Showing papers on "Bessel function published in 2000"

Dynamic symmetries at the critical point

Abstract: A new class of dynamic symmetries is introduced. It is suggested that an element of this class, associated with zeros of Bessel functions, be used to describe spectra of nuclei at or around the critical point of the U(5)-SO(6) shape phase transition, and, in general, spectra of systems undergoing a (second order) phase transition between the algebraic structures U(n-1) and SO(n).

Atom guiding along Laguerre-Gaussian and Bessel light beams

Abstract: We study and analyse the optical dipole potential for guiding atoms in various forms of both Laguerre-Gaussian and Bessel light beams. We find that Laguerre-Gaussian beams of high azimuthal index are advantageous for focusing atoms. We identify that high-order Bessel beams offer significant advantages for atom transport over extended distances when compared to Laguerre-Gaussian light beams. Possible future experimental methods to channel atoms into a high-order Bessel beam via a Laguerre-Gaussian light beam are discussed.

Propagation-invariant wave fields with finite energy.

Abstract: Propagation invariance is extended in the paraxial regime, leading to a generalized self-imaging effect. These wave fields are characterized by a finite number of transverse self-images that appear, in general, at different orientations and scales. They possess finite energy and thus can be accurately generated. Necessary and sufficient conditions are derived, and they are appropriately represented in the Gauss-Laguerre modal plane. Relations with the following phenomena are investigated: classical self-imaging, rotating beams, eigen-Fourier functions, and the recently introduced generalized propagation-invariant wave fields. In the paraxial regime they are all included within the generalized self-imaging effect that is presented. In this context we show an important relation between paraxial Bessel beams and Gauss-Laguerre beams.

Gaussian Fluctuation for the Number of Particles in Airy, Bessel, Sine, and Other Determinantal Random Point Fields

Abstract: We prove the Central Limit Theorem (CLT) for the number of eigenvalues near the spectrum edge for certain Hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the Gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions. As another corollary of the Costin–Lebowitz Theorem we prove the CLT for the empirical distribution function of the eigenvalues of random matrices from classical compact groups.

Unified description of nondiffracting X and Y waves

Abstract: A unified spectral and temporal representation is introduced for nondiffracting waves. We consider a set of elementary broadband X waves that spans the commonly considered nondiffracting wave solutions. These basis X waves have a simple spectral representation that leads to expressions in closed algebraic form or, alternatively, in terms of hypergeometric functions. The span of the X waves is also closed with respect to all spatial and temporal derivatives and, consequently, they can be used to compose different types of waves with complex spectral and spatial properties. The unified description of Bessel-based nondiffracting waves is further extended to include singular Neumann and Hankel waves, or Y waves. We also discuss connections between the different known nondiffracting wave solutions, and their relations to the present unified approach.

A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation

Abstract: An explicit eighth algebraic order Bessel and Neumann fitted method is developed in this paper for the numerical solution of the Schrodinger equation. The new method has free parameters which are defined in order the method is fitted to spherical Bessel and Neumann functions. A variable-step procedure is obtained based on the newly developed method and the method of Simos [17]. Numerical illustrations based on the numerical solution of the radial Schrodinger equation and of coupled differential equations arising from the Schrodinger equation indicate that this new approach is more efficient than other well known methods.

Optical generation of focus wave modes.

Abstract: Localized wave solutions of free-space wave equation can be used in numerous applications where the localized transmission of electromagnetic energy is of major importance. However, an optical implementation of localized wave fields has not been accomplished yet, except for an ultrashort version of the Bessel beams or the so called Bessel-X pulses. We propose an approach to constructing realizable optical schemes for generation of localized wave fields. We show that wavelength dispersion of the cone angle of axicons and circular diffraction gratings can be used to generate good approximation to focus wave modes.

