Showing papers on "Bessel function published in 1979"

Properties of the zeros of the classical polynomials and of the Bbessel functions

Abstract: Many algebraic equations satisfied by the zeros of the classical polynomials and of the Bessel functions are reported. Some of them are collected from recent papers; several of them are new; most of them display remarkable diophantine features. Certain matrices constructed with arbitrary numbers rather than the zeros of special functions, but displaying analogous diophantine properties, are also exhibited.

Remarks on the Hamiltonian path integral in polar coordinates

Abstract: Problems associated with the derivation of the Hamiltonian path integral in polar coordinates are examined. First the use of the ill‐defined asymptotic formula of the modified Bessel function is pointed out. A procedure is proposed to justify its practical use, in which the mass m is complexified and the limit Imm→0 is taken after path integrations. Hereby a restriction is imposed on the class of allowed potentials. The difference between the Hamiltonian path integral so obtained and the phase space path integral formally defined is also discussed.

Explicit solution of the wave equation for arbitrary power potentials with application to charmonium spectroscopy

Abstract: We present an explicit and almost complete series solution of the Schrodinger equation for an arbitrary quark‐confining power potential with or without a weak Coulomb component or other corrections. In particular, we derive two pairs of high‐energy asymptotic expansions of the bound‐state eigenfunctions together with a corresponding expansion of the eigenvalue determined by the secular equation. We also obtain a pair of uniformly convergent expansions and discuss other types of solutions. Various properties of the solutions and eigenvalues are examined including the scattering problem of the cutoff potential and the behavior of Regge trajectories. Finally, the relevance of these investigations to the spectroscopy of heavy quark composites is discussed. In particular, we derive approximate expressions for leptonic decay rates. Examples are given to demonstrate the usefulness of these results for theoretical discussion and as alternatives for numerical integration techniques. A subsequent paper will deal with the normalization of the bound‐state wavefunctions and the corresponding derivation of explicit series expressions for certain decay rates.

An analytical model for the nearfield of a baffled piston transducer

Abstract: The linearized sound field from a baffled piston source (radius a, wavenumber k) in a dissipative fluid is considered. A simplified parabolic equation is derived (for ka ≫ 1) and solved analytically. The solution matches a plane collimated beam in the vicinity of the source and has the Bessel function directivity in the farfield. The nearfield‐farfield transition region is studied. The range of validity of the parabolic equation is discussed. Its exact solution is shown to be the first term of an expansion in powers of (ka)−2 for the solution of the Helmholtz equation. The higher‐order terms are secular at distances of order a(ka)1/3 from the piston. The analytical results obtained for the linearized field can be used to calculate the effects of nearfield oscillations on nonlinear effects generated in soundbeams. (For example, see Paper GG1, by the same authors, in this Program.) [Work supported in part by the Office of Naval Research.]

Eddy currents in a conducting cylinder with a crack

Abstract: We report calculations for the impedance of a long solenoid which surrounds a cylinder of conducting material containing a radial surface crack of constant depth. The calculation is accomplished by solving for the longitudinal ac magnetic field in the interior of the ’’cracked’’ cylinder in terms of an infinite series of cylindrical Bessel functions. All the coefficients in the series are determined in principle by boundary‐condtion requirements and the most significant terms are obtained numerically by truncation of the series. The resulting impedance is calculated for a wide range of values of the ratios of crack depth to radius and radius to skin depth. The results are tabulated in a form useful for nondestructive testing purposes.

On high precision methods for computing integrals involving Bessel functions

Abstract: The technique of Bakhvalov and Vasil'eva for evaluating Fourier integrals is generalized to integrals involving exponential and Bessel functions.

S -matrix formulation of the charged-particle scattering in the presence of a laser radiation field

Abstract: AS-matrix formulation of charged-particle scattering in the presence of a laser radiation field is given. An clastic-scattering Born series is obtained, that is a rather straightforward generalization of the corresponding series without field. A diagrammatic representation is given to the terms of the series. The presence of the Bessel functions in different combinations in the developed formalism makes to treat exactly the terms higher than the first-order one rather difficult. To make the formalism manageable, various approximate expressions are derived for the limiting cases of small and large arguments of the Bessel functions, that correspond, respectively, to weak field and/or high frequencies and to strong fields and/or low frequencies. In these limits, the calculation of corrections to the first-order results is approximately as difficult as when there is no field. In the weak-field case, the scattering matrix elements (of any order in the scattering potential) are proportional to (A0/ω)l,l being the net number of exchanged photons, A0 and ω, respectively, the amplitude and the frequency of the radiation field. Besides this, A0/ω≪ 1. Hence in this limit multiphoton exchanges result to bo quite unlikely. In the strong-field case, when A0≪ 1, each matrix element is proportional to (ω/A0)N/12,N being the order in the scattering potential. Now the field amplitude appears at the denominators, thus contributions from highN are expected to be small. Contrary to the weak-field case, for a givenN the analytic expressions of the matrix elements depend

