On uniform asymptotic expansion of a class of integral transforms

Abstract

Asymptotic expansions for the integrals of the type

$$F(x,a) = \int\limits_0^a {K(xt)f(t)dt,x \to \infty }$$

which hold uniformly in a when either 0≤a≤δ or when δ≤a<∞ for some δ>0, are obtained. It is assumed that f has an algebraic singularity at the origin and K(t) t^λ−1, λ>0 is locally absolutely integrable in [0, ∞). In general, the asymptotic expansion of F(x,a) when a → 0+ cannot be obtained directly from the corresponding expansion when a is bounded away from zero. In some cases, a similar situation may arise as a → ∞. Analytic continuation of the incomplete Mellin transform of K provides a unified approach to this problem.