Uniform Asymptotic Expansion Research Articles

Correction to "Diffraction by a Half-Plane with Two Face Impedance Uniform Asymptotic Expansion for Plane Wave and Arbitrary Line Source Incidence"

Correction to "Diffraction by a Half-Plane with Two Face Impedance Uniform Asymptotic Expansion for Plane Wave and Arbitrary Line Source Incidence"

Compensation of Diffraction Effects in SAW Filters Using the Uniform Asymptotic Expansion

Abstruct-The uniform asymptotic expansion (UAE) previously developed for the calculation of the diffracted SAW beam profile is extended to the calculation of diffraction effects in the frequency response of a SAW filter. The method provides a fast and accurate method of calculating the effects of diffraction in the frequency response of SAW filters even in the very near field region (z < 2.5 A,,). Accuracy and speed comparisons between the UAE and a direct numerical integration algorithm are given. For transducer separations greater than 25 X. the relative difference between the two methods is less than 0.1 dB in the frequency range of interest. In addition, the computation speed of the UAE is greater than that of the numerical integration algorithm by at least a factor of 60 and continues to increase with increasing transducer separation. This technique is used in conjunction with well-known wide-band compensation procedures to design diffraction compensated SAW filters. Experimental results of diffraction compensated SAW filters constructed on 128 rotated Y-cut X-propagating LiNbO, are given.

Geometrical Theory of Diffraction(GTD) Part II: Extensions and future trends of the theory

An analysis of the extensions and improvements of GTD since its early time is given. Especially the theoretical foundations of two powerful methods, namely the Boundary Layer Method and the Uniform Asymptotic Expansion Method, are outlined. Aspects of these theories which are of growing interest at present, for solving electromagentic diffraction and radiation problems are pointed out. The resulting solutions are described in a general framework including perfectly conducting and imperfectly conducting objects condition. Their limitations are discussed and indications for further improvements are given. Some new non published results on higher-order edges and coated metallic bodies are also included. The author concludes with a brief description of future trends in the development of GTD.

Computation of the incomplete gamma function ratios and their inverse

An algorithm is given for computing the incomplete gamma function ratios P ( a , x ) and Q&gt; ( a , x ) for a ⪈ 0, x ⪈ 0, a + x ≠ 0. Temme's uniform asymptotic expansions are used. The algorithm is robust; results accurate to 14 significant digits can be obtained. An' extensive set of coefficients for the Temme expansions is included. An algorithm, employing third-order Schröder iteration supported by Newton-Raphson iteration, is provided for computing x when a , P ( a , x ), and Q ( a , x ) are given. Three iterations at most are required to obtain 10 significant digit accuracy for x .

Uniform Asymptotic Expansions for Prolate Spheroidal Functions with Large Parameters

By application of the theory for second order linear differential equations with a turning point and a regular (double pole) singularity developed by Boyd and Dunster (this Journal, 17 (1986), pp. 422–450) uniform asymptotic expansions are obtained for prolate spheroidal functions for large $\gamma$. The results are uniformly valid for ${{0 \leqslant \mu ^2 } / {\gamma ^2 \leqslant 1 + A}}$ and for ${{A' \leqslant \lambda } / {\gamma ^2 \leqslant A''}}$, where A, $A'$ and $A''$ are arbitrary real constants such that $0 \leqq A < A' \leqq A'' < \infty $. An asymptotic relationship between $\lambda $, $\mu $, $\gamma $ and the characteristic exponent $\nu $ is then derived from the approximations for the spheroidal functions. All the error terms associated with the approximations have explicit bounds given.

Transitional Approximations and the Autler-Townes Effect for Pulsed Lasers

The absorption of a weak probe beam by an atomic system strongly driven by a pulsed laser exhibits several maxima and zeros as the probe is tuned. This is the Autler-Townes spectrum for a pulsed pump, and has been discussed in the context of a uniform asymptotic expansion previously. This work concentrates on the behaviour of the extreme maximum as a function of the flatness of the pump beam profile. Transitional asymptotic expansions are used to derive properties of the spectrum, by comparison with the canonical integrals of catastrophe theory.

