Showing papers on "Bessel function published in 1968"

Boundary value problems of mathematical physics.

Abstract: VOLUME I. Preface to the classics edition Preface 1. The Green's function 2. Introduction to linear spaces 3. Linear integral equations 4. Spectral theory of second-order differential operators Appendix A. Static and dynamic problems for dtrings and membranes Static and fynamic problems for beams and plates The equation of heat conduction Appendix B. Bessel functions Wronskian Relationships The modified Bessel function The behavior of cylinder functions at zero and at infinity Index. VOLUME II. Preface to the classics edition Preface 5. Distributions and generalized solutions 6. Potential theory 7. Equations of evolution 8. Variational and related methods Appendix A. Spherical harmonics Appendix B. Asymptotic expansions Suggested additional readings Index.

Theory of a Two-Dimensional Ising Model with Random Impurities. I. Thermodynamics

Abstract: Recent experiments demonstrate that at the Curie temperature the specific heat may be a smooth function of the temperature. We propose that this effect can be due to random impurities and substantiate our proposal by a study of an Ising model containing such impurities. We modify the usual rectangular lattice by allowing each row of vertical bonds to vary randomly from row to row with a prescribed probability function. In the case that this probability is a particular distribution with a narrow width, we find that the logarithmic singularity of Onsager's lattice is smoothed out into a function which at ${T}_{c}$ is infinitely differentiable but not analytic. This function is expressible in terms of an integral involving Bessel functions and is computed numerically.

Design of monopulse antenna difference patterns with low sidelobes

Abstract: The flexibility of modern monopulse radar antenna systems makes possible the independent optimization of sum and difference patterns. The two parameter difference pattern, developed here for the circular aperature antenna, is designed to have nearly equal sidelobes similar to those of the Taylor sum pattern. The difference pattern is asymptotic to a model difference pattern which has the greatest slope (angle sensitivity) for a given sidelobe level. The model function is unrealizable because it has uniform sidelobes which are infinite in extent. The two parameter difference pattern is realizable and is expressed in a Fourier — Bessel series of N terms in a manner similar to Taylor's treatment of the sum pattern. The other parameter, A, controls sidelobe level. Comprehensive tables of the Fourier — Bessel coefficients are given for both the circular aperture series and the difference pattern series. Directivity and angle sensitivity are investigated and found to have maximum values that decrease as sidelobe level decreases. The monopulse system performance using the asymptotic difference pattern and the Taylor sum pattern compares favorably with a maximum likelihood angle estimation system. Development of a line source difference pattern is presented in the appendix.

Analytical Evaluation of Multicenter Integrals of r12−1 with Slater‐Type Atomic Orbitals. III. (2–2)‐Type Three‐Center Integrals

Abstract: The general three-center one-electron nuclear-attraction integral with integer-n Slater-type orbitals is evaluated analytically by letting the orbital exponent of a 1s orbital in the analytical formula for two-electron three-center electron-repulsion integrals tend to infinity. The result is an infinite sum in which the internuclear angles appear in spherical harmonics, and the internuclear distances appear in modified spherical Bessel functions and exponential-type integrals.

New Aspects of the Bipolar Expansion and Molecular Multicenter Integrals

Abstract: Standard bipolar expansions contain radial functions Jl1l2 L(r1, r2, R) which depend on three variables, r1, r2, R. Using representations of Bessel functions as series of Laguerre polynomials, the J's are expanded in terms of functions of the individual variables. As a result, the inverse distance between two points is brought into the form r12−1 = ∑ nlllml ∑ n2l2m2 Fn1l1m1, n2l2m2(R, θ, Ψ) Yl1m1(θ1, φ1)fn1l1(r1)Yl2m2(θ2, θ2)fn2l2(r2). This modified bipolar expansion is used to derive expressions for multicenter electron repulsion and nuclear attraction integrals. The method is particularly suitable for Gaussian orbitals expressed with spherical harmonics and yields compact expressions directly. For Slater‐type orbitals, the multicenter energy integrals appear as series involving only integrals of the two‐center overlap type. The one‐center and multipole limits of the bipolar expansion are examined.

