Uniformly asymptotic expansions for an integral with a large and a small parameter
Fritz Joseph Ursell · Mathematical Proceedings of the Cambridge Philosophical Society · 1987
Abstract The integral involves a large real parameter N and a small real parameter ∈. Its asymptotic behaviour is non-uniform when N → ∞ and ∈ → 0. Thus, when ∈ > 0 is kept fixed and N → ∞, the integral decays exponentially at a rate depending on ∈; when ∈ → 0 the integral tends to which decays algebraically when N → ∞. It is shown that several distinct uniformly asymptotic expansions can be obtained which each involve an infinite set of functions of the combination Certain related integrals are also treated.