Analytical study and numerical experiments for true and spurious eigensolutions of a circular cavity using the real‐part dual BEM

Abstract: It has been found recently that the multiple reciprocity method (MRM) (Chen and Wong. Engng. Anal. Boundary Elements 1997; 20(1):25–33; Chen. Processings of the Fourth World Congress on Computational Mechanics, Onate E, Idelsohn SR (eds). Argentina, 1998; 106; Chen and Wong. J. Sound Vibration 1998; 217(1): 75–95.) or real-part BEM (Liou, Chen and Chen. J. Chinese Inst. Civil Hydraulics 1999; 11(2):299–310 (in Chinese)). results in spurious eigenvalues for eigenproblems if only the singular (UT) or hypersingular (LM) integral equation is used. In this paper, a circular cavity is considered as a demonstrative example for an analytical study. Based on the framework of the real-part dual BEM, the true and spurious eigenvalues can be separated by using singular value decomposition (SVD). To understand why spurious eigenvalues occur, analytical derivation by discretizing the circular boundary into a finite degree-of-freedom system is employed, resulting in circulants for influence matrices. Based on the properties of the circulants, we find that the singular integral equation of the real-part BEM for a circular domain results in spurious eigenvalues which are the zeros of the Bessel functions of the second kind, Y (kρ), while the hypersingular integral equation of the real-part BEM results in spurious eigenvalues which are the zeros of the derivative of the Bessel functions of the second kind, Yn′(kρ). It is found that spurious eigenvalues exist in the real-part BEM, and that they depend on the integral representation one uses (singular or hypersingular; single layer or double layer) no matter what the given types of boundary conditions for the interior problem are. Furthermore, spurious modes are proved to be trivial in the circular cavity through analytical derivations. Numerically, they appear to have the same nodal lines of the true modes after normalization with respect to a very small nonzero value. Two examples with a circular domain, including the Neumann and Dirichlet problems, are presented. The numerical results for true and spurious eigensolutions match very well with the theoretical prediction. Copyright © 2000 John Wiley & Sons, Ltd.

Monotonicity and bounds on Bessel functions.

Abstract: I survey my recent results on monotonicity with respect to order of general Bessel functions, which follow from a new identity and lead to best possible uniform bounds. Application may be made to the ‘spreading of the wave packet’ for a free quantum particle on a lattice and to estimates for perturbative expansions. On my arrival as a graduate student at Berkeley in September 1964, I was amused to see a Volkswagen Beetle with Schrödinger’s equation written on it drive past. (I don’t recall if it was the time-dependent or time-independent equation.) As I stood in line to enroll, a table off to the side with a ‘Free Speech’ banner caught my eye. Soon were to begin the student demonstrations which culminated in Vietnam war protests. I managed to complete the typing of my thesis in 1969 even as tear gas wafted in through the open window. I had asked Eyvind Wichmann if he would supervise my Ph.D. studies, and after checking that Emilio Segrè had given a good report on my oral examination, he agreed to take me on. I’d like to thank Eyvind for helping to make my stay at Berkeley a successful one.

Master formulas for two‐ and three‐center one‐electron integrals involving Cartesian GTO, STO, and BTO

Abstract: As a first application of the shift operators method we derive master formulas for the two- and three-center one-electron integrals involving Gaussians, Slater, and Bessel basis functions. All these formulas have a common structure consisting in linear combinations of polynomials of differences of nuclear coordinates. Whereas the polynomials are independent of the type (GTO, BTO, or STO) of basis functions, the coefficients depend on both the class of integral (overlap, kinetic energy, nuclear attraction) and the type of basis functions. We present the general expression of polynomials and coefficients as well as the recurrence relations for both the polynomials and the whole integrals. Finally, we remark on the formal and computational advantages of this approach. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 83–93, 2000

Explicit Zeta Functions for Bosonic and Fermionic Fields on a Noncommutative Toroidal Spacetime

Abstract: Explicit formulas for the zeta functions $\zeta_\alpha (s)$ corresponding to bosonic ($\alpha =2$) and to fermionic ($\alpha =3$) quantum fields living on a noncommutative, partially toroidal spacetime are derived. Formulas for the most general case of the zeta function associated to a quadratic+linear+constant form (in {\bf Z}) are obtained. They provide the analytical continuation of the zeta functions in question to the whole complex $s-$plane, in terms of series of Bessel functions (of fast, exponential convergence), thus being extended Chowla-Selberg formulas. As well known, this is the most convenient expression that can be found for the analytical continuation of a zeta function, in particular, the residua of the poles and their finite parts are explicitly given there. An important novelty is the fact that simple poles show up at $s=0$, as well as in other places (simple or double, depending on the number of compactified, noncompactified, and noncommutative dimensions of the spacetime), where they had never appeared before. This poses a challenge to the zeta-function regularization procedure.

