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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 degeneracydoi:10.1098/rspa.2013.0109 fatcat:4ypd6dwmkfabnda4bu4ah5qpaa