scholarly journals Composite Asymptotic Expansions and Difference Equations

2015 ◽  
Vol Volume 20 - 2015 - Special... ◽  
Author(s):  
Augustin Fruchard ◽  
Reinhard Schäfke
Keyword(s):  
Asymptotic Expansion ◽  
Small Parameter ◽  
Difference Equations ◽  
Asymptotic Expansions ◽  
Inverse Function ◽  
Complex Domain ◽  
Square Root ◽  
Step Size ◽  
International Audience

International audience Difference equations in the complex domain of the form y(x+ϵ)−y(x)=ϵf(y(x))/y(x) are considered. The step size ϵ>0 is a small parameter, and the equation has a singularity at y=0. Solutions near the singularity are described using composite asymptotic expansions. More precisely, it is shown that the derivative v′ of the inverse function v of a solution (the so-called Fatou coordinate) admits a Gevrey asymptotic expansion in powers of the square root of ϵ, denoted by η, involving functions of y and of Y=y/η. This also yields Gevrey asymptotic expansions of the so-called Écalle-Voronin invariants of the equation which are functions of epsilon. An application coming from the theory of complex iteration is presented. On considère des équations aux différences dans le plan complexe de la forme y(x+ϵ)−y(x)=ϵf(y(x))/y(x). Le pas de discrétisation ϵ>0 est un petit paramètre, et l'équation a une singularité en y=0. On décrit les solutions près de la singularité en utilisant des développements asymptotiques combinés. Plus précisément, on montre que la dérivée v′ de la fonction réciproque (appelée coordonnée de Fatou) v d'une solution admet un développement asymptotique Gevrey en puissances de la racine carrée de ϵ, notée η, et faisant intervenir des fonctions de y et de Y=y/η. On obtient également des développements asymptotiques Gevrey des invariants d'Écalle-Voronin de l'équation, qui sont des fonctions de ϵ. Une application venant de la théorie de l'itération complexe est présentée.

scholarly journals Asymptotic Description and the Diagnostic Properties of Low-Degree Stellar P-Modes

1998 ◽  
Vol 185 ◽  
pp. 391-392
Author(s):  
I.W. Roxburgh ◽  
S.V. Vorontsov
Keyword(s):  
Asymptotic Expansion ◽  
Small Parameter ◽  
Asymptotic Expansions ◽  
Turning Point ◽  
Higher Order ◽  
Asymptotic Description ◽  
Low Degree ◽  
Diagnostic Capabilities ◽  
Diagnostic Properties

Standard asymptotic descriptions of stellar p-modes use 1/ω as a small parameter; either the degree ℓ is kept constant in the asymptotic expansions (e.g. Tassoul 1980), or the parameter , which specifies the position of the inner turning point (e.g. Vorontsov 1991). At low degree, due to the strong effects of gravity perturbations, these expansions are known to produce rather poor results, even when developed to higher order (Roxburgh and Vorontsov 1994). Here, we employ an alternative asymptotic expansion, capable of providing much better accuracy at low degree-with as a small parameter instead of 1/ω, and test its diagnostic capabilities.

scholarly journals Prime geodesics and averages of the Zagier L-series

2021 ◽  
pp. 1-24
Author(s):  
OLGA BALKANOVA ◽  
DMITRY FROLENKOV ◽  
MORTEN S. RISAGER
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Error Term ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Prime Geodesic Theorem ◽  
Error Terms ◽  
Average Size ◽  
Real Quadratic

Abstract The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

Asymptotic Expansions in the Exponent: a Compound Poisson Approach

1997 ◽  
Vol 29 (02) ◽  
pp. 374-387 ◽  
Author(s):  
V. Čekanavičius
Keyword(s):  
Asymptotic Expansion ◽  
Large Deviations ◽  
Poisson Distribution ◽  
Asymptotic Expansions ◽  
Compound Poisson ◽  
Poisson Measures ◽  
Poisson Approximations

The accuracy of the Normal or Poisson approximations can be significantly improved by adding part of an asymptotic expansion in the exponent. The signed-compound-Poisson measures obtained in this manner can be of the same structure as the Poisson distribution. For large deviations we prove that signed-compound-Poisson measures enlarge the zone of equivalence for tails.

