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This paper presents a quadrature method for evaluating layer potentials intwo dimensions close to periodic boundaries, discretized using the trapezoidalrule. It is an extension of the method of singularity swap quadrature, whichrecently was introduced for boundaries discretized using compositeGauss-Legendre quadrature. The original method builds on swapping the targetsingularity for its preimage in the complexified space of the curveparametrization, where the source panel is flat. This allows the integral to beefficiently evaluated using an interpolatory quadrature with a monomial basis.In this extension, we use the target preimage to swap the singularity to apoint close to the unit circle. This allows us to evaluate the integral usingan interpolatory quadrature with complex exponential basis functions. This iswell-conditioned, and can be efficiently evaluated using the fast Fouriertransform. The resulting method has exponential convergence, and can be used toaccurately evaluate layer potentials close to the source geometry. We reportexperimental results on a simple test geometry, and provide a baseline Juliaimplementation that can be used for further experimentation.

Singularity swap quadrature for nearly singular line integrals on closed curves in two dimensions