Integrals Of Bessel Functions Research Articles

On Jacobi functions and multiplication theorems for integrals of Bessel functions

In this paper, the product of a Jacobi polynomial and function is shown to generate the Jacobi polynomial. This type of expansion was previously known for all the classical orthogonal polynomials except the Jacobi. The result is then used to obtain generalizations to Watson's [10] multiplication theorem involving integrals of Bessel functions. The integrals considered are of the form ∝ ∞ 0 t 1− λ J v ( tx 1 ) J μ ( tx 2 ) J σ ( tx 3 ) J τ ( tx 4 ) dt .

Stability of continuous double-plate systems

Stability characteristics of circular double-plate systems which are interconnected by n intermediate, concentric, and elastic ring supports are considered in this paper. In particular, for annular isotropic plates with different loads at the inner and outer edges, closed-form solutions in terms of integrals of Bessel and Lommel functions are developed for the symmetric (m = 0) and the first asymmetric (m = l) modes. Also, it is shown that, for isotropic complete plates and annular plates with equal loads at the inner and outer edges, the solutions will reduce to the well-known solutions in terms of Bessel functions. Several new buckling characteristics of the continuous double-plate systems are illustrated by numerical results obtained for isotropic, fixed-fixed, and fixedfree systems with one intermediate elastic connection.

Positive Integrals of Bessel Functions

It is shown that some new and some already known positivity results for integrals of Bessel functions and for generalized hypergeometric functions can be easily obtained by writing the integrals and functions either as a sum of squares of Bessel functions with positive coefficients or as a fractional integral of such a sum. In particular, this method is used to prove that $\int_0^x {(x - t)^\lambda t^{\lambda + {1 / 2}} J_\alpha (t)dt > 0,\quad 0 \leqq \lambda \leqq \alpha - {1 / 2},\quad \alpha > 1/2, \quad x > 0,} $ and to give a simple proof of Steinig’s recent result that the Lommel function $S_{\mu ,\nu } (x) > 0$ for $x > 0$ if, $\mu = {1 / 2}$ and $ - {1 / 2} < \nu < {1 / 2}$, or if $\mu > {1 / 2}$ and $ - \mu \leqq \nu \leqq \mu $.

The solution of theoretical seismic problems by best rational function approximations for Laplace transform inversion

Abstract In a recent paper, the first author has developed a method of computation of “best” rational function approximations ḡn(p) to a given function f̄(p) of the Laplace transform operator p. These approximations are best in the sense that analytic inversion of ḡn(p) gives a function gn(t) of the time variable t, which approximates the (generally unknown) inverse f(t) of f̄(p in a minimum least-squares manner. Only f̄(p) but not f(t) is required to be known in order to carry out this process. n is the “order” of the approximation, and it can be shown that as n tends to infinity gn(t) tends to f(t) in the mean. Under suitable conditions on f(t) the convergence is extremely rapid, and quite low values of n (four or five, say) are sufficient to give high accuracy for all t ≧ 0. For seismological applications, we use geometrical optics to subtract out of f(t) its discontinuities, and bring it to a form in which the above inversion method is very rapidly convergent. This modification is of course carried out (suitably transformed) on f̄(p), and the discontinuities are restored to f(t) after the inversion. An application is given to an example previously treated by the first author by a different method, and it is a certain vindication of the present method that an error in the previously given solution is brought to light. The paper also presents a new analytical method for handling the Bessel function integrals that occur in theoretical seismic problems related to layered media.

Disturbance wave propagation on a semi-infinite one-dimensional lattice containing a mass defect

We use a simple techique to study the influence of a “weak” mass defect on the propagation of a general disturbance on a semi-infinite one-dimensional monoatomic lattice. The location of the deffective mass is assumed arbitrary. Result are expressed as combinations of Bessel functions and integrals of Bessel functions. A numerical example demonstrates that the influence of the defect appears to modulate the response of the otherwise perfect chain. This modulation vanishes in the limit as t → ∈.

