Asymptotic Expansions Of Integrals Research Articles

Resolution of singularities, asymptotic expansions of integrals and related phenomena

Resolution of singularities, asymptotic expansions of integrals and related phenomena

Problems and Techniques

The three papers in this issue are concerned with asymptotic expansions of integrals, calculus of variations, and singular perturbation techniques. 1. Among integral transforms, the Fourier transform is arguably the best known, but there are many others, including Laplace, Stieltjes, Mellin, Hankel, and Poisson transforms. They can all be represented as $I(x)\equiv\int_0^{\infty}{f(t)h(xt)dt}$. If x is an asymptotic parameter, which means that x is close to zero or very large, then one may be able to approximate $I(x)$ by an asymptotic expansion. In his paper “Asymptotic Expansions of Mellin Convolution Integrals,” José López presents a general and simple method to generate asymptotic expansions for $I(x)$ that encompasses many existing methods as special cases. 2. Analysis of “slope stability” is an important aspect of geology and soil mechanics: How likely is a sloped terrain (an embankment, a dam) to succumb to erosion and turn into a landslide? And how should the slope profile be modified to prevent further sliding? Analysis of slope stability is one of the applications envisioned by Enrique Castillo, Antonio Conejo, and Ernesto Aranda in their paper “Sensitivity Analysis in Calculus of Variations. Some Applications.” They propose to perform slope stability by means of a sensitivity analysis in the calculus of variations. In the context of slope stability, for instance, one might want to determine how sensitive the slope safety factor is to changes in the slope profile or to changes in soil strength. The mathematical problem comes down to this: How sensitive to changes in parameters are the following quantities: objective function values, primal solutions, and dual solutions? The authors show how to express the sensitivities in terms of partial derivatives, and how to compute them numerically by solving a boundary value problem. 3. In their paper “High-Frequency Oscillations of a Sphere in a Viscous Fluid near a Rigid Plane,” Richard Chadwick and Zhijie Liao model the operation of atomic force microscopes. An atomic force microscope (AFM) is a high resolution scanning device for imaging at nanoscales. Unlike conventional microscopes, which use light, an AFM scans a specimen by “feeling” the surface with a mechanical probe, which in this case consists of a sphere attached to a cantilever. When the specimens are delicate biological samples, such as tissues of the inner ear, they are scanned in a fluid, and the AFM is employed in “tapping mode.” In tapping mode, the sphere is forced to oscillate up and down. When it approaches the specimen, its oscillations are changed by hydrodynamic forces. This provides information about properties of the specimen. To understand the hydrodynamic interactions taking place between sphere and wall, one can model the situation as a fluid that contains a sphere oscillating close to a rigid wall. The problem becomes singular when the sphere approaches the wall. The authors revive singular perturbation techniques developed in the 1960s to derive asymptotic expansions for the forces acting on the sphere.

ACTION AND PERIOD OF HOMOCLINIC AND PERIODIC ORBITS FOR THE UNFOLDING OF A SADDLE-CENTER BIFURCATION

At a saddle-center bifurcation for Hamiltonian systems, the homoclinic orbit is cusp shaped at the nonlinear nonhyperbolic saddle point. Near but before the bifurcation, orbits are periodic corresponding to the unfolding of the homoclinic orbit, while after the bifurcation a double homoclinic orbit is formed with a local and global branch. The saddle-center bifurcation is dynamically unfolded due to a slowly varying potential. Near the unfolding of the homoclinic orbit, the period and action are analyzed. Asymptotic expansions for the action, period and dissipation are obtained in an overlap region near the homoclinic orbit of the saddle-center bifurcation. In addition, the unfoldings of the action and dissipation functions associated with zero energy orbits (periodic and homoclinic) near the saddle-center bifurcation are determined using the method of matched asymptotic expansions for integrals.

Dispersive hyperasymptotics and the anharmonic oscillator

Hyperasymptotic summation of steepest-descent asymptotic expansions of integrals is extended to functions that satisfy a dispersion relation. We apply the method to energy eigenvalues of the anharmonic oscillator, for which there is no known integral representation, but for which there is a dispersion relation. Hyperasymptotic summation exploits the rich analytic structure underlying the asymptotics and is a practical alternative to Borel summation of the Rayleigh–Schrodinger perturbation series.

Développement asymptotique d'intégrales oscillantes doubles…

In this Note, we shall consider asymptotic expansions of integrals I(λ)=∫ Dg· e iλf dx , where D is a bidimensional bounded domain, f and g are two C ∞ functions, and the set γ of stationary points of the phase f is a C ∞ simple curve in D . In the contrary to the previous results, the expansions realized here are more explicit and obtained under weaker conditions and in more general situations than in the case where the curve γ cuts tangentially the boundary of D. Moreover, the influence of the geometry of γ is taken into account.

