Explicit representation of the green’s function for the three-dimensional exterior Helmholtz equation

We construct a sequence of solutions of the exterior Helmholtz equation such that their restrictions form an orthonormal basis on a given surface. The dependence of the coefficients of these functions on the coefficients of the surface are given by an explicit algebraic formula. In the same way, we construct an explicit normal derivative of the Dirichlet Green’s function. We also construct the Dirichlet-to-Neumann operator. We prove that the normalized coefficients are uniformly bounded from zero.