Few-optical-cycle bessel-gauss pulsed beams in free space

Abstract: We introduce a new family of nonseparable, pulselike and beamlike solutions of the wave equation in the paraxial approximation with pseudonondiffracting behavior. They are the pulsed versions of the Bessel-Gauss beams by Gori et al., and encompass as particular cases the diffraction-free Bessel-X pulses, isodiffracting pulses, and, in the many-cycle limit, Bessel and Gaussian beams. Unlike Bessel-X waves, these solutions carry finite energy but retain nondiffracting behavior over a finite propagation distance, and could be physically produced with mode-locked toroidal resonators.

Numerical and asymptotic aspects of parabolic cylinder functions

Abstract: Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver's results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered.

The Askey-Wilson function transform

Abstract: In this paper we present an explicit (rank one) function transform which contains several Jacobi-type function transforms and Hankel-type transforms as degenerate cases. The kernel of the transform, which is given explicitly in terms of basic hypergeometric series, thus generalizes the Jacobi function as well as the Bessel function. The kernel is named the Askey-Wilson function, since it provides an analytical continuation of the Askey-Wilson polynomial in its degree. In this paper we establish the $L^2$-theory of the Askey-Wilson function transform, and we explicitely determine its inversion formula.

On the superluminal propagation of X-shaped localized waves

Abstract: The generation and propagation of superluminal X-shaped pulses is investigated. We demonstrate that such pulses can be modelled using a spectral approach that produces time-limited Bessel beams. Special attention is given to calculating the velocities of the modelled pulsed Bessel beams. The velocities of the peaks of the resulting pulses depend on the shapes of the spatio-temporal distributions of the applied time-windows. The generation of pulsed Bessel beams is investigated for various set-ups; including circular arrays, annular slits and axicons. It is shown that superluminal pulsed Bessel beams undergo a delayed generation before they are launched; henceforth, the peak of these pulses travels at speeds exceeding that of light.

On the strong and weak solutions of stochastic differential equations governing Bessel processes

Abstract: We prove that the δ-dimensional Bessel process (δ > 1) is a strong solution of a stochastic differential equation of the special form. The purpose of this paper is to investigate whether there exist other (weak and strong) solutions of these equations. This leads us to the conclusion that Zvonkin's theorem cannot be extended to stochastic differential equations with an unbounded drift.

An X wave transform

Abstract: Limited diffraction beams such as X waves can propagate to an infinite distance without spreading if they are produced with an infinite aperture and energy. In practice, when the aperture and energy are finite, these beams have a large depth of field with only limited diffraction. Because of this property, limited diffraction beams could have applications in medical imaging, tissue characterization, blood flow velocity vector imaging, nondestructive evaluation of materials, communications, and other areas such as optics and electromagnetics. In this paper, a new transform, called X wave transform, is developed. In the transform, any well behaved solutions to the isotropic-homogeneous wave equation or limited diffraction beams can he expanded using X waves as basis functions. The coefficients of the expansions can be calculated with the properties that X waves are orthogonal. Examples are given to demonstrate the efficacy of the X wave transform. The X wave transform reveals an intrinsic relationship between any well behaved solutions to the wave equation and X waves, including limited diffraction beams. This provides a theoretical foundation to develop new limited diffraction beams or solutions to the wave equation that may have practical usefulness.

