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Showing papers on "Asymptotology published in 1968"




Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of determining logarithmic, as well as polynomial, asymptotic estimates for certain convergent integrals containing parameters.
Abstract: In this paper, we consider the problem of determining logarithmic, as well as polynomial, asymptotic estimates for certain convergent integrals containing parameters. We state and prove an asymptotic theorem which gives the logarithmic asymptotic behavior of a convergent integral where any subset of the parameters becomes large while the remaining parameters remain bounded. This theorem is then applied to the photon and electron self‐energy graphs of quantum electrodynamics.

39 citations




Journal ArticleDOI
TL;DR: In this article, it was shown that a divergent Taylor series for analytic functions at an isolated singularity does not represent the function in a full deleted neighborhood, but only in certain sectors.
Abstract: The principle of analytic continuation makes it possible to calculate effectively the corresponding convergent power series about all points where a holomorphic continuation exists. However, the nature of the function near its singularities cannot be so readily deduced from the series (1.1). Often series expansions about such singular points do exist, and sometimes it is possible to calculate them explicitly from the coefficients of the convergent expansions about a regular point. These expansions may even be power series. Nevertheless, they differ from the familiar convergent Taylor series in several decisive respects. The most important new feature is that they represent the function ƒ only in an asymptotic sense. To explain this concept, let us assume, for simplicity, that the singularity occurs at z = oo. To say that the function ƒ is asymptotically represented by the series 23r^o rZ~, in symbols ƒ(z)~ 2^r°l0 £r* ', as z—> oo, means that, for all N, the error committed in replacing ƒ (z) by the sum of the first N terms of the power series is 0(z~), as z—> oo. Such a series may well be divergent, in fact, it usually is. If so, another important feature enters the picture: A divergent asymptotic series for an analytic function at an isolated singularity never represents the function in a full deleted neighborhood, but only in certain sectors. There exists a substantial body of theory for the \"connection problem\" just described, namely the problem of finding asymptotic expansions about a singular point from a given convergent expansion for the same function about a regular point. I shall not say much

17 citations


Journal ArticleDOI
TL;DR: In this paper, the asymptotic properties of a class of nonlinear boundary value problems are studied, and very simple approximate analytic solutions are obtained, and their accuracy is illustrated by showing their good agreement with the exact numerical solution of the problem.
Abstract: The asymptotic properties of a class of nonlinear boundary‐value problems are studied. For large values of a parameter, the differential equation is of the singular‐perturbation type, and its solution is constructed by means of matched asymptotic expansions. In two special cases, very simple approximate analytic solutions are obtained, and their accuracy is illustrated by showing their good agreement with the exact numerical solution of the problem.

12 citations


Journal ArticleDOI
TL;DR: In this article, a matched asymptotic expansion for the eigenvalues arising in perturbations about the Blasius solution was proposed, and good agreement was obtained with the numerical results of Libby.
Abstract: A further term is found in the asymptotic expansion suggested by Stewartson for the eigenvalues arising in perturbations about the Blasius solution. The method employed is that of matched asymptotic expansions, and good agreement is obtained with the numerical results of Libby.

10 citations





Journal ArticleDOI
TL;DR: In this article, the exact master equation of Prigogine and Resibois is derived for inhomogeneous as well as homogeneous systems from the following postulate: the singularities of the analytically continued Liouville resolvent nearest the real axis are isolated simple poles arising from irreducible vacuum-to-vacuum transitions.
Abstract: The exact master equation of Prigogine and Resibois is derived for inhomogeneous as well as homogeneous systems. The asymptotic master equation is obtained from the following postulate: The singularities of the analytically continued Liouville resolvent nearest the real axis are isolated simple poles arising from “irreducible vacuum‐to‐vacuum” transitions. The asymptotic distribution function found in this way differs from the one obtained previously. If the asymptotic distribution function at time t1, fa(t1) is taken as a new initial value in the exact master equation, and the subsequent asymptotic solution is calculated for a time t2 later, the result is fa(t1 + t2).