Rigorous justification of the Friedlander–Keller formulas

In the paper an asymptotic expansion (as $\kappa\to\infty$, $\kappa$ the wave number) is proved for the Green function of the problem of diffraction by a smooth convex body in two cases: when one of the points of the source or observer lies on the boundary while the other is an arbitrary distance from the boundary and also when both points lie off the boundary but not far from it. The two-dimensional Dirichlet problem is considered.