Bessel Functions Of Large Order Research Articles

Computation of Hyperspherical Bessel Functions

AbstractIn this paper we present a fast and accurate numerical algorithm for the computation of hyperspherical Bessel functions of large order and real arguments. For the hyperspherical Bessel functions of closed type, no stable algorithm existed so far due to the lack of a backwards recurrence. We solved this problem by establishing a relation to Gegenbauer polynomials. All our algorithms are written in C and are publicly available at Github [https://github.com/lesgourg/class_public]. A Python wrapper is available upon request.

Modal analysis of a multilayer dielectric microsphere in a free-space ambient medium.

An analysis of the eigenmodes associated with a dielectric sphere embedded within one or more spherical layers of varying permittivity is presented. A matrix formulation is applied to the general multilayer problem, and the solution is validated by comparison to the complex transcendental equations derived for the homogeneous dielectric spherical resonator and a single-layer spherical dielectric resonator. Although only lossless cases are considered for simplicity, the formulation presented is applicable to resonators (with loss or gain) via the replacement of real permittivities with complex permittivities. The matrix formulation presented for the multilayer dielectric sphere follows a previously published development with the inclusion of some significant corrections. A Wronskian filter effective index approximation is developed for the estimation of Bessel functions of large order (expanding the functionality of the models to evaluation of whispering gallery modes), which is easily integrated into the presented models. Numerical simulations are generated utilizing MATLAB to solve for the complex roots of the characteristic equation, amplitude reflection coefficient, surface impedance of the dielectric layers, Q-factor, and field patterns for both TE and TM modes. Results are presented for single-layer and multilayer spherical geometries and compared to previously published results.

AN INVESTIGATION OF UNIFORM EXPANSIONS OF LARGE ORDER BESSEL FUNCTIONS IN GRAVITATIONAL WAVE SIGNALS

In this work, we extend the analytic treatment of Bessel functions of large order and/or argument. We examine uniform asymptotic Bessel function expansions and show their accuracy and range of validity. Such situations arise in a variety of applications, particularly the Fourier transform (FT) of the gravitational wave (GW) signal from a pulsar, global parameter space correlations of a coherent matched filtering search for continuous GWs from isolated neutron stars and tomographic reconstruction of GW LISA sources. The uniform expansion we consider here is found to be valid in the entire range of the argument.

Exactification of the method of steepest descents: the Bessel functions of large order and argument

The Hadamard expansion procedure applied to Laplace–type integrals taken along contours in the complex plane enables an exact description of the method of steepest descents. This mode of expansion is illustrated by the evaluation of the Bessel functions J v (? x) and Y v (v x) of large order and argument when x is bounded away from unity. The limit x → 1 , corresponding to the coalescence of the active saddles in the integral representations of the Bessel functions, translates into a progressive loss of exponential separation between the different levels of the Hadamard expansion, which renders computation in this limit more difficult. It is shown how a simple modification to this procedure can be employed to deal with the coalescence of the active saddles when x → 1 .

Improvements in the phase-amplitude method for calculating free–free gaunt factors and spherical bessel functions of high angular momentum

Standard quantum mechanical calculations of free–free gaunt factors can be computationally exacting due to the fine radial mesh required to resolve the oscillations in the continuum wavefunctions. Introducing a wavefunction representation in terms of amplitude and phase allows a coarser radial mesh leading to considerable speed up. However, special treatments of solving the non-linear differential equation for the amplitude and evaluating the radial dipole integrals are required. These algorithmic details are presented and contrasted to earlier schemes. A non-recursive method for calculating spherical Bessel functions of large order is also given.

On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order

For the coefficients An(‘) and Bn(‘), that occur in the uniform asymptotic expansions of Bessel functions of large order, we give asymptotic expansions as n M X. The coefficients in these asymptotic expansions are again Am>(‘) and Bm(‘), and the asymptotic base consists of functions Apq(n,‘), which can be seen as new generalizations of the Airy function. We describe the asymptotic behaviour of the functions Apq(n, ‘), as n to infinity, and we compute the Taylor-series expansions of Apq(n, ‘) at ‘ = 0. Two numerical examples are included.

Convergent Liouville–Green expansions for second-order linear differential equations, with an application to Bessel functions

A class of second-order linear differential equations with a large parameter u is considered. It is shown that Liouville–Green type expansions for solutions can be expressed using factorial series in the parameter, and that such expansions converge for Re ( u ) > 0, uniformly for the independent variable lying in a certain subdomain of the domain of asymptotic validity. The theory is then applied to obtain convergent expansions for modified Bessel functions of large order.