Target Patterns and Spirals in Planar Reaction-Diffusion Systems

Abstract: Summary. Solutions of reaction-diffusion equations on a circular domain are considered. With Robin boundary conditions, the primary instability may be a Hopf bifurcation with eigenfunctions exhibiting prominent spiral features. These eigenfunctions, defined by Bessel functions of complex argument, peak near the boundary and are called wall modes. In contrast, if the boundary conditions are Neumann or Dirichlet, then the eigenfunctions are defined by Bessel functions of real argument, and take the form of body modes filling the interior of the domain. Body modes typically do not exhibit pronounced spiral structure. We argue that the wall modes are important for understanding the formation process of spirals, even in extended systems. Specifically, we conjecture that wall modes describe the core of the spiral; the constant-amplitude spiral visible outside the core is the result of strong nonlinearities which enter almost immediately above threshold as a consequence of the exponential radial growth of the wall modes.

Thermodynamics of self-gravitating systems with softened potentials

Abstract: The microcanonical statistical mechanics of a set of self-gravitating particles is analyzed in a mean-field approach. In order to deal with an upper bounded entropy functional, a softened gravitational potential is used. The softening is achieved by truncating to N terms an expansion of the Newtonian potential in spherical Bessel functions. The order N is related to the softening at short distances. This regularization has the remarkable property that it allows for an exact solution of the mean-field equation. It is found that for N not too large the absolute maximum of the entropy coincides to high accuracy with the solution of the Lane-Emden equation, which determines the mean-field mass distribution for the Newtonian potential for energies larger than E(c) approximately -0.335GM(2)/R. Below this energy a collapsing phase transition, with negative specific heat, takes place. The dependence of this result on the regularizing parameter N is discussed.