Coulomb scattering in a laser field

Abstract: The problem of scattering by a local potential in the presence of an intense radiation field is studied for the case where the potential is Coulombic at great distances. The effect of the Coulomb tail on the asymptotic dynamics is accounted for here through a modification of the form of the wave functions which describe the time evolution of the system in initial and final states. This is in analogy with previous treatments of field-free Coulomb scattering. Starting from the time-dependent picture, the author obtains a time-independent formulation of the problem and then applies it to the derivation of a low-frequency approximation. In the simplest version of this approximation the transition amplitude is represented as the product of a known field-dependent factor (a Bessel function) and the physical field-free scattering amplitude, thus generalizing an earlier result of this type derived for the case of a short-range potential.

A non-integer Bessel uniform approximation for use in semiclassical collision theory

Abstract: A new uniform approximation for use in semiclassical collision theory has been derived. It involves a Bessel function of non-integer order as the canonical integral. The integer Bessel uniform approximation is contained as a special case. The breakdown of the integer Bessel approximation for elastic transitions at high energies is analysed. The new approximation is derived with the help of Sommerfeld's contour integral representation of a Bessel function. Both classically allowed and classically forbidden transitions are treated. Limiting cases of the non-integer Bessel uniform approximation and generalizations to multidimensional integrals are also considered.

Average transient temperature within a body heated by a disk heat source

Abstract: An exact solution is given for a transient average temperature in a semi-infinite body that is heated by a constant heat flux over a circular surface region and insulated elsewhere at the surface. The average temperature is for any circular region parallel to and directly below the heated disk. The solution has direct application to thermal conductance, electric contacts, laser heating, and other cases. Though studied by others, no exact solution has previously been given. The exact solution is compared with several approximate solutions and substantial differences are noted, particularly at small dimensionless times. In addition to effective recursive methods for evaluating the infinite series solution, some simple but very accurate expressions are given for the heated surface and other locations. The basic solution for the semiinfinite geometry with a heat flux boundary condition is shown to be a building block for a number of other boundary conditions for both semi-infinite and infinite geometries. Nomenclature A = contact area Cn = specific heat erfc(x; = complementary error function ierfc(x) = integrated error function J(<(r) = Bessel function I|<(r) = Bessel function k = thermal conductivity

Special functions, infinite divisibility and transcendental equations

Abstract: We establish integral representations for quotients of Tricomi ψ functions and of several quotients of modified Bessel functions and of linear combinations of them. These integral representations are used to prove the complete monotonicity of the functions considered and to prove the infinite divisibility of a three parameter probability distribution. Limiting cases of this distribution are the hitting time distributions considered recently by Kent and Wendel. We also derive explicit formulas for the Kent–Wendel probability density functions.

Application of a non-integer Bessel uniform approximation to inelastic molecular collisions

Abstract: A non-integer Bessel uniform approximation has been used to calculate transition probabilities for collinear atom-oscillator collisions. The collision systems used are a harmonic oscillator interacting via a Lennard-Jones potential and a Morse oscillator interacting via an exponential potential. Both classically allowed and classically forbidden transitions have been treated. The order of the Bessel function is chosen by a physical argument that makes use of information contained in the final-action initial-angle plot. Limitations of this procedure are discussed. It is shown that the non-integer Bessel approximation is accurate for elastic 0 →0 collisions at high collision energies, where the integer Bessel approximation is inaccurate or inapplicable.

On The Inference Of Crack Statistics From Observations On An Outcropping

Abstract: It is difficult to examine the cracks in a three-dimensional body; one is usually limited to observations on an outcropping, a cut, or a plane obtained by sectioning a sample. This paper considers two problems. The direct problem is to find the distribution of line segments in a plane section when the three-dimensional distribution of cracks is homogeneous, isotropic, and exponential. This distribution can be expressed in closed form by means of Hankel functions. It is shown that the distribution in a plane section is qualitatively different from the three-dimensional distribution in having a peak for a finite value of segment length, i.e., there is a most probable (non-zero) segment length. It is also concluded that the mean segment size in the plane is ..pi../2 times the mean crack diameter in three dimensions. This result is consistent with the well-known observation that small cracks have a lower probability of being intercepted by a plane than larger cracks. The number density of line segments is finally expressed in terms of the Hankel function of order zero. The indirect problem is to infer the three-dimensional distribution of cracks from the distribution on a section, which could be, for example, an outcropping. Thismore » problem is solved by deriving an integral equation relating the three-dimensional distribution of cracks and the distribution of line segments in a plane, and showing that it can be solved for an arbitrary distribution of segments on the out-cropping. The special case of the Hankel distribution leads to the exponential distribution in three dimensions; thus, thesolution method is verified. 3 figures.« less

