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Numerical Algorithms

, Volume 15, Issue 2, pp 207–225 | Cite as

Numerical algorithms for uniform Airy-type asymptotic expansions

  • N.M. Temme
Article

Abstract

Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we do not need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.

uniform asymptotic expansions turning points Airy-type expansions Bessel functions computation of special functions 41A60 34E20 33C10 65D20 

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Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • N.M. Temme
    • 1
  1. 1.CWIAmsterdamThe Netherlands

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