Topic
Asymptotology
About: Asymptotology is a research topic. Over the lifetime, 1319 publications have been published within this topic receiving 35831 citations.
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TL;DR: In this paper, an attack is made on the problem of determining the asymptotic behavior at high energies and momenta of the Green's functions of quantum field theory, using new mathematical methods from the theory of real variables.
Abstract: An attack is made on the problem of determining the asymptotic behavior at high energies and momenta of the Green's functions of quantum field theory, using new mathematical methods from the theory of real variables. We define a class ${A}_{n}$ of functions of $n$ real variables, whose asymptotic behavior may be specified in a certain manner by means of certain "asymptotic coefficients." The Feynman integrands of perturbation theory (with energies taken imaginary) belong to such classes. We then prove that if certain conditions on the asymptotic coefficients are satisfied then an integral over $k$ of the variables converges, and belongs to the class ${A}_{n\ensuremath{-}k}$ with new asymptotic coefficients simply related to the old ones. When applied to perturbation theory this theorem validates the renormalization procedure of Dyson and Salam, proving that the renormalized integrals actually do always converge, and provides a simple rule for calculating the asymptotic behavior of any Green's function to any order of perturbation theory.
485 citations
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476 citations
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TL;DR: In this article, a survey of tools used in asymptotic analysis for enumeration problems is presented, focusing on tools which are general, are easily applied, and give estimates of the form $a_n \sim f(n)$.
Abstract: This is an expository paper dealing with those tools in asymptotic analysis which are especially useful in obtaining asymptotic results in enumeration problems. Emphasis is on tools which are general, are easily applied, and give estimates of the form $a_n \sim f(n)$. Many examples are given to illustrate the usage of the various tools. It is assumed that a summation or a generating function for $a_n $ is explicitly or implicitly given.
443 citations