Diffraction by a half-plane with two face impedances uniform asymptotic expansion for plane wave and arbitrary line source incidence

A uniform asymptotic expansion (UAE) of Maliuzhinets' exact solution for incident plane wave diffraction by a half-plane with two face impedances has been obtained using Van der Waerden's method. This solution has been further extended to the case of arbitrary line source incidence using a heuristic approach.

Uniform Asymptotic Solutions of a Class of Second-Order Linear Differential Equations Having a Turning Point and a Regular Singularity, with an Application to Legendre Functions

The asymptotic behaviour, as a parameter $u \to \infty $, of solutions of second-order linear differential equations with a turning point and a regular (double pole) singularity is considered. It is shown that the solutions can be approximated by expressions involving Bessel functions in a region which includes both the turning point and the singularity. Explicit error bounds for the difference between the approximations and the exact solutions are established. The theory is applied to find uniform asymptotic expansions for Legendre functions.

A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points

For the uniform asymptotic expansion of certain types of contour integrals, one of whose critical points is an endpoint of the interval of integration, a method alternative to Bleistein's is introduced and numerically tested by way of a non-trivial physical example.

A uniform asymptotic expansion for the Jacobian elliptic functions n(u, k)

The usual two term approximation for the Jacobian elliptic functions n (u, k) when the modulusk is close to unity (see for example [3], [4], [5]) is shown to be nonuniform when |u|∼K, whereK is the quarter period. The nonuniformity is shown to be weak, with the remainder term becoming of the same order as the last retained term in the approximation. A formal application of the method of matched asymptotic expansions is then used to obtain a uniformly valid two term expansion fors n(u, k). Uniform expansions forc n(u, k) andd n(u, k) can also be developed in a similar manner.

A Uniform Asymptotic Expansion of the Jacobi Polynomials with Error Bounds

In a recent investigation of the asymptotic behavior of the Lebesgue constants for Jacobi polynomials, we encountered the problem of obtaining an asymptotic expansion for the Jacobi polynomials , as n → ∞, which is uniformly valid for θ in . The leading term of such an expansion is provided by the following well-known formula of “Hilb's type” [13, p. 197]:(1.1)where α &gt; – 1, β real and ; c and are fixed positive numbers. Note that the remainder in (1.1) is always θ1/2O(n–3/2).

Scattering of low-frequency acoustic waves by baffled membranes and plates

The method of matched asymptotic expansions is used to study the scattering of plane, monochromatic, acoustic waves from baffled flexible surfaces in the limit as L/λ→0. Here, λ is the wavelength of the incident acoustic wave and L is a characteristic size of the flexible surface, such as its maximum diameter. The baffled surface is either a membrane or a thin plate. Uniform asymptotic expansions of the scattered fields are obtained for both surfaces as L/λ→0. Thus, for example, it is valid in the nearfields and farfields of the flexible surface. The method is applied to obtain the fields scattered by rectangular membranes and plates. It is found that plates are more efficient scatterers than ‘‘similar’’ membranes if the plates are sufficiently thin.

Scattering by a parabolic cylinder—A uniform asymptotic expansion

A uniform asymptotic expansion in the variable, determining the location of the observer for the fields scattered by a perfectly conducting parabolic cylinder, is derived. This expansion can be applied to scattering by arbitrary smooth convex surfaces of variable curvature. The accuracy of numerical results is examined.

Schrödinger semigroups for vector fields

Suppose H is the Hamiltonian that generates time evolution in an N-body, spin-dependent, nonrelativistic quantum system. If r is the total number of independent spin components and the particles move in three dimensions, then the Hamiltonian H is an r×r matrix operator given by the sum of the negative Laplacian −Δx on the (d=3N)-dimensional Euclidean space Rd plus a Hermitian local matrix potential W(x). Uniform higher-order asymptotic expansions are derived for the time-evolution kernel, the heat kernel, and the resolvent kernel. These expansions are, respectively, for short times, high temperatures, and high energies. Explicit formulas for the matrix-valued coefficient functions entering the asymptotic expansions are determined. All the asymptotic expansions are accompanied by bounds for their respective error terms. These results are obtained for the class of potentials defined as the Fourier image of bounded complex-valued matrix measures. This class is suitable for the N-body problem since interactions of this type do not necessarily decrease as ‖x‖→∞. Furthermore, this Fourier image class also contains periodic, almost periodic, and continuous random potentials. The method employed is based upon a constructive series representation of the kernels that define the analytic semigroup {e−zH‖Re z&amp;gt;0}. The asymptotic expansions obtained are valid for all finite coordinate space dimensions d and all finite vector space dimensions r, and are uniform in Rd×Rd. The order of expansion is solely a function of the smoothness properties of the local potential W(x).

