhomogeneities: analysis and improvement

The direct sampling method (DSM) has been introduced for non-iterative imaging of small inhomogeneities and is known to be fast, robust, and effective for inverse scattering problems. However, to the best of our knowledge, a full analysis of the behavior of the DSM has not been provided yet. Such an analysis is proposed here within the framework of the asymptotic hypothesis in the 2D case leading to the expression of the DSM indicator function in terms of the Bessel function of order zero and the sizes, shapes and permittivities of the inhomogeneities. Thanks to this analytical expression the limitations of the DSM method when one of the inhomogeneities is smaller and/or has lower permittivity than the others is exhibited and illustrated. An improved DSM is proposed to overcome this intrinsic limitation in the case of multiple incident waves. Then we show that both the traditional and improved DSM are closely related to a normalized version of the Kirchhoff migration. The theoretical elements of our proposal are supported by various results from numerical simulations with synthetic and experimental data.

High accuracy approximation for the modified Bessel function of fractional order I1/3(x)

The modified Bessel functions of fractional order I1/3(x) has been approximated by an analytic function containing rational and hyperbolic fumctions. A technique using both power series and assymptotic expansion has been used. The accuracy of the approximation is very high using only three parameters., and the largest relative error is smaller than 0,004.

Motions about a fixed point by hypergeometric functions: new non-complex analytical solutions and integration of the herpolhode

We study four problems in the dynamics of a body moving about a fixed point, providing a non-complex, analytical solution for all of them. For the first two, we will work on the motion first integrals. For the symmetrical heavy body, that is the Lagrange-Poisson case, we compute the second and third Euler angles in explicit and real forms by means of multiple hypergeometric functions (Lauricella, functions). Releasing the weight load but adding the complication of the asymmetry, by means of elliptic integrals of third kind, we provide the precession angle completing some previous treatments of the Euler-Poinsot case. Integrating then the relevant differential equation, we reach the finite polar equation of a special trajectory named the {\it herpolhode}. In the last problem we keep the symmetry of the first problem, but without the weight, and take into account a viscous dissipation. The approach of first integrals is no longer practicable in this situation and the Euler equations are faced directly leading to dumped goniometric functions obtained as particular occurrences of Bessel functions of order $-1/2$.

An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments

We describe a method for the rapid numerical evaluation of the Bessel functions of the first and second kinds of nonnegative real orders and positive arguments. Our algorithm makes use of the well-known observation that although the Bessel functions themselves are expensive to represent via piecewise polynomial expansions, the logarithms of certain solutions of Bessel’s equation are not. We exploit this observation by numerically precomputing the logarithms of carefully chosen Bessel functions and representing them with piecewise bivariate Chebyshev expansions. Our scheme is able to evaluate Bessel functions of orders between 0 and 1,000,000,000 at essentially any positive real argument. In that regime, it is competitive with existing methods for the rapid evaluation of Bessel functions and has at least three advantages over them. First, our approach is quite general and can be readily applied to many other special functions which satisfy second order ordinary differential equations. Second, by calculating the logarithms of the Bessel functions rather than the Bessel functions themselves, we avoid many issues which arise from numerical overflow and underflow. Third, in the oscillatory regime, our algorithm calculates the values of a nonoscillatory phase function for Bessel’s differential equation and its derivative. These quantities are useful for computing the zeros of Bessel functions, as well as for rapidly applying the Fourier-Bessel transform. The results of extensive numerical experiments demonstrating the efficacy of our algorithm are presented. A Fortran package which includes our code for evaluating the Bessel functions is publicly available.

