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.
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M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Nat. Bur. Standards Appl. Series 55 (U.S. Government Printing Office, Washington, 1964).
D. Amos, Algorithm 644, A portable package for Bessel functions of a complex argument and nonnegative order, ACM Trans. Math. Software 12 (1986) 265–273.
T.M. Dunster, Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter, SIAM J. Math. Anal. 21 (1990) 995–1018.
D.W. Lozier and F.W.J. Olver, Numerical evaluation of special functions, in: Mathematics of Computation, 1943–1993: A Half-Century of Computational Mathematics, ed. W. Gautschi, AMS, Proc. Symp. Appl. Math. 48 (1994) pp. 79–125.
G. Matviyenko, On the evaluation of Bessel functions, Appl. Comp. Harm. Anal. 1 (1993) 116–135.
A.B. Olde Daalhuis and N.M. Temme, Uniform Airy type expansions of integrals, SIAM J. Math. Anal. 25 (1994) 304–321.
F.W.J. Olver, Asymptotics and Special Functions (Academic Press, New York, 1974, reprinted A.K. Peters, 1997).
N.M. Temme, On the computation of the incomplete gamma functions for large values of the parameters, in: Algorithms for Approximation, Proc. of the IMA Conference on Algorithms for the Approximation of Functions and Data, eds. J.C. Mason and M.G. Cox (Clarendon, Oxford, 1987) pp. 479–489.
N.M. Temme, Steepest descent paths for integrals defining the modified Bessel functions of imaginary order, Methods Appl. Anal. 1 (1994) 14–24.
N.M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics (Wiley, New York, 1996).
J. Wimp, Computation with Recurrence Relations (Pitman, Boston, 1984).
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Temme, N. Numerical algorithms for uniform Airy-type asymptotic expansions. Numerical Algorithms 15, 207–225 (1997). https://doi.org/10.1023/A:1019197921337
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DOI: https://doi.org/10.1023/A:1019197921337