Uniform Asymptotic Expansion Research Articles

VACUUM ENERGY IN SPHERICAL DOMAINS: WKB AND UNIFORM ASYMPTOTIC EXPANSIONS

We evaluate the finite part of the regularized zero-point energy for a massless scalar field confined in the interior of a D-dimensional spherical region. While some insight is offered into the dimensional dependence of the WKB approximations by examining the residues of the spectral-zeta-function poles, a mode-sum technique based on an integral representation of the Bessel spectral zeta function is applied with the help of uniform asymptotic expansions (u.a.e.’s).

Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function

Uniform asymptotic expansions are derived for solutions of the differential equation d 2 W/dζ 2 = ( u 2 ζ 2 + βu + ψ( u , ζ))W, which are uniformly valid for u real and large, β bounded (real or complex), and £ lying in a well-defined bounded or unbounded complex domain, which contains the origin. The function ψ(u, ζ) is assumed to be holomorphic in this domain, and is o ( u / ln( u )) uniformly as u —> oo. The approximations involve parabolic cylinder functions, and include explicit and realistic error bounds. The new theory is then applied to the complementary incomplete gamma function Γ( α , α x ), furnishing an asymptotic approximation which is uniformly valid for x lying in a complex domain which properly contains all x satisfying |arg(x)| oo.

A uniform asymptotic expansion for the jacobi polynomials with explicit remainder

A new asymptotic expansion is derived for the Jacobi polynonmial which holds uniformly for.An explicit expression is also given for the error term associated with the expansion. Our approach begins with a contour integral representation, followed by a suitable canonical transformation.

Vacuum energy in a spherically symmetric background field.

The vacuum energy of a scalar field in a spherically symmetric background field is considered. It is expressed through the Jost function of the corresponding scattering problem. The renormalization is discussed in detail and performed using the uniform asymptotic expansion of the Jost function. The method is demonstrated in a simple explicit example. {copyright} {ital 1996 The American Physical Society.}

Uniform asymptotic expansions of integrals: a selection of problems

On the occasion of the conference we mention examples of Stieltjes' work on asymptotics of special functions. The remaining part of the paper gives a selection of asymptotic methods for integrals, in particular on uniform approximations. We discuss several “standard” problems and examples, in which known special functions (error functions, Airy functions, Bessel functions, etc.) are needed to construct uniform approximations. Finally, we discuss the recent interest and new insights in the Stokes phenomenon. An extensive bibliography on uniform asymptotic methods for integrals is given, together with references to recent papers on the Stokes phenomenon for integrals and related topics.

A general formulation for asymptotic expansions with a pole near the saddle point

We describe the two uniform asymptotic expansions which already exist and which are commonly used in asymptotic electromagnetic theories, when a pole approaches the saddle point. Then, we establish a simple relation between these two expansions. By analyzing this relation, we give the general form of any asymptotic expansion with a pole near the saddle point. By using diffraction integrals, we give explicit examples and compare the corresponding numerical computations. Then we conclude by examining the specific problem of many coalescing poles singularities.

Dependence of the rate of convergence of the Rayleigh-Ritz method on a nonlinear parameter.

Numerical computation of optimum values for nonlinear parameters in a Rayleigh-Ritz variational trial function is considerably more difficult than numerical computation of optimum values for linear parameters. Thus, an analytic understanding of the mechanisms that determine these optimum values can be quite useful. Uniform asymptotic expansions can be used to explore these mechanisms for the nonlinear parameter that sets the length scale for a basis set. These uniform asymptotic expansions usually involve two or more different kinds of terms whose relative importance changes as the nonlinear parameter changes, with two different terms being equally important at the point where the nonlinear parameter has its optimum value. Interference effects between these different terms are typical, and tend to become most pronounced near the optimum value. These different kinds of terms arise from singularities of the wave function, from the neighborhood of the classical turning point for the basis functions, and/or from saddle points. Comparisons of theory with (numerical) experiment will be given for Rayleigh-Ritz calculations on three model problems that illustrate the kinds of terms listed above.

Semiclassical techniques for long‐range, low‐frequency propagation

The Chester, Friedman, and Ursell (CFU) uniform asymptotic expansion near caustics has been implemented as an adjunct to a ray‐tracing code. The method is applied to the propagation of 400 Hz sound from a 200‐m deep source in a Munk sound‐speed profile through a range of 300 km. Making the parabolic approximation in the ray trace, the validity of ‘‘semiclassical’’ wave functions with the CFU expansion near caustics can be determined by comparison with a parabolic equation code. Good results for the semiclassical wave function are found. If the CFU expansion is not used, the field is badly inaccurate over hundreds of meters in the vertical. A video of the propagation comparing the Wigner function with the ray depth‐angle plot is shown. [Work supported by ONR.]