Laplace type integrals: transformation to standard form and uniform asymptotic expansions

Integrals are considered which can be transformed into the Laplace integral \[ F λ ( z ) = 1 Γ ( λ ) ∫ 0 ∞ t λ − 1 e − z t f ( t ) d t {F_\lambda } \left ( z \right ) = \frac {1}{{\Gamma \left ( \lambda \right )}}\int _0^\infty {{t^{\lambda - 1}}{e^{ - zt}}f\left ( t \right )dt} \] , where f f is holomorphic, z z is a large parameter, μ = λ / z \mu = \lambda /z is a uniformity parameter, μ ≥ 0 \mu \ge 0 . A uniform asymptotic expansion is given with error bounds for the remainders. Applications are given for special functions, with a detailed analysis for a ratio of gamma functions. Further applications are mentioned for Bessel functions and parabolic cylinder functions. Analogue results are given for loop integrals in the complex plane.

Incomplete Laplace Integrals: Uniform Asymptotic Expansion with Application to the Incomplete Beta Function

The incomplete Laplace integral \[ \frac{1}{{\Gamma (\lambda )}}\int_\alpha ^\infty {t^{\lambda - 1} e^{ - zt} f(t)dt} \] is considered for large values of z. Both $\lambda $ and $\alpha $ are uniformity parameters in $[0,\infty )$. The basic approximant is an incomplete gamma function, that is, the above integral with $f = 1$. Also, a loop integral in the complex plane is considered with the same asymptotic features. The asymptotic expansions are furnished with error bounds for the remainders in the expansions. The results of the paper combine four kinds of asymptotic problems considered earlier. An application is given for the incomplete beta function. The present investigations are a continuation of earlier works of the author for the above integral with $\alpha = 0$. The new results are significantly based on the previous case.

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

An asymptotic expansion including error bounds is given for polynomials {P n, Qn} that are biorthogonal on the unit circle with respect to the weight function (1−eiθ)α+β(1−e−iθ)α−β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function2 F 1(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.

A uniform asymptotic expansion of a typical diffraction integral with many coalescing simple pole singularities and a first-order saddle point

The asymptotic evaluation of a typical diffraction integral with many coalescing simple pole singularities and a first-order saddle point is examined. A uniform solution is obtained by means of the Taylor expansion.

A series expansion for the first positive zero of the Bessel functions

It is shown that the first positive zero j v , l {j_{v,l}} of the Bessel function J v ( x ) {J_v}(x) is given by \[ j v , l = 2 ( v + 1 ) 1 / 2 [ 1 + ( v + 1 ) 4 − 7 ( v + 1 ) 2 96 + 49 ( v + 1 ) 3 1152 − 8363 ( v + 1 ) 4 276480 + ⋯ ] {j_{v,l}} = 2{(v + 1)^{1/2}}\left [ {1 + \frac {{(v + 1)}}{4} - \frac {{7{{(v + 1)}^2}}}{{96}} + \frac {{49{{(v + 1)}^3}}}{{1152}} - \frac {{8363{{(v + 1)}^4}}}{{276480}} + \cdots } \right ] \] for − 1 &gt; v &gt; 0 - 1 &gt; v &gt; 0 .

Plane-wave diffraction by a wedge--A spectral domain approach

The canonical problem of plane wave diffraction by a wedge in the context of the spectral domain approach which exploits the relationship between the induced current on a scatterer and its far field is investigated. It is shown how the exact solution to the wedge diffraction problem can be manipulated in a form which enables one to interpret the far scattered field as the Fourier transform of the physical optics (PO) current on the two faces of the wedge augmented by the fringe current near the tip of the wedge. A uniform asymptotic expansion for the total field which slightly modifies the Ansatz in the uniform asymptotic theory of electromagnetic edge diffraction is constructed.