In this paper, we present a method solving the problem of the asymptotic expansion of the integral I(λ) = D g(x) e iλf (x) dx (λ → +∞), in the case when D is a bounded domain in R n (n ≥ 2), and the set S of stationary points of the phase f is a hypersurface. This problem was considered in the literature, in the two-dimensional case, where it is required that the Laplacian f of the phase f does not vanish on S, and the curve S cuts transversely ∂D. It will be seen that the order of degeneracy

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... normal derivatives of f , with respect to the surface S, plays a key role in solving the problem. We shall develop complete asymptotic expansions when this order is constant along S, and show that the problem leads to the use of special functions in the other case. on July 19, 2018 http://rspa.royalsocietypublishing.org/ Downloaded from Preliminary transformations Throughout this paper, we use the notation T δ (W) = {x + ty x ∈ R 3 : x ∈ W, |t| ≤ δ},