scholarly journals On the Asymptotic Behavior of the Coefficients of Asymptotic Power Series and Its Relevance to Stokes Phenomena

1991 ◽  
Vol 22 (2) ◽  
pp. 524-542 ◽  
Author(s):  
G. K. Immink
Keyword(s):  
Asymptotic Behavior ◽  
Power Series ◽  
Asymptotic Power ◽  
Stokes Phenomena

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 Stokes phenomena in discrete Painlevé II

2017 ◽  
Vol 473 (2198) ◽  
pp. 20160539 ◽  
Author(s):  
N. Joshi ◽  
C. J. Lustri ◽  
S. Luu
Keyword(s):  
Power Series ◽  
Asymptotic Behaviour ◽  
Complex Plane ◽  
Painlevé Equation ◽  
Asymptotic Solutions ◽  
Series Expression ◽  
Painleve Equation ◽  
Rapid Switching ◽  
Power Series Methods ◽  
Stokes Phenomena

We consider the asymptotic behaviour of the second discrete Painlevé equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region of the complex plane. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. We subsequently apply exponential asymptotic techniques to investigate such phenomena, and obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the behaviour of these asymptotic solutions shares a number of features with the tronquée and tri-tronquée solutions of the second continuous Painlevé equation.

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.

scholarly journals Asymptotic behavior of successive coefficients of some power series

1962 ◽  
Vol 6 (4) ◽  
pp. 620-629 ◽  
Author(s):  
A. M. Garsia ◽  
S. Orey ◽  
E. Rodemich
Keyword(s):  
Asymptotic Behavior ◽  
Power Series

Asymptotic behavior of eigenenergies of nonpolynomial oscillator potentials V(x) = x2N + (λx m1)/(1 + gx m2)

2012 ◽  
Vol 90 (6) ◽  
pp. 585-592 ◽  
Author(s):  
Asiri Nanayakkara
Keyword(s):  
Asymptotic Behavior ◽  
Power Series ◽  
Recurrence Relations ◽  
Energy Spectra ◽  
Action Variable ◽  
Analytic Expressions ◽  
Quantum Action ◽  
Asymptotic Energy ◽  
Polynomial Potentials ◽  
Positive Integers

Analytic semiclassical energy expansions of nonpolynomial oscillator (NPO) potentials V(x) = x2N + (λx[Formula: see text])/(1 + gx[Formula: see text]) are obtained for arbitrary positive integers N, m1, and m2, and the real parameters λ and g using the asymptotic energy expansion (AEE) method. Because the AEE method has been previously developed only for polynomial potentials, the method is extended with new types of recurrence relations. It is then applied to the preceding general NPO to obtain expressions for quantum action variable J in terms of E and the parameters of the potential. These expansions are power series in energy and the coefficients of the series contain parameters λ and g explicitly. To avoid the singularities in the potential we only consider the cases where both λ and g are non-negative at the same time. Using the AEE expressions, it is shown that, for certain classes of NPOs, if potentials have the same N, and the same m1 – m2 or m1 – 2m2 then they have the same asymptotic eigenspectra. It was also shown that for certain cases, both λ and –λ as well as g and –g will produce the same asymptotic energy spectra. Analytic expressions are also derived for asymptotic level spacings of general NPOs in terms of λ and g.

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