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Related Experiment Videos

Airey's Converging Factor.

F D Murnaghan1

  • 16202 Sycamore Road, Baltimore, Maryland 21212.

Proceedings of the National Academy of Sciences of the United States of America
|February 1, 1972
PubMed
Summary
This summary is machine-generated.

Asymptotic series approximations can be improved by adjusting the least term. A converging factor method refines calculations for functions like the exponential integral.

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Background:

  • Asymptotic series are used for function approximation, particularly for large arguments.
  • These series have terms that initially decrease but eventually increase, requiring truncation.
  • The 'least term' is the first of potentially two equal-valued adjacent terms at the minimum.

Purpose of the Study:

  • To investigate methods for improving approximations derived from asymptotic series.
  • To define and determine the converging factor for asymptotic series.
  • To present a method for calculating the converging factor for the exponential integral.

Main Methods:

  • Analysis of the behavior of terms in asymptotic series.
  • Definition of the converging factor based on the least term.

Related Experiment Videos

  • Development of a method to determine coefficients of the converging factor series.

    • Main Results:

    • The sum of initial terms up to the least term provides an approximation.
    • Modifying the least term (e.g., halving if terms alternate sign) can improve accuracy.
    • Airey's converging factor for the exponential integral's asymptotic series starts with 1/2.

    Conclusions:

    • The converging factor offers a systematic way to enhance asymptotic series approximations.
    • The presented method allows for the determination of these crucial factors.
    • Accurate calculation of the exponential integral for large negative arguments is facilitated.