Showing papers on "Bessel function published in 1970"

Theory of the Derivative Spectrometer

Abstract: The theory for sinusoidally modulated derivative spectrometers shows that the transfer function for the mth derivative is proportional to the Bessel function of the first kind of order m. The instrument function is expressed in terms of the Chebyshev polynomial of order m or in terms of a weighted-mean mth derivative of the spectral flux within the interval of modulation.

General purpose matrix processor with convolution capabilities

Abstract: Method and apparatus of computing a generalized convolution of values from two matrices of complex values Ao through Am and Bo through Bn respectively. The formula used in the computation of each complex vector element Ck of the generalized convolution is WHERE P and U specify the increment for each succeeding element involved in a single convolution from each sequence respectively, Q and V specify the increments between first elements of successive convolution coefficients, in each sequence, respectively, and R and W specify the first pair of elements used in forming Co. PC specifies the number of Ck's to be computed. This computation has wide applicability to such allied mathematical operations as vector and matrix algebra, linear programming and a wide variety of transformation weighting and skirting operations such as Bessel function weighting, Hanning windows, complex Kernal transformations, and fast Fourier transforms. In addition, the apparatus described has capability to compute various special cases of the generalized equation involving vectors of real values only.

Integral‐Transform Gaussian Functions for Heliumlike Systems

Abstract: The one‐dimensional Laplace transform of the Gaussian exp (−r2x), g(r)=L{G(x)}= ∫ 0∞e−r2xG(x)dx, was used to generate functions which could be useful as basis sets for atomic and molecular calculations. A particular choice of the weighting function G(x) led to functions of the form g(r) = (qr)νKν(qr), where Kν(qr) are modified Bessel functions of the second kind. These functions were used as basis functions for the helium isoelectronic series and accounted for 98.98% (H−), 99.89% (He), 99.96% (Li+), 99.98% (Be++), 99.996% (O6+) of the Hartree‐Fock energy.

Characteristic Functions for Time-Average Holography

Abstract: This paper presents an analysis of the fringes obtained by time-average holographic interferometry of a generalized time-dependent optical phase function. The generalized optical phase function considered is the sum of a series of sinusoidal functions of time having arbitrary amplitudes, frequencies, and relative phases. Characteristic functions are determined for various optical phase functions of interest in time-average holography. In general, the characteristic functions are sums of products of Bessel functions (zero order and higher orders) and exponential phase factors. Rationally and irrationally related frequencies are included in this analysis. An example of vibrating string is considered, to illustrate the application of the results of this paper to objects vibrating at a multitude of frequencies.

Multipliers of fourier integrals and bounds of convolution in spaces with mixed norms. applications

Abstract: Multipliers of Fourier integrals and estimates of integrals of potential type in spaces with mixed norms are considered in this paper. As applications we prove theorems of imbedding type for spaces of Bessel potentials with density from .

Exact solutions of SH wave equation for inhomogeneous media

Abstract: The SH wave equation in a vertically inhomogeneous elastic medium has been considered. It is possible to determine all vertical inhomogeneities for which the SH wave equation can be solved in terms of standard transcendental functions. The inhomogeneities where either shear-wave velocity, modulus of rigidity, or density is constant have also been considered. A few simple inhomogeneities and their solutions in terms of hypergeometric, Whittaker, Bessel and exponential functions have been given.

On the capacity of the circular disc condenser at small separation

Abstract: A pair of identical circular discs, held at equal and opposite potentials, forms a condenser whose capacity C depends on the ratio e of separation against diameter. The determination of an asymptotic expansion for C when e is small poses an axisymmetric boundary-value problem for harmonic functions that has engaged the attention of numerous investigators over a long span of time. It is a simple matter to construct a Fredholm integral equation of the first kind for the charge density ± σ on the discs, in terms of which the potential field and the capacity are implicitly determined, but the equation is unsuitable if e ≪ 1. Integral equations of the second kind and of the dual variety have also been proposed as a means of securing a more manageable formulation of the boundary-value problem. An elementary approximation follows from the hypothesis that the charge density is almost the same as though the discs were of infinite extent, except for a region close to the edges, and leads to the result C ∼ l/8e as e → 0. Kirchhoff considerably improved on this crude estimate by suggesting a plausible edge correction which yields two further terms for C, of orders log ∈ and a constant, respectively, and his results have been rigorously established by the more refined analysis of Hutson. In the present work an integral equation of the first kind for the distribution of potential off the discs is derived and utilized to obtain an approximation for C when e is small, reproducing the result of Kirchhoff and Hutson. Furthermore, an estimate of the error provides explicit details regarding the next term in the asymptotic expansion of C, which is of the order e(log e)2.

