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

作者：Benaissa A^*; Roger C

来源：Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 2013, 469(2155): 20130109.

DOI：10.1098/rspa.2013.0109

• asymptotic expansion
• oscillatory integral
• stationary points
• hypersurface

In this paper, we present a method solving the problem of the asymptotic expansion of the integral I(lambda)= integral(D) g(x) e(i lambda f(x)) dx (lambda -%26gt; + infinity), in the case when D is a bounded domain in R-n (n %26gt;= 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 Delta f of the phase f does not vanish on S, and the curve S cuts transversely partial derivative D. It will be seen that the order of degeneracy of 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.

• 出版日期2013-7-8