This paper describes a numerical method for the solution of the Laplace and the Helmholtz equations in the exterior of a smooth open arc in , which is based on layer potentials. It is well known that with a cosine change of variable, the integral equation can be transformed into another equation where the kernel is periodic. This equation is well posed only for Sobolev spaces of even functions, and hence numerical methods must take the parity into account.

The main contribution of this paper is a modified equation that, for even data, provides the same solution as the even equation, but is well posed for the usual Sobolev spaces of periodic functions. Thus, standard discretization methods can be used. In particular, the author considers a quadrature method and a collocation method, where the collocation points are shifted away from the grid. With a suitable choice of the shift parameter, the leading terms in the error of the potential can be shown to cancel, and superconvergence is established. The discussion concludes with two numerical experiments, which confirm the theoretical error bounds.