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Zbl 1053.33003
Fabijonas, Bruce R.; Olver, F. W. J.
On the reversion of an asymptotic expansion and the zeros of the Airy functions. (English)
[J] SIAM Rev. 41, No. 4, 762-773 (1999). ISSN 0036-1445; ISSN 1095-7200/e

Summary: The general theories of the derivation of inverses of functions from their power series and asymptotic expansions are discussed and compared. The asymptotic theory is applied to obtain asymptotic expansions of the zeros of the Airy functions and their derivatives, and also of the associated values of the functions or derivatives. A Maple code is constructed to generate exactly the coefficients in these expansions. The only limits on the number of coefficients are those imposed by the capacity of the computer being used and the execution time that is available. \par The sign patterns of the coefficients suggest open problems pertaining to error bounds for the asymptotic expansions of the zeros and stationary values of the Airy functions.

MSC 2000:: *33C10 Cylinder functions, etc.
33F05 Numerical approximation of special functions
41A60 Asymptotic problems in approximation
65D20 Computation of special functions

Keywords: Airy functions; asymptotic expansions; error bounds; inversion theorems; Lagrange's reversion theorem; phase principle; principle of the argument; symbolic computation; zeros

Cited in: Zbl 0998.33005

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