Asymptotic Expansions of Integral Transforms with Oscillatory Kernels: A Generalization of the Method of Stationary Phase
Richard A. Handelsman, Norm Bleistein · SIAM Journal on Mathematical Analysis · 1973
Integrals with integrands of the form $H(\lambda \phi (t))f(t)$ are considered for $\lambda \to \infty $ and $H(t)$ oscillatory for large argument. It is shown that the set of critical points for such integrals includes zeros of the phase function $\phi $ as well as all of those that arise in the analysis of the standard integrals of Fourier type, that is, for the special case where $H(t) = \exp \{ it\} $. The contribution to the asymptotic expansion from each type of critical point is derived. In particular, a formula is obtained which generalizes the stationary phase formula associated with Fourier-type integrals.