  This paper presents a quadrature method for evaluating layer potentials in
two dimensions close to periodic boundaries, discretized using the trapezoidal
rule. It is an extension of the method of singularity swap quadrature, which
recently was introduced for boundaries discretized using composite
Gauss-Legendre quadrature. The original method builds on swapping the target
singularity for its preimage in the complexified space of the curve
parametrization, where the source panel is flat. This allows the integral to be
efficiently evaluated using an interpolatory quadrature with a monomial basis.
In this extension, we use the target preimage to swap the singularity to a
point close to the unit circle. This allows us to evaluate the integral using
an interpolatory quadrature with complex exponential basis functions. This is
well-conditioned, and can be efficiently evaluated using the fast Fourier
transform. The resulting method has exponential convergence, and can be used to
accurately evaluate layer potentials close to the source geometry. We report
experimental results on a simple test geometry, and provide a baseline Julia
implementation that can be used for further experimentation.
