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Nonstandard Asymptotic Analysis

  • Book
  • © 1987

Overview

Authors:
  1. Imme Berg
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Part of the book series: Lecture Notes in Mathematics (LNM, volume 1249)

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Table of contents (6 chapters)

  1. Front Matter

    Pages I-IX
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  2. Four examples of nonstandard reasoning in asymptotics

    • Imme van den Berg
    Pages 2-43

  3. Asymptotic expressions for the remainders associated to expansions of type \(\sum\limits_{n = 0}^\infty { c_n \frac{{z^n }}{{n!}}, } \sum\limits_{n = 0}^\infty { c_n z^n and } \sum\limits_{n = 0} { c_n n!z^n }\), where cn+p/cn → c

    • Imme van den Berg
    Pages 46-71

  4. Asymptotic expressions for the remainders associated to expansions of type \(\sum\limits_{n = 0}^\infty { c_n \frac{{z^n }}{{n!}}, } \sum\limits_{n = 0}^\infty { c_n z^n and } \sum\limits_{n = 0} { c_n n!z^n }\): Critical regions, uniform behaviour

    • Imme van den Berg
    Pages 72-103

  5. External sets

    • Imme van den Berg
    Pages 106-141

  6. Approximation lemma's

    • Imme van den Berg
    Pages 142-151

  7. Shadow expansions

    • Imme van den Berg
    Pages 152-176

  8. Back Matter

    Pages 177-187
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Keywords

About this book

This research monograph considers the subject of asymptotics from a nonstandard view point. It is intended both for classical asymptoticists - they will discover a new approach to problems very familiar to them - and for nonstandard analysts but includes topics of general interest, like the remarkable behaviour of Taylor polynomials of elementary functions. Noting that within nonstandard analysis, "small", "large", and "domain of validity of asymptotic behaviour" have a precise meaning, a nonstandard alternative to classical asymptotics is developed. Special emphasis is given to applications in numerical approximation by convergent and divergent expansions: in the latter case a clear asymptotic answer is given to the problem of optimal approximation, which is valid for a large class of functions including many special functions. The author's approach is didactical. The book opens with a large introductory chapter which can be read without much knowledge of nonstandard analysis. Here the main features of the theory are presented via concrete examples, with many numerical and graphic illustrations. N

Bibliographic Information

  • Book Title: Nonstandard Asymptotic Analysis

  • Authors: Imme Berg

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/BFb0077577

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1987

  • Softcover ISBN: 978-3-540-17767-8Published: 08 April 1987

  • eBook ISBN: 978-3-540-47810-2Published: 15 November 2006

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XII, 192

  • Topics: Mathematical Logic and Foundations

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