Asymptotic expansions of Mellin convolutions by means of analytic continuation

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Abstract

We present a method for deriving asymptotic expansions of integrals of the form ∫0∞ f (t) h (xt) d t for small x based on analytic continuation. The expansion is given in terms of two asymptotic sequences, the coefficients of both sequences being Mellin transforms of h and f. Many known and unknown asymptotic expansions of important integral transforms are derived trivially from the approach presented here. This paper reconsiders earlier work of McClure and Wong [Explicit error terms for asymptotic expansions of Stieltjes transforms, J. Inst. Math. Appl. 22 (1978) 129-145; Exact remainders for asymptotic expansions of fractional integrals, J. Inst. Math. Appl. 24 (1979) 139-147] and Asymptotic approximations of integrals, Academic Press, New York, 1989. Chaps. 5, where elements of distribution theory are used, and Wong [Explicit error terms for asymptotic expansions of Mellin convolutions, J. Math. Anal. Appl. 72(2) (1979) 740-756], where, as in the present paper, the asymptotic expansions are obtained without the use of distributions. In this paper we re-derive the expansions given in Wong [Explicit error terms for asymptotic expansions of Mellin convolutions, J. Math. Anal. Appl. 72(2) (1979) 740-756] by using a different approach and we obtain new results which are not present in Wong [Explicit error terms for asymptotic expansions of Mellin convolutions, J. Math. Anal. Appl. 72(2) (1979) 740-756]: a proof of the asymptotic character of the expansions and accurate error bounds. © 2006 Elsevier B.V. All rights reserved.

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López, J. L. (2007). Asymptotic expansions of Mellin convolutions by means of analytic continuation. Journal of Computational and Applied Mathematics, 200(2), 628–636. https://doi.org/10.1016/j.cam.2006.01.019

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