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

AbstractThe 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, Watson's lemma and the integration by parts technique applied to Laplace's and a special family of Fourier's transforms become corollaries of the term-by-term integration method