Two-Dimensional Stationary Phase Approximation: Stationary Point at a Corner

Asymptotic expansions are derived for the double integral \[ \iint_D {g(x,y)\exp (iNf(x,y))dx\,dy}, \] as $N \to + \infty $, where $f(x,y)$ has a stationary point at a corner of the boundary of D. Two different methods are given, one for the case of local extrema and one for saddlepoints. In the case of saddlepoints, our method allows the boundary of D to be tangent to a level curve of f.