Uniformly Valid Expansions for Laplace Integrals

Uniformly valid asymptotic expansions for a class of parameter-dependent Laplace integrals involving coalescing saddle points are obtained by a direct method based on matched asymptotic expansion theory. Bessel functions of large order are treated as an example.

Spherical bessel functions of large order

This note introduces functions b n ( x), related to spherical Bessel functions j n ( x) and y n ( x). They are scaled so that they are bounded functions of n and polynomially bounded functions of x, and therefore avoid the problems of underflow and overflow which are so common with Bessel functions. They can be generated from a stable recurrence relation for which starting values are readily computable.

Scalar analysis of nonsymmetrically distorted umbrella reflector

An approximation technique for analyzing nonsymmetrically distorted umbrella reflectors is presented which offers savings in computer time by a factor of twice the number of gores. The surface distortions are described by incorporating an additional path-length phase factor into the illumination function. By expanding this phase factor in a finite Fourier series and neglecting Bessel functions of large order and small argument, the field integral reduces to a compact expression integrated over only half of a gore.

XIX.—On the Determination of the General Term in Green-type Expansions

SynopsisA formal method is developed for deriving a series expansion of the general term in Green-type expansions. The technique is exemplified by detailed calculations for modified Bessel functions of large order.

An asymptotic expansion for the Green's function of nonrelativistic potential-scattering theory and the asymptotic character of the born series for the jost function for large complex angular momenta

From a generalization of the Langer uniform asymptotic formula for Bessel functions of large order, an asymptotic expansion is constructed for the scattering Green's function. From this expansion a simple uniform bound for the Green's function follows, which contains the well-known nonuniform 1/k bound. This uniform bound is then used to study the asymptotic properties for large complex angular momenta of the Born series for the Jost function in nonrelativistic potential scattering theory for a great number of potentials. Among other things the following following result is obtained. If the potential has a meromorphic continuation, with a finite number of complex poles, into the half-plane Rer>0 satisfying the condition\(\int\limits_\upsilon ^\alpha {\left| {rV\left( r \right)dr} \right|< \infty ,} \int\limits_\alpha ^\infty {\left| {V\left( r \right)dr} \right|< \infty , 0}< a< \infty \), on all rays in this half-plane, then the Born series for the Jost function is an asymptotic series in the Erdelyi sense and the Jost function tends to 1 when |λ|→∞ in any direction in the region |arg λ|\s-π/2. Thus the Sommerfeld-Watson transformation is allowed for such potentials. For parallels to the imaginaryr-axis the above condition may be slightly relaxed to include, for instance, the Yukawa potential. Our method also gives some information about the necessary conditions to be imposed on the potential in order to make the Sommerfeld-Watson transformation permissible. Although strict mathematical proof is lacking, the following conclusions have been reached. The potential must have a smooth connection between small and large values ofr, and it must not decrease faster than exponentially whenr»∞, in order to make the Sommerfeld-Watson transformation permissible.

Derivation of Green-type, transitional, and uniform asymptotic expansions from differential equations I. General theory, and application to modified Bessel functions of large order

The problem of deriving Green-type asymptotic solutions from differential equations of general form d 2 y /dz 2 = X(a 2 &gt;, z)y , for large values of a 2 , is reformulated. Combination of this formulation with the method of Mellin transforms leads further to a particularly convenient procedure for finding asymptotic expansions valid in transitional regions, and general uniform expansions. The methods are illustrated by detailed calculations for modified Bessel functions.

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.

An extension of the method of steepest descents

ABSTRACTIn the integralthe functions g(z), f(z, α) are analytic functions of their arguments, and N is a large positive parameter. When N tends to ∞, asymptotic expansions can usually be found by the method of steepest descents, which shows that the principal contributions arise from the saddle points, i.e. the values of z at which ∂f/∂z = 0. The position of the saddle points varies with α, and if for some a (say α = 0) two saddle points coincide (say at z = 0) the ordinary method of steepest descents gives expansions which are not uniformly valid for small α. In the present paper we consider this case of two nearly coincident saddle points and construct uniform expansions as follows. A new complex variable u is introduced by the implicit relationwhere the parameters ζ(α), A(α) are determined explicitly from the condition that the (u, z) transformation is uniformly regular near z = 0, α = 0 (see § 2 below). We show that with these values of the parameters there is one branch of the transformation which is uniformly regular. By taking u on this branch as a new variable of integration we obtain for the integral uniformly asymptotic expansions of the formwhere Ai and Ai′ are the Airy function and its derivative respectively, and A(α), ζ(α) are the parameters in the transformation. The application to Bessel functions of large order is briefly described.