Fraunhofer and refractive scattering of heavy ions in strong laser fields

Until recently the potential scattering of a charged particle in a laser field received attention exclusively in atomic physics. The differential cross-section of laser-assisted electron-atom collisions for n emitted or absorbed photons is provided by a simple law which casts the result as a product between the field-free value and the square of the Bessel function of order n with its argument containing the effect of the laser in a non-perturbative way. From the experimental standpoint, laser-assisted electron-atom collisions are important because they allow the observation of multiphoton effects even at moderate laser intensities. The aim of this study is to calculate the nucleus-nucleus differential cross section in the field of a strong laser with wavelengths in the optical domain such that the low-frequency approximation is fulfilled. We investigate the dependence of the n-photon differential cross-section on the intensity, photon energy and shape of the pulse for a projectile/target combination at a fixed collision energy which exhibits a superposition of Fraunhofer and refractive behavior. We also discuss the role of the laser perturbation on the near and farside decomposition in the angular distribution, an issue never discussed before in the literature. We apply a standard optical model approach to explain the experimental differential cross-section of the elastic scattering of 4He on 58Ni at a laboratory energy E = 139 MeV and resolve the corresponding farside/nearside (F/N) decomposition in the field-free case. We give an example of reaction in which Fraunhofer diffraction and refractive rainbow hump effects are easily recognized in the elastic angular distribution. Next, we apply the Kroll-Watson theorem, in order to determine the n -photon contributions to the cross-section for continuous-wave (cw) and modulated pulses. In the elastic scattering of heavy ions in a radiation field of low intensity, the amplitude drops by orders of magnitude with respect to the unperturbed case once the exchange of photons is initiated. For intensities approaching $I=10^{17}$ W/cm2 multiphoton effects become important. In the case of short laser pulses we conclude that the strength of n-photon contribution increases with the pulse duration.

Coulomb Scattering of a Slow Quantum Particle in a Space of Arbitrary Dimension

We assume that a charged quantum particle moves in a space of dimension d = 2, 3,... and is scattered by a fixed Coulomb center. We derive and study expansions of the wave function and all radial functions of such a particle in integer powers of the wave number and in Bessel functions of a real order. We prove that finite sums of such expansions are asymptotic approximations of the wave functions in the low-energy limit.

Pointwise Asymptotics for Orthonormal Polynomials at the Endpoints of the Interval via Universality

Abstract We show that universality limits and other bounds imply pointwise asymptotics for orthonormal polynomials at the endpoints of the interval of orthonormality. As a consequence, we show that if $\mu $ is a regular measure supported on $\left [ -1,1\right ] $, and in a neighborhood of 1, $ \mu $ is absolutely continuous, while for some $\alpha>-1$, $\mu ^{\prime }\left ( t\right ) =h\left ( t\right ) \left ( 1-t\right )^{\alpha }$, where $ h\left ( t\right ) \rightarrow 1$ as $t\rightarrow 1-$, then the corresponding orthonormal polynomials $\left \{ p_{n}\right \} $ satisfy the asymptotic $$ \lim_{n\rightarrow \infty }\frac{p_{n}\left( 1-\frac{z^{2}}{2n^{2}}\right) }{ p_{n}\left( 1\right) }=\frac{J_{\alpha }^{\ast }\left( z\right) }{J_{\alpha }^{\ast }\left( 0\right) } $$uniformly in compact subsets of the plane. Here $J_{\alpha }^{\ast }\left ( z\right ) =J_{\alpha }\left ( z\right ) /z^{\alpha }$ is the normalized Bessel function of order $\alpha $. These are by far the most general conditions for such endpoint asymptotics.

Direct sampling method for anomaly imaging from scattering parameter

In this paper, we develop a fast imaging technique for small anomalies located in homogeneous media from S-parameter data measured at dipole antennas. Based on the representation of S-parameters when an anomaly exists, we design a direct sampling method (DSM) for imaging an anomaly and establishing a relationship between the indicator function of DSM and an infinite series of Bessel functions of integer order. Simulation results using synthetic data at f=1GHz of angular frequency are illustrated to support the identified structure of the indicator function.

Uniform analytic approximation of Wigner rotation matrices

We derive the leading asymptotic approximation, for low angle {\theta}, of the Wigner rotation matrix elements $d^j_{m_1m_2}(\theta)$, uniform in $j,m_1$ and $m_2$. The result is in terms of a Bessel function of integer order. We numerically investigate the error for a variety of cases and find that the approximation can be useful over a significant range of angles. This approximation has application in the partial wave analysis of wavepacket scattering.

