Multidimensional stationary phase approximation: boundary stationary point

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)e^iλf(x) dx (x ∈ ℝⁿ), where a stationary point of f(x) lies on the boundary of D.