scholarly journals Asymptotic expansions of Lambert series and related q-series

2017 ◽  
Vol 13 (08) ◽  
pp. 2097-2113 ◽  
Author(s):  
Shubho Banerjee ◽  
Blake Wilkerson
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Theta Functions ◽  
Lambert Series ◽  
Pochhammer Symbol ◽  
Complete Asymptotic Expansion ◽  
Polygamma Functions ◽  
Jacobi Theta Functions ◽  
Relative Errors ◽  
Asymptotic Forms

We study the Lambert series [Formula: see text], for all [Formula: see text]. We obtain the complete asymptotic expansion of [Formula: see text] near [Formula: see text]. Our analysis of the Lambert series yields the asymptotic forms for several related [Formula: see text]-series: the [Formula: see text]-gamma and [Formula: see text]-polygamma functions, the [Formula: see text]-Pochhammer symbol and the Jacobi theta functions. Some typical results include [Formula: see text] and [Formula: see text], with relative errors of order [Formula: see text] and [Formula: see text] respectively.

Effect of compact inclusion on the natural frequencies of vibrations of a pipe string in a well

2021 ◽  
pp. 73-77
Author(s):  
A.M. Svalov ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Small Parameter ◽  
Natural Frequencies ◽  
Oscillation Period ◽  
Drill String ◽  
Analytical Relation ◽  
Longitudinal Vibrations ◽  
Bottom Hole ◽  
Tubing String ◽  
Analytical Relations

The influence of small-size inclusion of pipes in a well column on the natural frequency of its longitudinal vibrations is investigated. Using the asymptotic expansion in a small parameter, an analytical relation is obtained that describes the change in the period of the column oscillations in the form of some additional small term to the period of the homogeneous column oscillations. Numerical calculations show that the obtained analytical relations almost accurately describe the oscillation period of a column with a massive compact inclusion, while its difference from the oscillation period of a homogeneous column is within ~20%. The results obtained can be useful for preventing resonant phenomena in the drill string when drilling wells, as well as for optimal use of the longitudinal vibrations of the tubing string to influence the bottom-hole zones of producing wells.

On obtaining limiting values of a class of infinite products

2018 ◽  
Vol 102 (555) ◽  
pp. 428-434
Author(s):  
Stephen Kaczkowski
Keyword(s):  
Difference Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Diffusion Theory ◽  
Numerical Technique ◽  
Applied Mathematics ◽  
Flow Analysis ◽  
Step Size ◽  
Infinite Products ◽  
Fluid Flow Analysis

Difference equations have a wide variety of applications, including fluid flow analysis, wave propagation, circuit theory, the study of traffic patterns, queueing analysis, diffusion theory, and many others. Besides these applications, studies into the analogy between ordinary differential equations (ODEs) and difference equations have been a favourite topic of mathematicians (e.g. see [1] and [2]). These applications and studies bring to light the similar character of the solutions of a difference equation with a fixed step size and a corresponding ODE.Also, an important numerical technique for solving both ordinary and partial differential equations (PDEs) is the method of finite differences [3], whereby a difference equation with a small step size is utilised to obtain a numerical solution of a differential equation. In this paper, elements of both of these ideas will be used to solve some intriguing problems in pure and applied mathematics.

scholarly journals Asymptotic expansions for distribution of sums quasi-lattice random variables

2011 ◽  
Vol 52 ◽  
pp. 359-364
Author(s):  
Algimantas Bikelis ◽  
Kazimieras Padvelskis ◽  
Pranas Vaitkus
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Independent Random Variables

Althoug Chebyshev [3] and Edeworth [5] had conceived of the formal expansions for distribution of sums of independent random variables, but only in Cramer’s work [4] was laid a proper foundation of this problem. In the case when random variables are lattice Esseen get the asymptotic expansion in a new different form. Here we extend this problem for quasi-lattice random variables.  

scholarly journals Uniform renewal theory with applications to expansions of random geometric sums

2007 ◽  
Vol 39 (4) ◽  
pp. 1070-1097 ◽  
Author(s):  
J. Blanchet ◽  
P. Glynn
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Renewal Theory ◽  
Random Variable ◽  
Higher Order ◽  
Order Correction ◽  
Diffusion Approximations ◽  
Geometric Sums ◽  
Correction Terms

Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

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