The optimal digital filters of sine and cosine transforms for geophysical transient electromagnetic method

The sine and cosine transforms implemented with digital filters have been used in the Transient electromagnetic methods for a few decades. Kong (2007) proposed a method of obtaining filter coefficients, which are computed in the sample domain by Hankel transform pair. However, the curve shape of Hankel transform pair changes with a parameter, which usually is set to be 1 or 3 in the process of obtaining the digital filter coefficients of sine and cosine transforms. First, this study investigates the influence of the parameter on the digital filter algorithm of sine and cosine transforms based on the digital filter algorithm of Hankel transform and the relationship between the sine, cosine function and the ±1/2 order Bessel function of the first kind. The results show that the selection of the parameter highly influences the precision of digital filter algorithm. Second, upon the optimal selection of the parameter, it is found that an optimal sampling interval s also exists to achieve the best precision of digital filter algorithm. Finally, this study proposes four groups of sine and cosine transform digital filter coefficients with different length, which may help to develop the digital filter algorithm of sine and cosine transforms, and promote its application.

Topological Derivative for Imaging of Thin Electromagnetic Inhomogeneity: Least Condition of Incident Directions

It is well-known that using topological derivative is an effective noniterative technique for imaging of crack-like electromagnetic inhomogeneity with small thickness when small number of incident directions are applied. However, there is no theoretical investigation about the configuration of the range of incident directions. In this paper, we carefully explore the mathematical structure of topological derivative imaging functional by establishing a relationship with an infinite series of Bessel functions of integer order of the first kind. Based on this, we identify the condition of the range of incident directions and it is highly depending on the shape of unknown defect. Results of numerical simulations with noisy data support our identification.

Detection of small inhomogeneities via direct sampling method in transverse electric polarization

Various studies have confirmed the possibility of identifying the location of a set of small inhomogeneities via a direct sampling method; however, when their permeability differs from that of the background, their location cannot be satisfactorily identified. However, no theoretical explanation for this phenomenon has been verified. In this study, we demonstrate that the indicator function of the direct sampling method can be expressed by the Bessel function of order one of the first kind and explain why the exact locations of inhomogeneities cannot be identified. Numerical results with noisy data are exhibited to support our examination.

Discrete Bessel functions and partial difference equations

We introduce a new class of discrete Bessel functions and discrete modified Bessel functions of integer order. After obtaining some of their basic properties, we show that these functions lead to fundamental solutions of the discrete wave equation and discrete diffusion equation.

The Evaluation of Derivatives of Double Barrier Options of the Bessel Processes by Methods of Spectral Analysis

The paper deals with the spectral methods to calculate the value of the double barrier option generated by the Bessel diffusion process. This technique enables us to calculate the option price in the form of a Fourier-Bessel series with the corresponding ratio. The authors propose a simple method to estimate options using the Green’s expansion function for boundary value problem for a singular parabolic equation. Thus, the accuracy of the estimation coincides with the accuracy of the convergence of the Fourier-Bessel series. In this paper, the authors use the spectral theory to calculate the price of derivatives of financial assets considering that the processes are described by Markov and can be considered in Hilbert spaces. In this work, the authors use the diffusion process to find derivatives prices by introducing them through the Bessel functions of first kind. They also examine the Sturm-Liouville problem where the boundary conditions utilize the Bessel functions and their derivatives. All assumptions lead to analytical formulae that are consistent with the empirical evidence and, when implemented in practice, reflect adequately the passage of processes on stock markets. The authors also focus on the financial flows generated by Bessel diffusion processes which are presented in the system of Bessel functions of the first order under the condition that the linear combination of the flow and its spatial derivative are taken into account. Such a presentation enables us to calculate the market value of a share portfolio, provides the measurement of internal volatility in the market at any given time, and allows us to investigate the dynamics of the stock market. The splitting of Green’s function in the system of Bessel functions is presented by an analytical formula which is convenient for calculating the price level of options.