Advanced mathematical methods in science and engineering

Abstract: Ordinary Differential Equations DEFINITIONS LINEAR DIFFERENTIAL EQUATIONS OF FIRST ORDER LINEAR INDEPENDENCE AND THE WRONSKIAN LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION OF ORDER N WITH CONSTANT COEFFICIENTS EULER'S EQUATION PARTICULAR SOLUTIONS BY METHOD OF UNDETERMINED COEFFICIENTS PARTICULAR SOLUTIONS BY THE METHOD OF VARIATIONS OF PARAMETERS ABEL'S FORMULA FOR THE WRONSKIAN INITIAL VALUE PROBLEMS Series Solutions of Ordinary Differential Equations INTRODUCTION POWER SERIES SOLUTIONS CLASSIFICATION OF SINGULARITIES FROBENIUS SOLUTION Special Functions BESSEL FUNCTIONS BESSEL FUNCTION OF ORDER ZERO BESSEL FUNCTION OF AN INTEGER ORDER N RECURRENCE RELATIONS FOR BESSEL FUNCTIONS BESSEL FUNCTIONS OF HALF ORDERS SPHERICAL BESSEL FUNCTIONS HANKEL FUNCTIONS MODIFIED BESSEL FUNCTIONS GENERALIZED EQUATIONS LEADING TO SOLUTIONS IN TERMS OF BESSEL FUNCTIONS BESSEL COEFFICIENTS INTEGRAL REPRESENTATION OF BESSEL FUNCTIONS ASYMPTOTIC APPROXIMATIONS OF BESSEL FUNCTIONS FOR SMALL ARGUMENTS ASYMPTOTIC APPROXIMATIONS OF BESSEL FUNCTIONS FOR LARGE ARGUMENTS INTEGRALS OF BESSEL FUNCTIONS ZEROES OF BESSEL FUNCTIONS LEGENDRE FUNCTIONS LEGENDRE COEFFICIENTS RECURRENCE FORMULAE FOR LEGENDRE POLYNOMIALS INTEGRAL REPRESENTATION FOR LEGENDRE POLYNOMIALS INTEGRALS OF LEGENDRE POLYNOMIALS EXPANSIONS OF FUNCTIONS IN TERMS OF LEGENDRE POLYNOMIALS LEGENDRE FUNCTION OF THE SECOND KIND QN(X) ASSOCIATED LEGENDRE FUNCTIONS GENERATING FUNCTION FOR ASSOCIATED LEGENDRE FUNCTIONS RECURRENCE FORMULAE FOR Pnm INTEGRALS OF ASSOCIATED LEGENDRE FUNCTIONS ASSOCIATED LEGENDRE FUNCTION OF THE SECOND KIND Qnm Boundary Value Problems and Eigenvalue Problems INTRODUCTION VIBRATION, WAVE PROPAGATION OR WHIRLING OF STRETCHED STRINGS LONGITUDINAL VIBRATION AND WAVE PROPAGATION IN ELASTIC BARS VIBRATION, WAVE PROPAGATION AND WHIRLING OF BEAMS WAVES IN ACOUSTIC HORNS STABILITY OF COMPRESSED COLUMNS IDEAL TRANSMISSION LINES (TELEGRAPH EQUATION) TORSIONAL VIBRATION OF CIRCULAR BARS ORTHOGONALITY AND ORTHOGONAL SETS OF FUNCTIONS GENERALIZED FOURIER SERIES ADJOINT SYSTEMS BOUNDARY VALUE PROBLEMS EIGENVALUE PROBLEMS PROPERTIES OF EIGENFUNCTIONS OF SELF-ADJOINT SYSTEMS STURM-LIOUVILLE SYSTEM STURM-LIOUVILLE SYSTEM FOR FOURTH-ORDER EQUATIONS SOLUTION OF NON-HOMOGENEOUS EIGENVALUE PROBLEMS FOURIER SINE SERIES FOURIER COSINE SERIES COMPLETE FOURIER SERIES FOURIER-BESSEL SERIES FOURIER-LEGENDRE SERIES Functions of a Complex Variable COMPLEX NUMBERS ANALYTIC FUNCTIONS ELEMENTARY FUNCTIONS INTEGRATION IN THE COMPLEX PLANE CAUCHY'S INTEGRAL THEOREM CAUCHY'S INTEGRAL FORMULA INFINITE SERIES TAYLOR'S EXPANSION THEOREM LAURENT'S SERIES CLASSIFICATION OF SINGULARITIES RESIDUES AND RESIDUE THEOREM INTEGRALS OF PERIODIC FUNCTIONS IMPROPER REAL INTEGRALS IMPROPER REAL INTEGRAL INVOLVING CIRCULAR FUNCTIONS IMPROPER REAL INTEGRALS OF FUNCTIONS HAVING SINGULARITIES ON THE REAL AXIS THEOREMS ON LIMITING CONTOURS INTEGRALS OF EVEN FUNCTIONS INVOLVING LOG X INTEGRALS OF FUNCTIONS INVOLVING Xa INTEGRALS OF ODD OR ASYMMETRIC FUNCTIONS INTEGRALS OF ODD OR ASYMMETRIC FUNCTIONS INVOLVING LOG X INVERSE LAPLACE TRANSFORMS Partial Differential Equations of Mathematical Physics INTRODUCTION THE DIFFUSION EQUATION THE VIBRATION EQUATION THE WAVE EQUATION HELMHOLTZ EQUATION POISSON AND LAPLACE EQUATIONS CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS UNIQUENESS OF SOLUTIONS THE LAPLACE EQUATION THE POISSON EQUATION THE HELMHOLTZ EQUATION THE DIFFUSION EQUATION THE VIBRATION EQUATION THE WAVE EQUATION Integral Transforms FOURIER INTEGRAL THEOREM FOURIER COSINE TRANSFORM FOURIER SINE TRANSFORM COMPLEX FOURIER TRANSFORM MULTIPLE FOURIER TRANSFORM HANKEL TRANSFORM OF ORDER ZERO HANKEL TRANSFORM OF ORDER nu GENERAL REMARKS ABOUT TRANSFORMS DERIVED FROM THE FOURIER INTEGRAL THEOREM GENERALIZED FOURIER TRANSFORM TWO-SIDED