Uniform q-series asymptotics for staircase polygons

We present a uniform asymptotic expansion for the area-perimeter generating function of staircase polygons by calculating the asymptotic behaviour of the alternating q-series 1 phi 1(0;y;q,x) as q to 1- from a new integral representation. This leads to a direct calculation of the scaling function for this model.

Uniform asymptotic expansions for oblate spheroidal functions II: negative separation parameter λ

In [3], uniform asymptotic expansions were derived for solutions of the oblate spheroidal wave equation (z2 − 1)d2p/dz2 + 2zdp/dz − (λ + μ2/(z2 − 1)) p = 0, for the case where the parameter μ is real and non-negative, the separation parameter λ is real and positive, and γ is purely imaginary (γ = iu). As u → ∞, uniform asymptotic expansions were derived involving elementary, Airy and Bessel functions, these being valid in certain subdomains of the complex z plane. In this paper the complementary case, where λ is real and negative, is considered. Asymptotic expansions are derived which are valid in certain subdomains of the half-plane |arg (z)| ≦ π/2, uniformly valid for u → ∞ with λ /u2 fixed and negative, and 0 ≦ μ/u ≦ − ½λ /u2 − δ, where δ is an arbitrary positive constant. Explicit error bounds are available for all the approximations.

Singular Perturbations and the Coupled/Quasi-Static Approximation in Linear Thermoelasticity

A uniform asymptotic expansion in the inertial constant e is given for one-dimensional linear thermoelasticity by a two-timing method. The result shows that the usual coupled/quasi-static approximation $(\epsilon = 0)$ is not uniformly asymptotically correct even to lowest order. The interaction between diffusion and wave propagation is clearly displayed. The proof of uniform asymptotic validity to order $\epsilon ^2 $ is accomplished by energy estimates.

Nonlinear interaction of surface waves in a basin covered with broken ice

The nonlinear interaction of periodic traveling waves of the first and second harmonics in a constant-depth uniform fluid covered with broken ice is considered. Uniform asymptotic expansions up to third-order values for the velocity potential of the fluid and the elevation of the basin surface are found by means of the multivariable expansion procedure. The dependence of the wave perturbations on the thickness of the ice and the interacting-harmonic characteristics is analyzed.

Acoustic-wave nonlinearity in stimulated Brillouin scattering

Stimulated Brillouin scattering is investigated here under conditions characterized by high optical pump intensity. Calculations are carried out at 1.3 and 3.8 μm with pump intensities equal to approximately three times the threshold. Such strong optical forcing leads to significant self-distortion of the density profile in the material wave serving as the optical grating. The method of multiple scales is used to find a uniform asymptotic expansion in the coupling. At the first perturbation level the resonant portion of the problem yields abridged equations for the incident and the scattered waves with frequencies ωL and ωS = ωL − Ω and for the grating and its harmonics with frequencies nΩ. The nonresonant portion of the first perturbation yields field components with the frequencies ωL + nΩ and ωS − nΩ for n = 1, 2, …. Numerical solutions of the truncated abridged system are found by a method of Chebychev collocation and indicate a reduction in the observed phonon lifetime and a broadening of the linewidth with increasing amplitude.

Spectral degeneracy in the one-dimensional Anderson problem: A uniform expansion in the energy band.

A uniform quantitative description of the properties of the one-dimensional Anderson model is obtained by mapping that problem onto an infinitely quasidegenerate master equation. This quasidegeneracy is identified as the source of the small-denominator problem encountered before in investigations of this problem. An appropriate quasidegenerate perturbation theory is developed to obtain a uniform asymptotic expansion, in powers of the strength of the noise, for the probability distribution function of the ratio of the value of the wave function at neighboring sites. Well known results, such as those obtained by Thouless, Kappus and Wegner, and Derrida and co-workers are reproduced and systematic corrections to these results as well as some more results are found. In particular, we find internal layers in the above-mentioned distribution function for values of the energy given by E=2 cos\ensuremath{\pi}\ensuremath{\alpha} with \ensuremath{\alpha} rational. We also find crossovers in the behavior of the distribution function (and consequently in quantities derived from it) near the band-edge and band-center regions. The properties of the model in the band-edge region were studied by us in detail in a previous publication [Phys. Rev. B 47, 1918 (1992)].