The asymptotic solution of linear second order differential equations in a domain containing a turning point and a regular singularity

Asymptotic solutions of the differential equation d1 2wjdz2 = {u2z~2(z0—z) pi(z) +z ~2ql(z)} w, for large positive values of u are examined; P 1 (z) AND Q 1 (Z) are regular functions of the complex variable z in a domain in which P 1 (z) does not vanish. The point z = 0 is a regular singularity of the equation and a branch-cut extending from z = 0 is taken through the point Z=Z O which is assumed to lie on the positive real z axis. Asymptotic expansions for the solutions of the equation, valid uniformly with respect to z in domains including Z=0, Z 0+-iO are derived in terms of Bessel functions of large order. Expansions given by previous theory are not valid at all these points. The theory can be applied to the Legendre functions.

The asymptotic expansion of bessel functions of large order

New expansions are obtained for the functions Iv{yz), ) and their derivatives in terms of elementary functions, and for the functions J v(vz), Yv{vz), H fvz) and their derivatives in terms of Airy functions, which are uniformly valid with respect to z when | | is large. New series for the zeros and associated values are derived by reversion and used to determine the distribution of the zeros of functions of large order in the z-plane. Particular attention is paid to the complex zeros of 7„(z) and the Hankel functions when the order n is an integer or half an odd integer, and for this purpose some new asymptotic expansions of the Airy functions are derived. Tables are given of complex zeros of Airy functions and other quantities which facilitate the rapid calculation of the smaller complex zeros of 7„(z), 7'(z), and the Hankel functions and their derivatives, when 2 n is an integer, to an accuracy of three or four significant figures.

Saddlepoint Approximations in Statistics

It is often required to approximate to the distribution of some statistic whose exact distribution cannot be conveniently obtained. When the first few moments are known, a common procedure is to fit a law of the Pearson or Edgeworth type having the same moments as far as they are given. Both these methods are often satisfactory in practice, but have the drawback that errors in the tail regions of the distribution are sometimes comparable with the frequencies themselves. The Edgeworth approximation in particular notoriously can assume negative values in such regions. The characteristic function of the statistic may be known, and the difficulty is then the analytical one of inverting a Fourier transform explicitly. In this paper we show that for a statistic such as the mean of a sample of size $n$, or the ratio of two such means, a satisfactory approximation to its probability density, when it exists, can be obtained nearly always by the method of steepest descents. This gives an asymptotic expansion in powers of $n^{-1}$ whose dominant term, called the saddlepoint approximation, has a number of desirable features. The error incurred by its use is $O(n^{-1})$ as against the more usual $O(n^{-1/2})$ associated with the normal approximation. Moreover it is shown that in an important class of cases the relative error of the approximation is uniformly $O(n^{-1})$ over the whole admissible range of the variable. The method of steepest descents was first used systematically by Debye for Bessel functions of large order (Watson [17]) and was introduced by Darwin and Fowler (Fowler [9]) into statistical mechanics, where it has remained an indispensable tool. Apart from the work of Jeffreys [12] and occasional isolated applications by other writers (e.g. Cox [2]), the technique has been largely ignored by writers on statistical theory. In the present paper, distributions having probability densities are discussed first, the saddlepoint approximation and its associated asymptotic expansion being obtained for the probability density of the mean $\bar{x}$ of a sample of $n$. It is shown how the steepest descents technique is related to an alternative method used by Khinchin [14] and, in a slightly different context, by Cramer [5]. General conditions are established under which the relative error of the saddlepoint approximation is $O(n^{-1})$ uniformly for all admissible $\bar{x}$, with a corresponding result for the asymptotic expansion. The case of discrete variables is briefly discussed, and finally the method is used for approximating to the distribution of ratios.

On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order

is possible, so that the resulting forms represent asymptotically a pair of solutions of the equation given. Moreover, each of these forms is then known to represent one and the same solution so long as p remains in a region of the complex plane in which its pure imaginary part is either invariably greater than or invariably less than some constant. The forms (2) are evidently of oscillatory or exponential type according as 42(x) is positive or negative on the x interval under account, and on different intervals in which +2(x) is of opposite sign the forms are respectively of the two opposing types. Between two such intervals a change in the character of any solution must, therefore, take place. It is evident, however, that the manner of this change is not deducible from the forms (2), for whatever the magnitude of p these forms fail to remain significant at a zero of 02(X).

On the Asymptotic Solutions of Ordinary Differential Equations, With an Application to the Bessel Functions of Large Order

On the Asymptotic Solutions of Ordinary Differential Equations, With an Application to the Bessel Functions of Large Order