On integrals of Bessel functions related to Weber's second exponential integral

Two closely related integrals of Bessel functions are discussed: $$\int\limits_0^\infty {t^{1 - 2m} e^{ - {{t^2 } \mathord{\left/ {\vphantom {{t^2 } {2z}}} \right. \kern-\nulldelimiterspace} {2z}}} [J_n (t)]^2 } dt;\int\limits_0^z {u^{ - l} e^{ - u} I_n (u)} du.$$ Whenn, m, l are integers (m andl=1,2, ...,n), both integrals can be written as closed expressions in modified Bessel functionsI k (z). The results are interpreted in terms of hypergeometric seriesp F q. Series expansions inz and in 1/z are given.

A Method for Computing Bessel Function Integrals

Infinite integrals involving Bessel functions are recast, by means of an Abel transform, in terms of Fourier integrals. As there are many efficient numerical methods for computing Fourier integrals, this leads to a convenient way of approximating Bessel function integrals.

A method for computing Bessel function integrals

Infinite integrals involving Bessel functions are recast, by means of an Abel transform, in terms of Fourier integrals. As there are many efficient numerical methods for computing Fourier integrals, this leads to a convenient way of approximating Bessel function integrals.

On the Collisional Transfer of Rovibronic and Translational Energy between Differently Excited Like Molecules

In line with the principle of the first Born approximation, we give a simple treatment of the comprehensive transfer of rotation, vibration, and electronic and translational (rovibronic-translational) energy in the collisions between (electronically) differently excited like molecules. It is shown that the electronic exchange degeneracy between the collision partners gives rise to first-order (long-range) dispersion forces which may result in large energy transfer cross sections. The formalism is for diatomic or linear molecules, with or without electronic angular momentum, and may be easily adapted to symmetric-top molecules. A first-order, general (electronic) multipole-multipole interaction is used as the scattering potential, which is written as a contraction of tensors. The transition matrix elements are evaluated by Racah's method, showing the explicit dependence on the conserved total angular momentum as well as on the angular momenta of the individual molecules and the angular momentum of the relative orbital motion. The relative motion is treated as a plane wave. Coupled representations with definite total angular momentum and alternatively uncoupled representations of the wavefunction (in the Appendix) are used to derive the matrix elements and cross-sections. The general case of small vibronic energy discrepancies is considered. Differential cross sections as well as total cross sections are given explicitly. Under the assumption of long-range electronic interaction and fast electronic transition, the cross sections are shown to be proportional to Franck-Condon overlap integrals, multipole rotational transition line strengths, and Bessel function integrals for the relative motion. The necessary reduced matrix elements for the multipole transition and for the relative motion are computed and discussed. Assuming the validity of the Born-Oppenheimer approximation, i.e., no vibronic interaction and no rotation-electronic interaction, etc., the effect of permuting the nuclei of like molecules of the same nuclear spin is shown to be unimportant if the permanent dipoles of the states are small. This treatment of energy transfer should be applicable to differently excited states (of like molecules) between which there are large multipole transition moments, which interact in the first order because of the exchange degeneracy.

Bessel function expansions of incomplete Lipschitz-Hankel integrals

TWO methods are described for constructing Neumann series for incomplete Lipschitz-Hankel integrals of Bessel functions of the first kind.

Evaluation in closed form of integrals of products or spherical bessel functions related to the bipolar expansion

The bipolar expansion obtained by Ruedenberg by a Fourier transform method leads to radial functions which are integrals of a product of three spherical Bessel functions. It is shown that these integrals can be evaluated in closed form by means of the theory of generalized functions.