On constructing distribution functions; with applications to lommel polynomials and bessel functions

Abstract: where the polynomials On(x) are recursively defined by: - 1(x) = 0, O(x) = 1, and (I-A) n+ (x) = (x - anX)-(x) - bnn- i(X) (n > 0). This study begins by showing how to obtain such a function +(x) for certain classes of sequences {an}l and {bn}l . Then we apply our results to obtain a distribution function for the modified Lommel polynomials (thus answering a question of Dickinson, [10, p. 121]) and to obtain some information about Bessel functions as a function of their order.

Graeffe''s method for eigenvalues

Abstract: Let an entire functionF(z) of finite genus have infinitely many zeros which are all positive, and take real values for realz. Then it is shown how to give two-sided bounds for all the zeros ofF in terms of the coefficients of the power series ofF, in fact in terms of the coefficients obtained byGraeffe's algorithm applied toF. A simple numerical illustration is given for a Bessel function.

Applications of a Theorem by A. B. Shidlovski

Abstract: Shidlovski’s deep theorem on Siegel E -functions satisfying systems of linear differential equations is applied in this paper to the study of the arithmetic properties of the partial derivatives C k (z) = 1/k!{∂/∂v} k J v (z)∣ v=0 ( k = 0,1,2,3) of the Bessel function J 0 (z) . As a by-product, expressions involving Euler’s constant γ and the constant ζ(3) are obtained for which the transcendency can be established.

Inverse Functions of the Products of Two Bessel Functions and Applications to Potential Scattering

Abstract: Inverse functions of products of two Bessel functions jl(xy)jm(xy) are determined for the cases m = l, l + 1, and l + 2. Integral representations for these inverse functions in terms of Neumann functions are given, and some of the simplest ones are expressed in terms of trigonometric functions.We show how one may obtain an integral representation for any well‐behaved function in terms of products of two Bessel functions, with the help of these inverse functions and also outline some of their applications to potential scattering. In particular, we demonstrate the usefulness of the inverse functions in determining the potential explicitly from the phase shifts in the Born approximation.

Improvement in recurrence techniques for the computation of Bessel functions of integral order

Abstract: 1. R. G. Hart, \"A close approximation related to the error function,\" Math. Comp., v. 20, 1966, pp. 600-602. 2. H. L. Gray, \"A limiting case of the G-transformation,\" SIAM J. Numer. Anal. (To appear.) 3. H. L. Gray & W. R. Schucany, \"A new rational approximation to Mills' ratio,\" /. Amer. Statist. Assoc. (To appear.) 4. C. Hastings, Approximations for Digital Computers, Princeton Univ. Press, Princeton, N. J., 1955, p. 167. MR 16, 963.

Some new re-lines with special functions for their solution

Abstract: An elementary transformation is used to reduce to a simple form, the Telegrapher's equation of a general tapered re-line. With suitable interrelationships between the distribution of r and c, the transformed equation is then solved in terms of the special functions such as Hermite, Bessel, confluent hypergeometric and Whittaker functions, resulting in a number of new classes of distributions. The effect of leakage is also discussed.

Toroidal and poloidal field representation for convective flow within a sphere

Abstract: In spherical regions, an arbitrary solenoidal velocity field can be developed into a series of toroidal and poloidal fields. These fundamental vector fields, expressed in terms of spherical Bessel functions and spherical harmonics, have certain orthogonality properties which prove useful in treating convective flow problems within spheres. The utility of this velocity field representation is demonstrated by considering the stability of a nonuniformly heated fluid in a spherical cavity. A variational principle is presented, equivalent to the eigenvalue problem for the critical Rayleigh number (the stability criterion). This principle forms the basis for an approximate method of determining upper bounds to the critical Rayleigh number. It is found that a class of three‐dimensional disturbances is more unstable than either the simplest poloidal (axisymmetric) disturbance mode or the simplest toroidal (two‐dimensional) disturbance mode. The numerical results are compared with previously published analyses.

On van der Pol's equation

Abstract: The family of trajectories (in the phase plane) of van der Pol's equation is a one parameter family of curves; no explicit analytical expression for it is known. The same is true for the orthogonal trajectories of this family. The differential equation of the orthogonal curves is solved here exactly in terms of modified Bessel functions. Two curves of the orthogonal family are found to be especially simple, and a graph is given which would permit plotting the solutions of van der Pol's equation with an accuracy higher than that attainable with the usual methods.

