Full low‐frequency asymptotic expansion for second‐order elliptic equations in two dimensions

The present paper contains the low‐frequency expansions of solutions of a large class of exterior boundary value problems involving second‐order elliptic equations in two dimensions. The differential equations must coincide with the Helmholtz equation in a neighbourhood of infinity, however, they may depart radically from the Helmholtz equation in any bounded region provided they retain ellipticity. In some cases the asymptotic expansion has the form of a power series with respect to k 2 and k 2 (ln k + a) −1 , where k is the wave number and a is a constant. In other cases it has the form of a power series with respect to k 2 , coefficients of which depend polynomially on In k . The procedure for determining the full low‐frequency expansion of solutions of the exterior Dirichlet and Neumann problems for the Helmholtz equation is included as a special case of the results presented here.