Nonarchimedean Fields and Asymptotic Expansions

Nonarchimedean Fields and Asymptotic Expansions

1st Edition - January 1, 1975

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  • Authors: A. H. Lightstone, Abraham Robinson
  • eBook ISBN: 9781483257440

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Description

North-Holland Mathematical Library, Volume 13: Nonarchimedean Fields and Asymptotic Expansions focuses on the connection between nonarchimedean systems and the orders of infinity and smallness that are related with the asymptotic behavior of a function. The publication first explains nonarchimedean fields and nonstandard analysis. Discussions focus on the method of mathematical logic, ultrapower construction, principles of permanence, internal functions, many-sorted structures, nonarchimedean fields and groups, and fields with evaluation. The text then discusses the Euler-Maclaurin expansions and the formal concept of asymptotic expansions. Topics include a generalized criterion for asymptotic expansions, asymptotic power series, Watson's Lemma, asymptotic sequences, and the Euler-Maclaurin formula. The manuscript examines Popken space, including asymptotically finite functions, convergence, norm, algebraic properties of the norm, and Popken's description of the norm. The text is a dependable reference for mathematicians and researchers interested in nonarchimedean fields and asymptotic expansions.

Table of Contents


  • Preface

    Chapter 1. Nonarchimedean Fields

    1. Many-Sorted Structures

    2. Nonarchimedean Groups

    3. Nonarchimedean Fields

    4. Fields with Valuation

    5. Development of Metric

    6. Hardy Fields

    7. The Field ʆ

    Chapter 2. Nonstandard Analysis

    1. The Method of Mathematical Logic

    2. The Languages of R and *R

    3. Filters

    4. The Ultrapower Construction

    5. Proof of Łos's Lemma

    6. *R is Sequentially Comprehensive

    7. Principles of Permanence

    8. Continuity in R

    9. Internal functions

    10. Continuity in *R

    Chapter 3. The Field pR

    1. Maximal Ideals

    2. The Field pR

    3. Valuation

    4. Convergence

    5. Series

    6. p-Series

    7. Iotas and megas

    Chapter 4. Functions in pR

    1. The Function Concept

    2. More Functions

    3. Continuity

    4. S-Continuity

    5. Functions pf and Continuity

    6. Differentiation

    Chapter 5. Euler-Maclaurin Expansions

    1. Introduction

    2. The Euler—Maclaurin Formula

    3. Some Examples

    Chapter 6. Asymptotic Expansions - The Formal Concept

    1. Asymptotic Sequences; Asymptotic Expansions

    2. Asymptotic Power Series

    3. Nonstandard Criterion for Asymptotic Expansions

    4. Watson's Lemma

    5. Other Scales

    6. A Generalized Criterion for Asymptotic Expansions

    Chapter 7. Popken Space

    1. Asymptotically Finite Functions

    2. Convergence

    3. Norm

    4. Algebraic Properties of the Norm

    5. Popken's Description of the Norm

    6. More Properties of P

    7. Asymptotic Expansions in P

    8. More About Norms

    Bibliography

    Index

Product details

  • No. of pages: 214
  • Language: English
  • Copyright: © North Holland 1975
  • Published: January 1, 1975
  • Imprint: North Holland
  • eBook ISBN: 9781483257440

About the Authors

A. H. Lightstone

Abraham Robinson

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