Nonarchimedean Fields and Asymptotic Expansions - 1st Edition - ISBN: 9780720424591, 9781483257440

Nonarchimedean Fields and Asymptotic Expansions

1st Edition

Authors: A. H. Lightstone Abraham Robinson
eBook ISBN: 9781483257440
Imprint: North Holland
Published Date: 1st January 1975
Page Count: 214
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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

Details

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

About the Author

A. H. Lightstone

Abraham Robinson