Transient Interactions of Spherical Acoustic Waves and a Cylindrical Elastic Shell

Abstract: The three‐dimensional transient interaction between spherical acoustic waves with infinitely steep wavefronts and a circular cylindrical elastic shell of infinite length is investigated. The incident spherical wave is transformed into cylindrical partial waves by using the addition theorem for the modified Bessel function. The governing wave equation and equations of motion of the shell are solved by a series expansion‐Laplace‐Fourier transform technique. The transformed solution of the problem is obtained in closed form exact within the limit of series solution imposed by the Gibb's phenomenon. The physical solution is obtained by an accurate numerical scheme for the two‐fold inverse Laplace‐Fourier transforms. Detailed numerical results are obtained for the transient response of the shell and some quantitative effects of the sphericality of the incident waves on the response of the shell are also revealed.

A Fortran Computer Program for Calculating the Oblate Spheroidal Radial Functions of the First and Second Kind and Their First Derivatives

Abstract: : The solutions of the Helmholtz wave equation in prolate spheroidal coordinates can be obtained by separation of variables. The subject of this report is a Fortran computer program called PRAD which calculates numerical values to the solutions of the resulting ordinary differential equation for the 'radial' coordinate. The printed output of PRAD consists of radial functions of the first and second types, their first derivatives, the separation constants or eigenvalues, and an accuracy check. The report describes the computer program PRAD and briefly reviews the theory of prolate spheroidal wave functions. A computer listing of PRAD along with some sample output is included in an appendix. (Author)

Some Extensions of Lardner’s Relations between ${}_0 F_3 $ and Bessel Functions

Abstract: The even and odd parts of a hypergeometric ${}_p F_q $ series are expressed as ${}_{2p} F_{2q + 1} $-series, and inversely, a restricted ${}_{2p} F_{2q + 1} $-series is expressed in terms of a pair of ${}_p F_q $-series. Generalizations are obtained by decomposing Meijer’s G-function into two G-functions which are not, however, always even or odd. More fundamentally, it is shown that any solution of a generalized hypergeometric differential equation with restricted parameters can be expressed in terms of solutions of an equation with order half as large. The results are illustrated by applications to Bessel functions, Kelvin functions, generalized Fresnel integrals, and restricted ${}_4 F_3 $-series with unit argument. A number of restricted G-functions are expressed in terms of more familiar functions. A reducibility criterion is used to identify cases in which fourth order differential equations governing vibrations of beams and deformations of shells can be solved in terms of Bessel functions or ${}_1 ...

Zeros of the modified Hankel function

Abstract: (t) with q and x real, and x > 0. This equation occurs in certain physical problems, such as in the determination of the bound states for an inverse square potential with hard core in SchrSdinger equation. The modified Bessel function of third kind K, (z) is a regular function of z in all the z plane cut along the negative real axis. For any fixed z (z 4~ 0), K, (z) is an entire function of v. K, (z) is connected with the Hankel functions through

Comparison of SCF and kν Functions for the Helium Series

Abstract: Wavefunctions for He, Li+, Be2+, and O6+ are presented. They were determined by using reduced modified Bessel functions of the second kind, kν(qr). The z dependence of energies calculated using such functions for high values of z is found to be the same as for the Hartree‐Fock functions.

Small-signal high-frequency response of the insulated-gate field-effect transistor

Abstract: A complete solution is presented for the small-signal high-frequency response of an idealized model of the insulated-gate field-effect transistor. The y parameters are found by solving Bessel's equation and are plotted as functions of signal frequency and the quiescent conditions. In addition to these general results, simple approximate results that apply only beyond the point of pinch-off are derived for operation at moderately low frequencies.