New backward recurrences for Bessel functions

Abstract: The recurrences for the coefficients of appropriate power series may be used with the Miller algorithm to evaluate J,(x) (lxI small), exKv(x) (Re x > 0, lxl large), and the modulus and phase of H$(l)(x) (Re x > 0, lxl large). The filrst converges slightly faster than the power series or the classical recurrence, but requires more arithmetic; the last three give both better ultimate precision and faster convergence than the corresponding asymptotic series. The analysis also leads to a formal continued fraction for Kv+ (x)/Kv(x) the convergence of which increases with Ixi. The procedures were tested numerically both for integer and fractional values of v, and for real and complex x. J. C. P. Miller [1] was the first modern worker to apply backward recurrence for evaluating sequences of functions ffk} when the recurrence connecting successive members was unstable for increasing k. He evaluated the modified Bessel functions Ik(x) by assigning values FN -1, F I 1 = 0, and used the recurrence to compute FNN41N FN-2, . . , FN. For N >> k, the FkN approached proportionality to Ik(x), and the proportionality constant CN could be evaluated using a generating function. Since 1952, the method has been applied to many families of functions, and the algorithm has been subjected to intensive analysis and refinement. Relatively recent surveys of the status of the method have been provided by Gautschi [2], [3], [4]. To use conventional backward recurrence methods, one needs a recurrence connecting successive elements of the sequence and, if the recurrence is homogeneous, some normalizing relation such as a generating function or a single function value. The algorithm must also, of course, converge, in the sense that the estimates for a particular element, fkN FN/ICN, approach the true value as N increases. In 1972, Thacher [5] pointed out that solution of linear differential equations with rational coefficients by the method of undetermined coefficients is a fruitful source of recurrences (albeit for the Taylor coefficients, instead of for the elements of a family of functions), and that the value of the function at any point within the circle of convergence of the series provides the required normalizing condition. Moreover, the analytic properties of the solutions of the differential equation provide useful clues to the convergence of the backward recurrence algorithm. Received May 31, 1977; revised March 9, 1978. AMS (MOS) subject classifications (1970). Primary 33A40, 33-04, 30A22, 65D20.

The inverse interior Dirichlet problem

Abstract: An infinite number of rotationally symmetrical functions Ψ (r,z) can be found which approximate a given function f (z) on the optical axis 0z (r and z being cylindrical polar coordinates). The Ψ (r,z) are given in the form of a finite Fourier‐Bessel series and it is shown how a solution that can be used in electron optical design can be selected by carefully assigning values to the following parameters: the number of terms in the series, the period of the series, and the amount of smoothing introduced. In this way a compromise is established between the quality of the axial approximation, the magnitude of potential gradients to be contended with in zonal regions, and the sizes of apertures allowed. Two‐foil, one‐foil, and open lenses can be modeled, and given axial potential functions can be accommodated which are either (i) analytic, (ii) continuous but with piecewise continuous z derivatives, or (iii) in the form of a set of experimentally determined values. The method requires no matrix inversion, the computer programming is of a simple nature, and memory requirements are modest enough to allow implementation of the program on small desk‐top computers.

JWKB Connection for Radial Wave Functions and Bound-State/Free-State Duality

Abstract: The oscillatory and non-oscillatory regions of radial wave functions are connected by the use of spherical Bessel functions or Coulomb wave functions. The resulting formulae for the magnitude of the wave function at the origin in terms of the energy spectrum are used to exhibit the appropriate form of boundstate/free-state duality.

Special functions associated with azimuthally magnetized ferrite rod phase shifters

Abstract: Solutions of the differential equation satisfied by the circularly symmetric electric field Eφ in an azimuthally magnetized ferrrite cylinder are given in terms of two infinite series which reduce to Bessel function J1(x) and Neumann function N1(x) in the absence of the dc magnetic field. These results differ from the solutions which have been used in literature associated with ferrite‐rod phase shifters. A numerical table is provided for the comparison between the present and the former solutions.

Fourier‐Bessel series solution for potential and intensity field of open electron optical systems with rotational symmetry, in terms of I0 Bessel functions

Abstract: A Fourier‐Bessel series representation in terms of I0 Bessel functions is given for the potential distribution in certain open Einzel‐ or immersion‐type electrostatic configurations with rotational symmetry described by φ (A,z) =F (z), 0

Mathematical Concepts and Methods in Science and Engineering Vol 11 : Integral Transforms in Science and Engineering

Abstract: Kurt Bernado Wolf 1979 New York: Plenum xiii + 489 pp price $32.50 In the first part of this book an introduction is given to complex vector analysis with emphasis on linear operators, their eigenvectors and eigenvalues, and the theory of finite transforms is applied to lattice structures such as crystals, electrical networks and signal sets. Fourier and Bessel series are then treated with applications to diffusive and elastic media, and this leads on to a detailed study of Fourier and other integral transforms.