Topological derivative-based technique for imaging thin inhomogeneities with few incident directions

Many non-iterative imaging algorithms require a large number of incident directions. Topological derivative-based imaging techniques can alleviate this problem, but lacks a theoretical background and a definite means of selecting the optimal incident directions. In this paper, we rigorously analyse the mathematical structure of a topological derivative imaging function, confirm why a small number of incident directions is sufficient, and explore the optimal configuration of these directions. To this end, we represent the topological derivative-based imaging function as an infinite series of Bessel functions of integer order of the first kind. Our analysis is supported by the results of numerical simulations.

Analytic approximation for the modified Bessel function I−2/3(x)

In the present work an analytic approximation to modified Bessel function of negative fractional order I−2/3(x) is presented. The validity of the approximation is for every positive value of the independent variable. The accuracy is high in spite of the small number (4) of parameters used. The approximation is a combination of elementary functions with rational ones. Power series and assymptotic expansions are simultaneously used to obtain the approximation.

Hipergeometric solutions to some nonhomogeneous equations of fractional order

In this paper a study is performed to the solution of the linear non homogeneous fractional order alpha differential equation equal to I0(x), where I0(x) is the modified Bessel function of order zero, the initial condition is f(0)=0 and 0 < alpha < 1. Caputo definition for the fractional derivatives is considered. Fractional derivatives have become important in physical and chemical phenomena as visco-elasticity and visco-plasticity, anomalous diffusion and electric circuits. In particular in this work the values of alpha=1/2, 1/4 and 3/4. are explicitly considered . In these cases Laplace transform is applied, and later the inverse Laplace transform leads to the solutions of the differential equation, which become hypergeometric functions.

Fractured–Injection–Well Performance Under Non–Newtonian, Power–Law Fluids

Summary A new source function developed in this work is used to predict well responses to evaluate long-term injectivity and late-time responses of fractured wells after the injection of a non-Newtonian, pseudoplastic fluid. The source function yields integrals of modified Bessel functions of fractional order that are evaluated numerically by techniques developed by the authors. Consideration of well responses after the injection of pseudoplastic fluids into reservoirs through fractured wells requires the solution of a nonlinear differential equation and examination of flow fields in two dimensions. Consequently, numerical methods have been the only recourse. This work explores the applicability of new tools developed in recent times by analytical means. Calculating well injectivity accurately and rapidly for this complex situation by the methods outlined here is particularly convenient. The expressions we derive may be readily applied to concepts such as the Peaceman (1978) probe radius and transient well injectivity. Predictions by use of the new solution are compared with numerical solutions in the literature.

The evaluation of derivatives of double barrier options of the Bessel processes by methods of spectral analysis

The paper deals with the spectral methods to calculate the value of the double barrier option generated by the Bessel diffusion process. This technique enables us to calculate the option price in the form of a Fourier-Bessel series with the corresponding ratio. The autors propose a simple method to estimate options using the Green’s expansion function for boundary value problem for a singular parabolic equation. Thus, the accuracy of the estimation coincides with the accuracy of the convergence of the Fourier-Bessel series. In this paper, the authors use the spectral theory to calculate the price of derivatives of financial assets considering that the processes described are by Markov and can be considered in Hilbert spaces. In this work, the authors use the diffusion process to find derivatives prices by introducing them through the Bessel functions of first kind. They also examine the Sturm-Liouville problem where the boundary conditions utilize the Bessel functions and their derivatives. All assumptions lead to analytical formulae that are consistent with the empirical evidence and, when implemented in practice, reflect adequately the passage of processes on stock markets. The authors also focus on the financial flows generated by Bessel diffusion processes which are presented in the system of Bessel functions of the first order under the condition that the linear combination of the flow and its spatial derivative are taken into account. Such a presentation enables us to calculate the market value of a share portfolio, provides the measurement of internal volatility in the market at any given time, and allows us to investigate the dynamics of the stock market. The splitting of Green’s function in the system of Bessel functions is presented by an analytical formula which is convenient for calculating the price level of options.

Proof of Bessel's Relation: Part I

Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (v = n); in spherical problems, one obtains half-integer orders (v = n ± ½). This paper claims that a Bessel function of higher order can be expressed by Bessel functions of lower orders.