Asymptotic expansions of integrals: The term-by-term integration method

The classical term-by-term integration technique used for obtaining asymptotic expansions of integrals requires the integrand to have an uniform asymptotic expansion in the integration variable. A modification of this method is presented in which the uniformity requirement is substituted by a much weaker condition. As we show in some examples, the relaxation of the uniformity condition provides the term-by-term integration technique a large range of applicability. As a consequence of this generality, Watsonʹs lemma and the integration by parts technique applied to Laplaceʹs and a special family of Fourierʹs transforms become corollaries of the term-by-term integration method.