Multidimensional stationary phase approximation: boundary stationary point

摘要

A necessary and sufficient condition is established for the existence of a nonsingular matrix which simplifies a linear form to a single coordinate and at the same time retains a quadratic form. A version of Morse's lemma is also derived. These results are then used in a rigorous derivation of the asymptotic expansion of the oscillatory integral I(λ) = ∫Dg(x) eiλf(x) dx (x \te; rn), where a stationary point of f(x) lies on the boundary of D.