scispace - formally typeset
Search or ask a question

Showing papers on "Asymptotology published in 1964"


Journal Article
TL;DR: In this article, the authors implique l'accord avec les conditions générales d'utilisation (http://www.compositio.org/legal.php).
Abstract: © Foundation Compositio Mathematica, 1962-1964, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

379 citations


Journal ArticleDOI
01 Jun 1964
TL;DR: In this article, the sufficiency of the above theorem without an assumption as to the sign of a(t) was established without any assumption on the degree of a (t) > 0, and a stronger asymptotic result for equation (1) under a stronger assumption was given.
Abstract: This theorem is an extension of a theorem of Atkinson [21. Theorem 1 of this paper establishes the sufficiency of the above theorem without an assumption as to the sign of a(t). A theorem somewhat like the theorem of Trench [3] is given in Theorem 2 but for a nonlinear equation. As an application of this theorem, a stronger asymptotic result (under a stronger assumption) for equation (1) is given in Theorem 3. Other theorems on the asymptotic behavior of (1) when a(t) > 0 are contained in [4 ]. THEOREM 1. If a(t) is continuous and

49 citations





Journal ArticleDOI
TL;DR: In this article, the problem of deriving Green-type asymptotic solutions from differential equations of general form d2 y /dz2 = X(a2>, z)y, for large values of a2, is reformulated.
Abstract: The problem of deriving Green-type asymptotic solutions from differential equations of general form d2 y /dz2 = X(a2>, z)y , for large values of a2, is reformulated. Combination of this formulation with the method of Mellin transforms leads further to a particularly convenient procedure for finding asymptotic expansions valid in transitional regions, and general uniform expansions. The methods are illustrated by detailed calculations for modified Bessel functions.

9 citations




Book
01 Jan 1964
TL;DR: A submitted manuscript is the version of the article upon submission and before peer-review as discussed by the authors, while a published version is the final layout of the paper including the volume, issue and page numbers.
Abstract: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication

6 citations


Journal ArticleDOI
TL;DR: In this paper, asymptotic expansions are obtained which describe the wave behaviour of particles, where a new variable (action) is introduced where a small parameter emerges as a constant ratio for the highest derivative.
Abstract: In this paper asymptotic expansions are obtained which describe the wave behaviour of particles. The essence of the method lies in the fact that a new variable (action) is introduced where a small parameter emerges as a constant ratio for the highest derivative. In this case it is convenient to investigate equations of wave mechanics with the help of a single asymptotic method. The new variable is introduced in order to obtain the necessary equation of the characteristics. Section 1 contains necessary information about the asymptotic method. In Section 2 on the basis of physical considerations the wave equations are given in five-dimensional form with action as the fifth coordinate. In Section 3 an asymptotic expansion in h is obtained and a Cauchy problem (dispersion problem) is solved. In Section 4 a connection between the asymptotic expansion and perturbation theory is established and it is shown that the series in perturbation theory so constructed is part of some asymptotic series. A series of perturbation theory, as we can easily see, is to some extent a classical limit, when e and → 0, so that e 2 / c = const. In Section 5 it is shown that, in the case of Schrodinger' s equation, on the basis of the formulae obtained, a bounded wave impulse moves in accordance with equations which, in the limit of an exact impulse, are the Hamiltonian equations of classical dynamics. Also this case is equivalent to the case when → 0. The formulae obtained in Section 5 generalize Ehrenfest' s theorem, and the method enables us to calculate any dynamical variables which characterize the wave behaviour of the particles. Here we show that the statistical explanation of ordinary quantum mechanics can be avoided.

3 citations


Journal ArticleDOI
01 May 1964-Nature
TL;DR: In this paper, the distribution of the time to emptiness of a finite dam or reservoir might be investigated by the use of Wald's identity, which has recently been used by R. M. Phatarfod2 in the case of inputs with negative exponential distribution and is useful as an approximation in all cases where individual (independent) increments are fairly small in relation to the capacity of the dam.
Abstract: IN my contribution to the discussion at a symposium1 in 1957 on the theory of dams, I suggested that the distribution of the time to emptiness of a finite dam or reservoir might be investigated by the use of Wald's identity. This method has recently been used by R. M. Phatarfod2 in the case of inputs with negative exponential distribution. It seems worth noting how the asymptotic case of normal diffusion, which should be useful as an approximation in all cases where individual (independent) increments are fairly small in relation to the capacity of the dam, may be dealt with. This asymptotic theory has, moreover, an immediate extension to the case of correlated increments by our determining the effective diffusion per unit time in such a case (compare with my concluding remarks, loc. cit.).

Journal ArticleDOI