Development of an indirect method of deriving formulas for evaluating some improper integrals of Bessel functions

Development of an indirect method of deriving formulas for evaluating some improper integrals of Bessel functions

Analytical Solutions to Eddy-Current Probe-Coil Problems

Solutions have been obtained for axially symmetric eddy-current problems in two configurations of wide applicability. In both cases, the eddy currents are assumed to be produced by a circular coil of rectangular cross section, driven by a constant amplitude alternating current. One solution is for a coil above a semi-infinite conducting slab with a plane surface, covered with a uniform layer of another conductor. This solution includes the special cases of a coil above a single infinite plane conductor or above a sheet of finite thickness, as well as the case of one metal clad on another. The other solution is for a coil surrounding an infinitely long circular conducting rod with a uniformly thick coating of another conductor. This includes the special cases of a coil around a conducting tube or rod, as well as one metal clad on a rod of another metal. The solutions are in the form of integrals of first-order Bessel functions giving the vector potential, from which the other electromagnetic quantities of interest can be obtained. The coil impedance has been calculated for the case of a coil above a two-conductor plane. The agreement between the calculated and experimental values is excellent.

The Integration of Bessel Functions

Starting with the well-known differential and recurrence relations of Bessel functions, a formula is obtained by means of which the nth-order derivative of a Bessel function of order p can be expressed in terms of the Bessel function of order p and its first derivative, the function and its derivative being multiplied by polynomials in 1/x, x being the argument. By using the method in reverse, the integral of a Bessel function can be expressed in terms of the Bessel function and its derivative, which are multiplied by series in x if p is even, or polynomials in 1/x if p is odd. These formulae are more convenient for computation than the well-known formulae involving series of Bessel functions.

Bessel-Function Integrals Needed for Two Classes of Physical Problems

Bessel-Function Integrals Needed for Two Classes of Physical Problems

Numerical Solution of Theoretical Seismic Problems

A method is developed for the solution of problems of pulse propagation in elastic half-spaces. It is based on Laplace transform inversion by means of rational approximations applied to the factor multiplying the Bessel function in the integral representing the operational solution. The discontinuities (at arrival times of rays) are generated automatically by analytic inversion of the rational approximations, followed by the analytic evaluation of some Bessel function integrals.

Bessel-function integrals needed for two classes of physics problems

Bessel-function integrals needed for two classes of physics problems

Tables of Integrals of Fractional Order Bessel Functions

Tables of Integrals of Fractional Order Bessel Functions

Electric-Field-Induced Indirect Optical Absorption in Solids and the Fine Structure of the Stark Splitting

The problem of optical transitions in solids subject to a strong electric field ls discussed using time-dependent wave functions. Expressions for the change in transition rate for direct and indirect transitions are obtained in terms of integrals of Bessel functions. Moreover, using approximations for the Bessel functions, it is shown that the change in transition rate for direct transitions, ${T}_{d}$, and for indirect transitions, ${T}_{i}$, are given by ${T}_{d}={E}^{\frac{1}{3}}{F}_{d}\frac{({\ensuremath{\epsilon}}_{p}\ensuremath{-}{\ensuremath{\epsilon}}_{g0})}{{E}^{\frac{2}{3}}}$ ${T}_{i}={E}^{\frac{4}{3}}{F}_{i}\frac{({\ensuremath{\epsilon}}_{p}\ensuremath{-}{\ensuremath{\epsilon}}_{g0})}{{E}^{\frac{2}{3}}}$ where $E$ is the applied electric field, ${\ensuremath{\epsilon}}_{p}$ the excitation energy, and ${\ensuremath{\epsilon}}_{g0}$ the minimum energy gap. These formulas are valid both below and above the absorption edge. As a special case a machine computation of the change in the transition rate for Si with field applied in the [100] direction is presented. The results show a series of decaying peaks. In addition, a fine structure is predicted of the same kind as that predicted by Callaway for the Stark splitting. The details of the fine structure are analyzed and are shown to depend on the particular structure of the bands both near and far above the minimum energy gap. Difficulties in the experimental observation of the fine structure arising from inhomogeneity of the electric field are discussed.

Expansions of Integrals of Bessel Functions of Large Order

Series expansions in powers of 1/v of integrals over x of the product of the Bessel function J v(vx/y), [J v(vx/y)]2, or J v(vx/y) J v+1(vx/y) with an arbitrary function of a restricted class are developed.