Asymptotic Expansions for Oscillatory Integrals Using Inverse Functions

We treat finite oscillatory integrals of the form ∫ b a F (x) exp[ikG(x)]dx in which both F and G are real on the real line, are analytic over the open integration interval, and may have algebraic singularities at either or both interval end points. For many of these, we establish asymptotic expansions in inverse powers of k. No appeal to the theories of stationary phase or steepest descent is involved. We simply apply theory involving inverse functions and expansions for a Fourier coefficient ∫ b a φ(t) exp[ikt]dt. To this end, we have assembled several results involving inverse functions. Moreover, we have derived a new asymptotic expansion for this integral, valid when φ(t) = ∑ ajt σj .