On sound diffraction by an impedance wedge

Diffraction of waves by a wedge is a problem of modern acoustics whose mathematical solutions provide a theoretical basis in analyzing sound radiation and scattering phenomena. Considerable attention has previously been attracted to this class of diffraction problems, especially to the particular cases of a perfectly reflecting wedge and half-plane. The model of an impedance wedge [G. D. Malyuzhinets, 1958 Dokl. Akad. Nauk SSSR 121, 436 (1958) in Russian] is a more general and realistic one that gives new opportunities for treating acoustic diffraction by irregular surfaces [A. D. Pierce and W. J. Hadden, J. Acoust. Soc. Am. 63, 17 (1978); A. V. Osipov, Sov. Phys. Acoust. 36, 287 (1990)]. This paper investigates the wave field excited by a line source in the presence of a wedge with arbitrary values of face impedances. Based on the Malyuzhinets’ solution, various exact representations for the Green’s function are deduced to generate a system of convenient formulas for any relative positions of the source and point of observation. A new representation of the Malyuzhinets’ solution in the form of double series [A. V. Osipov, Sov. Phys. Acoust. 37, 381 (1991)] is exploited to describe the wave field of the line source placed in the vicinity of edge. In the opposite case of large distances from the edge, a uniform asymptotic expansion is presented. Applications of the obtained results to problems of sound attenuation by wedge-like barriers and sound radiation from a line source located at the edge of a corrugated surface are described.

A uniform asymptotic expansion for the shear-wave front in a layer

In studying transient waves in a layer, a Fourier-type integral arises. For a material point at a position in a neighborhood behind the shear-wave front, the phase function of this integral has a stationary point which approaches positive infinity. As a result, the classical method of stationary phase does not apply. A heuristic treatment has been suggested by Jones ( Q. J. Mech. Appl. Math. 17, 401–421 (1964)), but it seems to give incorrect results. In this paper, an asymptotic expansion is derived for this integral, which is uniformly valid in a neighborhood behind the shear-wave front. Our result shows that there is a jump in the asymptotic order of the vertical acceleration behind and ahead of the shear-wave front. It is also found that there is a transition from an order O(1) disturbance to an order O(t − 1 4 ) disturbance as the distance to the shear-wave front increases.

The instability time distribution behavior of slotted ALOHA

AbstractThis article deals with a quantification of the instability behavior of the slotted multiuser communication network system ALOHA. In (2) U. Schmid and the author could quantify the expected value E(Y) of the instability time Y = max (n: Xn = 0), where Xn denotes the backlog after n slots. Here it is shown that Y is approximately exponentially distributed. Moreover, uniform asymptotic expansions for ym = P[Y = m] and for the moments Mm = E(Ym) are given. © 1994 John Wiley & Sons, Inc.

Uniform asymptotic expansion of Charlier polynomials

Uniform asymptotic expansion of Charlier polynomials

Unfolding the high orders of asymptotic expansions with coalescing saddles: singularity theory, crossover and duality

We study the leading behaviour of the late coefficients (high ordersr) of asymptotic expansions in a large parameterk, for contour integrals involving a cluster of coalescing saddles, and thereby establish the form of the divergence of the expansions. The two principal cases are: 'saddle-to-cluster’, where the integral is through a simple saddle and its expansion diverges because of a distant cluster; and 'cluster-to-saddle', where the integral is through a cluster and its expansion diverges because of a distant simple saddle. In both, the large-rcoefficients are dominated by the 'factorial divided by power' familiar in asymptotics, but this changes its form as the saddles in the cluster are made to coalesce and separate by varying parametersA= {A1,A2....} in the integrand. The 'crossover' between different forms is described by a series of canonical integrals, built from the cuspoid catastrophe polynomials of singularity theory that describe the geometry of the coalescence. The arguments of these integrals involve not only theAbut also fractional powers ofr, which by a curious duality replace the powers of the original large parameterkwhich occur in uniform approximations involving these integrals. A by-product of the cluster-to-saddle analysis is a new exact formula for the coefficients of uniform asymptotic expansions.

Uniform formulas for infinite antenna current

A uniform asymptotic expansion of the time‐harmonic antenna current with large parameter In [2kz/(ka)2] is given. As usual, the expansion is based on Hallén's filament integral and is restricted by the condition z ≫ a but is otherwise uniform in kz. The use of averaging leads to the vanishing of the second term of the asymptotic series. Two different averaging procedures are used to derive two accurate formulas: The first one, which is very simple, is useful for practical applications. The second formula is of theoretical interest. An improved asymptotic series for large kz is derived to relate the averaging results to existing asymptotic results. Comparison of these two formulas and the exact antenna integral demonstrates that these formulas are more accurate than any other existing formulas for infinite cylindrical antenna current. The Hallén filament integral is found to approximate closely the exact integral even when the condition z ≫ a is violated. Discrepancies in the literature are discussed.