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Airey's Converging Factor
16202 Sycamore Road, Baltimore, Maryland 21212.
Summary
Asymptotic series approximations can be improved by adjusting the least term. A converging factor method refines calculations for functions like the exponential integral.
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.
- 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.
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