Connections between generalized singular functions and Bessel functions

The present work deals with the mathematical investigation of some generalizations of Bessel functions. The main motive of this paper is to show that the generating function can be employed efficiently to obtain certain results for special functions. The complex form of Bessel functions is introduced by means of the generating function. Certain enthralling properties for complex Bessel functions are investigated using the generating function method. By considering separately the real and the imaginary part of complex Bessel functions, we get respectively cosine-Bessel functions and sine-Bessel functions for which several novel identities and Jacobi–Anger expansions are established. Also, the generating function of degenerate Bessel functions is investigated and certain novel identities are obtained for them. A hybrid form of degenerate Bessel functions, namely, of degenerate Fubini–Bessel functions, is constructed using the replacement technique. Finally, the explicit forms of the real and the imaginary part of complex Bessel functions are established by a hypergeometric approach.

Some Integrals Involving Bessel Functions

Although it is known theoretically that certain infinite integrals of Bessel functions can be expressed in terms of elementary functions, the practical evaluation of such integrals was quite difficult due to the algebraic complexity of the expressions involved. A simple and elegant algebra is introduced here which allows these integrals to be calculated in an elementary way in terms of elementary functions. Some relationships are shown between the integrals involving Bessel functions and two-dimensional integrals over a circle of elementary functions involving distances between points. A comparison is made with existing results, and some of them were found in error (or were misprints).

In the theory of Herlofson only one resonance is predicted for a cylindrical plasma column irradiated by an electromagnetic wave having both its direction of propagation and electric field E perpendicular to the axis of the column, a mode which he designates as sagittal. Herlofson treats the problem by solving the wave equation in cylindrical coordinates and then imposing boundary conditions to find the frequency or frequencies for maximum scattering from the column. In his treatment, the modes which involve Bessel functions of order higher than unity have the same resonant frequency as that for the dipolar mode for which the order of the Bessel function is unity. No resonances at all are predicted for the parallel mode of excitation in which E is parallel to the axis of the column. These predictions are contrary to the experimental observations of Dattner and others for the saggital mode and also contrary to the observations reported by Willis and Petroff in which a spectrum of resonances is found for the parallel mode. Experiments by Boley have shown that the sagittal scattering for the higher order resonances is that appropriate for a dipole, that is, his experiments show that the field about the column for the higher-order modes is not quadropolar or sextupolar.

A gyrokinetic linearized exact (not model) Landau collision operator is derived by transforming the symmetric and conservative Landau form. The formulation obtains the velocity-space flux density and preserves the operator's conservative form as the divergence of this flux density. The operator contains both test-particle and field-particle contributions, and finite Larmor radius effects are evaluated in either Bessel function series or gyrophase integrals. While equivalent to the gyrokinetic Fokker-Planck form with Rosenbluth potentials [B. Li and D. R. Ernst, Phys. Rev. Lett. 106, 195002 (2011)10.1103/PhysRevLett.106.195002], the gyrokinetic conservative Landau form explicitly preserves the symmetry between test-particle and field-particle contributions, which underlies the conservation laws and the H theorem, and enables discretization with a finite-volume or spectral method to preserve the conservation properties numerically, independent of resolution. The form of the exact linearized field-particle terms differs from those of widely used model operators. We show the finite Larmor radius corrections to the field-particle terms in the exact linearized operator involve Bessel functions of all orders, while present model field-particle terms involve only the first two Bessel functions. This new symmetric and conservative formulation enables the gyrokinetic exact linearized Landau operator to be implemented in gyrokinetic turbulence codes for comparison with present model operators using similar numerical methods.

On the evaluation of infinite integrals involving Bessel functions

In this work we develop the Gaussian quadrature rule for weight functions involving powers, exponentials and Bessel functions of the first kind. Besides the computation based on the use of the standard and the modified Chebyshev algorithm, here we present a very stable algorithm based on the preconditioning of the moment matrix. Numerical experiments are provided and a geophysical application is considered.

Some results concerning the non-existence of complex and purely imaginary zeros of the transcendental equation F(z)J_v(z)+G(z)J'_v(z) = 0 are given, where J_v(z) is the Bessel function of first kind and order v (in general complex), J'_v(z) is the derivative of J_v(z) , and F(z) , G(z) are analytic functions. The obtained results improve and generalize previously known ones.

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.

A general and simple procedure is presented for evaluating matrix elements that involve Bessel functions. The method is based upon hypervirial relationships for systems subjected to Dirichlet boundary conditions.

Abstract. Certain algebraic solutions of the Klein-Gordon equation which involve Bessel functions are examined. It is demonstrated that these functions constitute an infinite series, each term of which is the solution of a boundary value problem involving a combination of source functions which comprise delta functions and their derivatives to infinite order. In addition, solutions to the homogeneous equation are constructed which comprise a continuous spectrum over non-integer order. These solutions are discussed in the context of wave propagation in isotropic cold plasma and the atmosphere.

Values for the principal plane halfpower beamwidths (HPBWs) of a well-known aperture distribution are re-examined. This article illustrates how a solution to the governing transcendental equations that involve Bessel functions can be simplified without any significant loss in accuracy by employing useful approximations. Moreover, the argument can be extended to any two orthogonal planes.

A method has been developed to calculate the scattering of slow electrons. This method along with the wellknown methods of HULTHÈN and KOHN are applied to calculate p-wave scattering in the normal hydrogen field. The new method avoids the ambiguity of HULTHEN'S quadratic equation and gives almost the same result. The interrelation of the three methods is studied and a proposal is being made how one can simultaneously satisfy the conditions L=∫ψ(H-E) ψ dτ=0, ∂, a k=—〈Rl⎜U⎟jl+〉 appearing in the three methods. The whole theoretical discussion can without difficulties be extended to the modified COULOMB potential, only the regular and singular spherical BESSEL functions are to be substituted by the regular and singular confluent hypergeometric functions of the COULOMB type respectively. The phase shifts and the wave functions calculated without exchange agree well with the numerically solved results of CHANDRASEKHAR and BREEN. The results including exchange terms agree reasonably well with the results of the numerical integration of the HARTREE-FOCK equation, which has been carried out here by E. TREFFTZ.

We show that the dynamics of an Ising spin chain in a transverse field conserves the number of domains (strings of down spins in an up-spin background) at discrete times. This enables the determination of the eigenfunctions of the time-evolution operator, and the dynamics of initial states with domains. The transverse magnetization is shown to be identically zero in all sectors with a fixed number of domains. For an initial state with a single string of down spins, the local magnetization, the equal-time and double-time spin-spin correlation functions, are calculated analytically as functions of time and the initial string size. The domain size distribution function can be expressed as a simple integral involving Bessel functions.

In this paper, we present two kinds of Hermite-type collocation methods for linear Volterra integral equations of the second kind with highly oscillatory Bessel kernels. One method is direct Hermite collocation method, which used direct two-points Hermite interpolation in the whole interval. The other one is piecewise Hermite collocation method, which used a two-points Hermite interpolation in each subinterval. These two methods can calculate the approximate value of function value and derivative value simultaneously. Both methods are constructed easily and implemented well by the fast computation of highly oscillatory integrals involving Bessel functions. Under some conditions, the asymptotic convergence order with respect to oscillatory factor of these two methods are established, which are higher than the existing results. Some numerical experiments are included to show efficiency of these two methods.