On Maitland's Generalised Bessel Function

Abstract: Maitland's generalised Bessel function [4] is defined by the equation where u is real and positive and v is any number real or complex. If u = 1, then (1.1) reduces to the form

Method of synthesis of phase-correcting networks

Abstract: A method is presented for determining the transfer functions of phase-corrective networks of both minimum-phase and all-pass types. The method is based on manipulation of a class of polynomials containing one or more variable parameters, which is derived from Bessel polynomials and represents basically an extension of the application of Bessel polynomials and related Bessel functions to solving the problem of approximation to the ideal constant-group-delay characteristic. Some simple phase-correcting networks, useful in those applications where the phase distortion occurs mainly at high frequencies, are first discussed, and the formulas and graphs for the evaluation of their group-delay responses are given. The extension of the method to allow for determination of the transfer function of the correcting network matching a prescribed phase characteristic is then presented. Both maximally fiat and equal-ripple types of approximation are considered, and explicit formulas for the evaluation of the coefficients of transfer functions of order n<6 are derived. The conditions on which the order of equalisation can be increased without affecting the complexity of the correcting network are also stated explicitly.

Frequency Spectrum of N Waves with Finite Rise Time

Abstract: Several investigators have derived expressions for the pressure spectrum of the sonic‐boom N wave Where the rise time is zero, Howes has shown that the pressure spectrum is a published spherical Bessel function The present analysis shows that the spectrum with a finite rise time, not exceeding one‐tenth of the duration, can be closely approximated by multiplying two spherical Bessel functions, or, equivalently, by adding their logarithmic curves

Exact solutions for some classes of tapered rc-lines

Abstract: A general transformation of the independent variable incorporating certain constants, which can be real or complex, dimensional or nondimensional, is used to transform the Telegrapher's equation of a general tapered rc-line. These constants permit with considerable ease the solution of the resulting equation in terms of known functions for a large number of tapered rc-lines. Solutions are obtained mainly in terms of the special functions like Bessel, Hermite, etc. The effect of leakage is also considered.

Spectral Representation of Vector Random Field

Abstract: A homogeneous multi-dimensional random field is represented as the moving average by means of the multi-dimensional Wiener integral, from which the Fourier spectral representation of the random field is derived. The spectral representation of a homogeneous and isotropic vector random field in 3-dimensional space is achieved using the result of the previous work on the generalized spherical Bessel function and vector harmonics. The correlation tensor and the spectral density tensor are expressed as the vector Hankel transforms of each other. The spectral representation of the random field is obtained in terms of the solid vector harmonics and the random spectral measures labelled by three quantum numbers. One of quantum numbers indicates a longitudinal and two transverse parts as well as their mutual independence. The expression is simplified considerably by introduction of a single random phase. Lastly, various vector random fields, including the potential, the solenoidal and the curl fields, are discusse...

Formulas for calculating bessel functions of integral order and complex argument

Abstract: : Formulas for calculating Bessel functions of integral order and complex argument are derived in this report. Calculations based on these formulas are not subject to the loss of significant figures which occurs in the Taylor and Neumann series when the argument is large and the order is small.

Synthesis of Concentric Ring Antenna Arrays Yielding Approximately Equal Side Lobes

Abstract: Based on the Haar's theorem, known in the general approximation theory, a method of synthesizing an optimum concentric ring array is presented. The far field pattern function of such an array is first formulated in terms of the zeroth-order first kind Bessel function. A numerical approach is used for obtaining a solution approximating any specified radiation pattern according to the “minimax” rather than the ordinary least-mean-square error criterion. With respect to a prespecified array size, the solution obtained has the following properties: (1) the maximum deviation between the synthesized and desired patterns is minimized, (2) the side lobes are approximately equal in level if the specified pattern is a Gaussian function, (3) a maximum directive gain can be realized, (4) a minimum number of elements required to achieve such a performance can be determined, and (5) the solution is unique under certain conditions. The theory and method to be presented are valid for arrays consisting of either isotropic sources or physical directional elements.

Some expansion formulae for G-function involving bessel functions

Abstract: In this paper we have derived some expansions for G-function involving Bessel functions with the help of the orthogonality property of Bessel functions.