LAPLACE TRANSFORM ONE-SIDED GENERALIZED FOURIER TRANSFORM LAPLACE TRANSFORM MELLIN TRANSFORM OPERATIONAL CALCULUS WITH LAPLACE TRANSFORMS SOLUTION OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS OPERATIONAL CALCULUS WITH FOURIER COSINE TRANSFORM OPERATIONAL CALCULUS WITH FOURIER SINE TRANSFORM OPERATIONAL CALCULUS WITH COMPLEX FOURIER TRANSFORM OPERATIONAL CALCULUS WITH MULTIPLE FOURIER TRANSFORM OPERATIONAL CALCULUS WITH HANKEL TRANSFORM Green's Functions INTRODUCTION GREEN'S FUNCTION FOR ORDINARY DIFFERENTIAL BOUNDARY VALUE PROBLEM GREEN'S FUNCTION FOR AN ADJOINT SYSTEM SYMMETRY OF THE GREEN'S FUNCTIONS AND RECIPROCITY GREEN'S FUNCTION FOR EQUATIONS WITH CONSTANT COEFFICIENTS GREEN'S FUNCTIONS FOR HIGHER ORDERED SOURCES GREEN'S FUNCTION FOR EIGENVALUE PROBLEMS GREEN'S FUNCTION FOR SEMI-INFINITE ONE DIMENSIONAL MEDIA GREEN'S FUNCTION FOR INFINITE ONE-DIMENSIONAL MEDIA GREEN'S FUNCTION FOR PARTIAL DIFFERENTIAL EQUATIONS GREEN'S IDENTITIES FOR THE LAPLACIAN OPERATOR GREEN'S IDENTITY FOR THE HELMHOLTZ OPERATOR GREEN'S IDENTITY FOR BI-LAPLACIAN OPERATOR GREEN'S IDENTITY FOR THE DIFFUSION OPERATOR GREEN'S IDENTITY FOR THE WAVE OPERATOR GREEN'S FUNCTION FOR UNBOUNDED MEDIA-FUNDAMENTAL SOLUTION FUNDAMENTAL SOLUTION FOR THE LAPLACIAN FUNDAMENTAL SOLUTION FOR THE BI-LAPLACIAN FUNDAMENTAL SOLUTION FOR THE HELMHOLTZ OPERATOR FUNDAMENTAL SOLUTION FOR THE OPERATOR, - 2 + mu2 CAUSAL FUNDAMENTAL SOLUTION FOR THE DIFFUSION OPERATOR CAUSAL FUNDAMENTAL SOLUTION FOR THE WAVE OPERATOR FUNDAMENTAL SOLUTIONS FOR THE BI-LAPLACIAN HELMHOLTZ OPERATOR GREEN'S FUNCTION FOR THE LAPLACIAN OPERATOR FOR BOUNDED MEDIA CONSTRUCTION OF THE AUXILIARY FUNCTION-METHOD OF IMAGES GREEN'S FUNCTION FOR THE LAPLACIAN FOR HALF-SPACE GREEN'S FUNCTION FOR THE LAPLACIAN BY EIGENFUNCTION EXPANSION FOR BOUNDED MEDIA GREEN'S FUNCTION FOR A CIRCULAR AREA FOR THE LAPLACIAN GREEN'S FUNCTION FOR SPHERICAL GEOMETRY FOR THE LAPLACIAN GREEN'S FUNCTION FOR THE HELMHOLTZ OPERATOR FOR BOUNDED MEDIA GREEN'S FUNCTION FOR THE HELMHOLTZ OPERATOR FOR HALF-SPACE GREEN'S FUNCTION FOR A HELMHOLTZ OPERATOR IN QUARTER-SPACE CAUSAL GREEN'S FUNCTION FOR THE WAVE OPERATOR IN BOUNDED MEDIA CAUSAL GREEN'S FUNCTION FOR THE DIFFUSION OPERATOR FOR BOUNDED MEDIA METHOD OF SUMMATION OF SERIES SOLUTIONS IN TWO DIMENSIONAL MEDIA Asymptotic Methods INTRODUCTION METHOD OF INTEGRATION BY PARTS LAPLACE'S INTEGRAL STEEPEST DESCENT METHOD DEBYE'S FIRST ORDER APPROXIMATION ASYMPTOTIC SERIES APPROXIMATION METHOD OF STATIONARY PHASE STEEPEST DESCENT METHOD IN TWO DIMENSIONS MODIFIED SADDLE POINT METHOD: SUBTRACTION OF A SIMPLE POLE MODIFIED SADDLE POINT METHOD: SUBTRACTION OF POLE OF ORDER N SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS FOR LARGE ARGUMENTS CLASSIFICATION OF POINTS AT INFINITY SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH REGULAR SINGULAR POINTS ASYMPTOTIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH IRREGULAR SINGULAR POINTS OF RANK ONE THE PHASE INTEGRAL AND WKBJ METHOD FOR AN IRREGULAR SINGULAR POINT OF RANK ONE ASYMPTOTIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH IRREGULAR SINGULAR POINTS OF RANK HIGHER THAN ONE ASYMPTOTIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH LARGE PARAMETERS Numerical Methods INTRODUCTION ROOTS OF NON-LINEAR EQUATIONS ROOTS OF A SYSTEM OF NON-LINEAR EQUATION FINITE DIFFERENCES NUMERICAL DIFFERENTIATION NUMERICAL INTEGRATION ORDINARY DIFFERENTIAL EQUATIONS: INITIAL VALUE PROBLEMS ORDINARY DIFFERENTIAL EQUATIONS: BOUNDARY VALUE PROBLEMS ORDINARY DIFFERENTIAL EQUATIONS: EIGENVALUE PROBLEMS PARTIAL DIFFERENTIAL EQUATIONS Appendix A: Infinite Series Appendix B: Special Functions Appendix C: Orthogonal Coordinate Systems Appendix D: Dirac Delta Functions Appendix E: Plots of Special Functions Appendix F: Vector Analysis Appendix G: Matrix Algebra References Answers Index Problems appear at the end of each chapter.

