Asymptotic expansion of multiple oscillatory integrals with a hypersurface of stationary points of the phase

In this paper, we present a method solving the problem of the asymptotic expansion of the integral, in the case whenD is a bounded domain in(n ≥2), and the setS of stationary points of the phasef 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 phasef does not vanish onS , and the curveS cuts transversely ∂D . It will be seen that the order of degeneracy of normal derivatives off , with respect to the surfaceS , plays a key role in solving the problem. We shall develop complete asymptotic expansions when this order is constant alongS , and show that the problem leads to the use of special functions in the other case.