Semiclassical approximation for bound states of the Schrödinger equation with a Coulomb-like potential

The eigenvalue problem for the radial Schrödinger equation with an ‘‘almost’’ Coulomb potential is considered. This problem provides the simplest example of a system whose classical trajectories have singularities. Thus the standard semiclassical quantization procedure (in the sense of Einstein–Brillouin–Keller–Maslov) cannot be applied straightforwardly to this situation. The equation under consideration has two transition points on the half axis 0<x<∞: a regular singular point at the origin and a turning point for some x0≳0; these points coalesce at low energy levels. The well-known comparison equation method provides uniform asymptotics for equations of this kind, but has the following shortcomings: it does not appeal to the corresponding classical mechanical problem and the formulas for the eigenvalues and eigenfunctions contain phase integrals which are not analytic with respect to the parameter characterizing the closeness of the transition points. In the present article we derive new formulas for the eigenfunctions in the form of inverse Fourier transforms of rapidly oscillating exponentials multiplied by some powers of x, together with simple formulas for the eigenvalues. These formulas are directly connected with the classical trajectories and do not have the shortcomings of the comparison equation method, i.e., all the functions entering the asymptotic expansions are analytic in x and energy. As a by-product, an asymptotic expansion of integrals having two stationary phase points which coalesce for some value of the parameter and tend to infinity when the parameter tends to zero is obtained.

A filter with a guaranteed asymptotic performance

In the case of low noise levels the optimal probability density function summarizing the available information about the state of a system can be accurately approximated by the product of a gaussian function and a linear function. The approximation preserves the ability to estimate to an accuracy of O ( λ -2 ) the expected value of any twice continuously differentiable function defined on the state space. The parameter λ depends on the noise level. If the noise level in the system is low then λ is large. A new filtering method based on this approximation is described. The approximating function is updated recursively as the system evolves with time, and as new measurements of the system state are obtained. The updates preserve the ability to estimate the expected values of functions to an accuracy of O ( λ -2 ). The new filter does not store previous measurements or previous approximations to the optimal probability density function. The new filter is called the asymptotic filter, because the definition of the filter and the analysis of its properties are based on the theory of asymptotic expansion of integrals of Laplace type. An analysis of the state propagation equations shows that the asymptotic filter performs better than a particular widely used suboptimal approximation to the optimal filter, the extended Kalman filter. The extended Kalman filter does not, in general, preserve the ability to estimate expected values to an accuracy of O ( λ -2 ). The computational cost of the asymptotic filter is comparable to that of the iterated extended Kalman filter.

Matched asymptotic expansions of integrals

Uniformly valid asymptotic expansions for integrals with coalescing critical points are obtained by finding inner or boundary layer expansions that match with standard Laplace method (outer) expansions. Simple algorithms for the terms of these expansions are established and programmed in MACSYMA. One of the applications is a new Bessel function expansion

SOME THEOREMS IN ASYMPTOTIC ANALYSIS

Basing ourselves on the analysis of magnitude of order, we strictly prove fundamental lemmas for asymptotic integral, including the cases of infinite region. Then a general formula for asymptotic expansion of integrals is given. Finally, we derive a sufficient condition for an ordinary differential equation to possess a solution of the Frobenius series type at finite irregular singularities or branching points.

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.

Asymptotic Expansions of Integrals with Oscillatory Kernels and Logarithmic Singularities

This paper is a follow-up to an earlier paper by Bleistein which derived asymptotic expansions of integral tranforms of functions with logarithmic singularities. That result dealt with exponentially decaying kernels. In this paper the results are expanded to include the case of general oscillatory kernels—e.g., Fourier or Hankel transforms. The results also include kernels such as Airy functions of negative argument and oscillatory Weber functions.

Reviews and abstracts - Asymptotic Expansions of Integrals

Reviews and abstracts - Asymptotic Expansions of Integrals

Numerical saddle point method

An asymptotic expansion of integrals containing an exponential integrand with a large parameter can be obtained by the well-known saddle point method. For integrands further containing Hankel functions a numerical double saddle point method is described herein, allowing the numerical computation of these integrals without using an asymptotic expansion.

A New Approach to Edge Diffraction

This paper presents a new method for computing the field produced when a wave is incident upon a plane screen. The field is given approximately by an integral over the screen, and our results are obtained by evaluating the integral asymptotically by means of stationary phase. This method is shown to be closely related to the geometrical theory of diffraction and results obtained here agree with the results of the geometrical theory in their common region of validity. Furthermore, by applying some recent results in the theory of uniform asymptotic expansions of integrals, the field can be computed at caustics, shadow boundaries and edges where the geometrical theory fails. Again results here agree with results obtained by other methods. Finally this work demonstrates that the results of geometrical diffraction theory actually represent an asymptotic approximation to an actual solution of the wave equation, although it has not been shown that this solution satisfies the boundary condition.

The remainder in Watson's lemma

Estimates are found for the remainder at the term of smallest modulus in the asymptotic expansions of integrals of a general type. When there is approximately a constant shift of phase between consecutive terms the remainder is easily estimated; in particular, in the case where the terms alternate in sign. The case where all term s have the same sign corresponds to a singularity on the positive real axis, and the same asymptotic expansion corresponds to different functions according as the integral is a principal value (in a somewhat extended sense) or on a loop. The two definitions may differ by a considerable multiple of the smallest term.