Molecular calculations with B functions

Abstract: A program for molecular calculations with B functions is reported and its performance is analyzed. All the one- and two-center integrals and the three-center nuclear attraction integrals are computed by direct procedures, using previously developed algorithms. The three- and four-center electron repulsion integrals are computed by means of Gaussian expansions of the B functions. A new procedure for obtaining these expansions is also reported. Some results on full molecular calculations are included to show the capabilities of the program and the quality of the B functions to represent the electronic functions in molecules.

A weighted uniform L^p-estimate of Bessel functions: A note on a paper of Guo

Abstract: An improved Guo’s uniform Lp estimate of Bessel functions is shown by using a uniform pointwise bound of Barceló and Córdoba. Recently, Guo has shown, [Guo, Theorem 3.5], the following uniform L estimate: ∫ ∞ 0 |Jν(x)|xdx ≤ C(p− 4)−1, ν ≥ 0, p > 4. (1) Here Jν(x) denotes the Bessel function of the first kind of order ν, cf. [W]. This estimate was proved first for ν = 0, 1, . . . , by means of a dual form of a Fourier restriction theorem for the plane unit circle and then extended to an arbitrary ν ≥ 0. The estimate was crucial in proving the main result of [Guo], Theorem 4.1. It was quite reasonable to expect a proof of (1) based on intrinsic properties of Bessel functions. Furthermore, it was natural to expect an estimate like (1) for a larger range of p’s by adding an appropriate power weight in the integral on the left side of (1). More precisely, it was natural to look for an inequality of the form ∫ ∞ 0 |Jν(x)|xdx ≤ C(p, a), ν ≥ 0, (2) with a constant C(p, a) > 0 depending only on p and a (we did not care about making the constant C(p, a) the best possible). Since Jν(x) = O(x−1/2), x → ∞, the necessary assumption on a to make the integral in (2) convergent at infinity for every single ν ≥ 0 is a < p/2− 1. On the other hand Jν(x) = O(x ), x → 0; hence the necessary assumption on a to make the integral in (2) convergent at zero for every ν ≥ 0 is a > −1. It is now interesting to note that Guo’s result, (1), shows that the assumption −1 < a < p/2 − 1 is also sufficient for (2) to hold in the case 0 < p ≤ 4. Indeed, assume ∫ ∞ 1 |Jν(x)|xdx ≤ Cq, ν ≥ 0, holds true for every q > 4 and consider p and a such that 0 < p ≤ 4 and a < p/2−1. Since 2(a+ 1) < p ≤ 4, we can choose s > 1 satisfying 2(a + 1)s < 4 < ps. Then, Received by the editors August 1, 1998 and, in revised form, November 11, 1998. 1991 Mathematics Subject Classification. Primary 33C10.