An Exact Solution for Wave-Form Distortion of Arbitrary Signals in Ideal Wave Guides

Abstract: An exact expression is derived for the transient response of an arbitrarily modulated signal propagating through an ideal wave guide. The solution is in the form of an infinite series of Bessel functions, the coefficients of which depend in a simple manner on the time derivatives of the input signal. It is shown that for the specific example of a step-modulated carrier signal (the only earlier case for which an exact solution is known) the general expression reduces to that derived in earlier works. A particular case of a ‘double exponential’ pulse is evaluated and briefly discussed.

Structural properties of the equilibrium solutions of riccati equations

Abstract: Structural properties of equilibrium solutions of quadratic matrix equation, using variational interpretation of associated Riccati equation, transform techniques and Parseval formula

Expansion Theorems for Magnetic-Resonance Line Shapes

Abstract: We have found an exact expression or expansion theorem for a free-induction-decay (FID) curve which involves all the moments of the corresponding cw absorption line as well as two arbitrary scale factors or parameters which may be chosen to optimize the convergence which is necessarily uniform The expression obtained is a generalization of Taylor's theorem known as a Newmann expansion, and it gives an FID curve as either an exponential or Gaussian damping factor times an infinite series of Bessel functions which may describe the oscillations characteristic of certain FID shapes Any one of the (infinitely) many Bessel-function expansions may be used to represent a particular FID curve, although there will be one which requires the fewest terms for a specified accuracy of approximation Application of these expansions to FID curves from calcium fluoride shows that it is possible to obtain an excellent fit to the data, using only the theoretical second and fourth moments for the expansion whose leading term corresponds to Abragam's trial function Furthermore, when several exact but different expansions were truncated to only three terms it was found that they were nearly equal to each other and to the data over a major portion of the decay for the optimum choice of the two scale factors Another application of these expansions would be the determination of the moments of a given FID curve, using the orthogonality integral for Bessel functions

Inversion of the spherical layer-matrix

Abstract: In their formulation of wave problems in a spherically layered medium, Phinney and Alexander have arrived at a layer-matrix M which forms the fundamental building block of their solution. In actual application of the theory, the inversion of M is needed for each assumed spherical shell. Since numerical inversion of the matrix M may introduce undesired accumulation of errors, an analytical inverse matrix M −1 is obtained. Using Wronskians and recurrence relations of the spherical Bessel functions, it is shown that the inverse matrix M −1 can be simplified enough to insure an improvement in economy and accuracy in machine computations. Some useful properties of the inverse matrix M −1 are discussed which reduce the amount of machine time even further.

On an eigenfunction expansion associated with a condition of radiation. II

Abstract: In this paper certain Bessel type eigenfunction expansions are developed by considering a non-seif-adjoint problem which involves a radiation type condition

Some results involving Fox’s H-function and Bessel function

Abstract: In this paper we have evaluated an integral involving Fox’s H-function and employed it to establish an expansion formula for the H-function involving Bessel functions.

Numerical techniques for finding -zeros of Hankel functions

Abstract: This paper is concerned with numerical procedures for the evaluation of the zeros, with respect to order, of Hankel functions and their derivatives in cases when the arguments of these functions are held fixed. Using Olver's asymptotic expansions, two auxiliary tables have been computed, one appropriate for real and the other for purely imaginary argument. These tables, included herein, permit the calculation of rather accurate approximations to the desired v-zeros for wide ranges of argument and index. Moreover, from the given tabular entries, the errors attendant with any approximate v-zero so determined can be easily estimated.

On integral operators with kernels involving Bessel functions (Correction and Addendum)

Abstract: 1. Introduction . The purpose of this paper is to correct a number of points in an earlier paper ( 1 ) of the same title, and to prove a number of new results which supplement those given in that paper. The notation and definitions of this paper are those of ( 1 ).

Fast generation of spherical Bessel functions with complex arguments

Abstract: A simple recurrence technique for fast generation of spherical Bessel functions of order n and real or complex arguments is presented. The technique is accurate and very efficient, especially when the order of the function is less than the magnitude of the argument.

The Born approximation for intermediate energy potential scattering

Abstract: A wave function is constructed from spherical Bessel functions within the framework of the Born approximation, which makes the method suitable for obtaining reliable results in intermediate energy potential scattering problems. With a correct evaluation of the S-wave phase shift, the method yields results which are in very good agreement with exact values for a particle scattered from a Yukawa well.