Asymptotic expansions of Mellin convolution integrals: An oscillatory case
Journal of Computational and Applied Mathematics
Full length article: The third Appell function for one large variable
Journal of Approximation Theory
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We present a new method for deriving asymptotic expansions of $\int_0^\infty f(t)h(xt)dt$ for small $x$. We only require that $f(t)$ and $h(t)$ have asymptotic expansions at $t=\infty$ and $t=0$, respectively. Remarkably, it is a very general technique that unifies a certain set of asymptotic methods. Watson's lemma and other classical methods, Mellin transform techniques, McClure and Wong's distributional approach, and the method of analytic continuation turn out to be simple corollaries of this method. In addition, the most amazing thing about it is that its mathematics are absolutely elemental and do not involve complicated analytical tools as the aforementioned methods do: it consists of simple “sums and subtractions." Many known and new asymptotic expansions of important integral transforms are trivially derived from the approach presented here.