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On The Reversion of an Asymptotic Expansion and the Zeros of the Airy Functions



Bruce R. Fabijonas, Frank W. Olver


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 capacity of the computer being used and the execution time that is available. 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.
Siam Review


airy functions, asymptotic expansions, error bounds, inversion theorems, Lagrange's reversion theorem, phase principle, principle of the argument, symbolic computation, zeros


Fabijonas, B. and Olver, F. (1999), On The Reversion of an Asymptotic Expansion and the Zeros of the Airy Functions, Siam Review, [online], (Accessed June 14, 2024)


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Created January 15, 1999, Updated February 19, 2017