Asymptotic power series

1994 ◽  
pp. 1-12
Author(s):  
Werner Balser
Keyword(s):  
Power Series ◽  
Asymptotic Power

The Wavefield Radiated Into an Elastic Half-Space by a Transducer of Large Aperture

1988 ◽  
Vol 55 (2) ◽  
pp. 398-404 ◽  
Author(s):  
John G. Harris
Keyword(s):  
Power Series ◽  
Plane Wave ◽  
Half Space ◽  
Wave Equations ◽  
Ultrasonic Transducer ◽  
Elastic Half Space ◽  
The Other ◽  
Circular Region ◽  
Asymptotic Power ◽  
Normal Traction

The wavefield radiated into an elastic half-space by an ultrasonic transducer, as well as the radiation admittance of the transducer coupled to the half-space, are studied. Two models for the transducer are used. In one an axisymmetric, Gaussian distribution of normal traction is imposed upon the surface, while in the other a uniform distribution of normal traction is imposed upon a circular region of the surface, leaving the remainder free of traction. To calculate the wavefield, each wave emitted by the transducer is expressed as a plane wave multiplied by an asymptotic power series in inverse powers of the aperture’s (scaled) radius. This reduces the wave equations satisfied by the compressional and shear potentials to their parabolic approximations. The approximations to the radiated waves are accurate at a depth where the wavefield remains well collimated.

Asymptotics of the Titchmarsh-Weyl m-coefficient for integrable potentials

1986 ◽  
Vol 103 (3-4) ◽  
pp. 347-358 ◽  
Author(s):  
Hans G. Kaper ◽  
Man Kam Kwong
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Asymptotic Behaviour ◽  
Potential Function ◽  
Error Bound ◽  
Power Series Expansion ◽  
Smoothness Condition ◽  
Asymptotic Power ◽  
Simple Method ◽  
Expansion Coefficients

This article is concerned with the asymptotic behaviour of m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation y“ + (λ − q(x))y = 0 on [0, ∞), as λ →∞ in a sector in the upper half of the complex plane. It is assumed that the potential function q is integrable near 0. A simplified proof is given of a result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(λ), and a sharper error bound is obtained. Theproof is then generalised to derive subsequent terms in the asymptotic expansion. It is shown that the Titchmarsh-Weyl m-coefficient admits an asymptotic power series expansion if the potential function satisfies some smoothness condition. A simple method to compute the expansion coefficients is presented. The results for the first few coefficients agree with those given by Harris [9].

scholarly journals Some Algebraic Properties Of Asymptotic Power Series

1956 ◽  
Vol 8 ◽  
pp. 220-224
Author(s):  
T. E. Hull
Keyword(s):  
Asymptotic Expansion ◽  
Power Series ◽  
Asymptotic Power ◽  
Algebraic Properties ◽  
Image Position ◽  
The Right ◽  
The Given ◽  
Class Of Functions

1. Introduction. Let us consider all power series of the formIt was shown first by Borel (1) that to each such series there corresponds a non-empty class of functions such that each function in the class has the given series as its asymptotic expansion about z = 0, the expansion being valid in a sector of the right half z-plane with vertex at the origin.

scholarly journals Asymptotic Power Series of Field Correlators

10.14311/1199 ◽  
2010 ◽  
Vol 50 (3) ◽  
Author(s):  
I. Caprini ◽  
J. Fischer ◽  
I. Vrkoč
Keyword(s):  
Power Series ◽  
Large Class ◽  
Complex Plane ◽  
Perturbation Expansion ◽  
Asymptotic Power ◽  
General Contour ◽  
Asymptotic Perturbation ◽  
Class Of Functions ◽  
Borel Type

We address the problem of ambiguity of a function determined by an asymptotic perturbation expansion. Using a modified form of theWatson lemma recently proved elsewhere, we discuss a large class of functions determined by the same asymptotic power expansion and represented by various forms of integrals of the Laplace-Borel type along a general contour in the Borel complex plane. Some remarks on possible applications in QCD are made.

Meromorphic functions with prescribed asymptotic behaviour, zeros and poles and applications in complex approximation

1999 ◽  
Vol 51 (1) ◽  
pp. 117-129
Author(s):  
A. Sauer
Keyword(s):  
Power Series ◽  
Asymptotic Behaviour ◽  
Series Expansion ◽  
Meromorphic Functions ◽  
Power Series Expansion ◽  
Complex Approximation ◽  
Asymptotic Power ◽  
Approximation Theorems ◽  
Zeros And Poles

AbstractWe construct meromorphic functions with asymptotic power series expansion in z−1 at ∞ on an Arakelyan set A having prescribed zeros and poles outside A. We use our results to prove approximation theorems where the approximating function fulfills interpolation restrictions outside the set of approximation.

Summability of asymptotic power series

1993 ◽  
Vol 7 (4) ◽  
pp. 239-250 ◽  
Author(s):  
Wolfgang B. Jurkat
Keyword(s):  
Power Series ◽  
Asymptotic Power

scholarly journals IONIZATION IN A 1-DIMENSIONAL DIPOLE MODEL

2008 ◽  
Vol 20 (07) ◽  
pp. 835-872 ◽  
Author(s):  
O. COSTIN ◽  
J. L. LEBOWITZ ◽  
C. STUCCHIO
Keyword(s):  
Power Series ◽  
Trigonometric Polynomial ◽  
Binding Potential ◽  
Dimensional Model ◽  
Asymptotic Power ◽  
One Dimensional ◽  
Dipole Radiation ◽  
Complete Ionization ◽  
Fixed Region ◽  
Model Atom

We study the evolution of a one-dimensional model atom with δ-function binding potential, subjected to a dipole radiation field E(t)x with E(t) a 2π/ω-periodic real-valued function. We prove that when E(t) is a trigonometric polynomial, complete ionization occurs, i.e. the probability of finding the electron in any fixed region goes to zero as t → ∞. For ψ(x, t = 0) compactly supported and general periodic fields, we decompose ψ(x, t) into uniquely defined resonance terms and a remainder. Each resonance is 2π/ω periodic in time and behaves like the exponentially growing Green's function near x = ±∞. The remainder is given by an asymptotic power series in t-1/2 with coefficients varying with x.

Sign in / Sign up

Export Citation Format

Share Document