Asymptotic expansions of Mellin convolutions by means of analytic continuation

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.