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Lightstone A. H. Robinson A. - Nonarchimedean fields and asymptotic expansions

Название: Nonarchimedean fields and asymptotic expansions
Автор: Lightstone A. H. Robinson A.
Категория: Математика
Тип: Книга
Дата: 31.12.2008 00:15:34
Скачано: 25
Описание: It has been known for many years that there is a close link between non-archimedean systems and the orders of infinity and of smallness that are associated with the asymptotic behaviour of a function. The present text provides a background for this connection from the point of view of nonstandard analysis. We have kept the argument at an elementary level and hope that the reader will find the book suitable as an introduction to nonstandard analysis as well as the theory of asymptotic expansions. The plan of the book is as follows. In the first chapter we introduce the notions of a nonarchimedean group and a nonarchimedean field and give several interesting examples of nonarchimedean fields. Chapter 2 contains an introduction to nonstandard analysis. The necessary resources from mathematical logic are brought in as we go along. In the following two chapters we link up the nonstandard models of analysis, themselves nonarchimedean fields, with a particular nonarchimedean field, here called jC, which was first studied by Levi-Civita and Ostrowski and, more recently, by Laugwitz. Unlike the nonstandard models of analysis, £ is canonical (i.e. unique), but unlike the former it cannot be studied by means of a transfer principle. We introduce a natural link between JC and the nonstandard models, the field P(R. In the last three chapters of the book, we study the fundamentals of asymptotic expansions. Instead of keeping the discussion at a purely theoretical level, we offer a (happy, we hope) melange of numerical examples and infinitesimals. In sum, we believe that we have at least realized the modest aim of showing that infinitesimals and infinitely large numbers form a natural background to asymptotics. This monograph is based on a draft by the second author (A.R.), while the final text is due to the first author (A.H.L.). We wish to express our thanks to the North-Holland Publishing Company for agreeing to publish the result of our joint effort in the "Mathematical Library". The first author is indebted to the Canada Council for a Leave Fellowship during 1971-1972, which made
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