On the use of Hadamard expansions in hyperasymptotic evaluation. I. Real variables

R. B. Paris

Abstract

We demonstrate that the Hadamard expansion, first introduced for the modified Bessel function I0(x) in 1908, can be used as a computational tool for hyperasymptotic evaluation. Such expansions, which are absolutely convergent series involving the incomplete gamma function, have early terms possessing a rapid asymptotic–like decay (when the variable is large) with late terms going over into a slow algebraic decay. Rearrangement of the slowly convergent tail shows that the Hadamard expansion can be converted into an absolutely convergent series representation whose late terms exhibit a rapid decay comparable with that of the early terms. The theory is illustrated by means of the confluent hypergeometric functions of real variable. Numerical examples involving the modified Bessel functions Iv(x) and Kv(x), when x > 0, are given to show the level of accuracy that can be achieved by this pseudoasymptotic procedure.

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