  We study the lattice Green's function (LGF) of the screened Poisson equation
on a two-dimensional rectangular lattice. This LGF arises in numerical
analysis, random walks, solid-state physics, and other fields. Its defining
characteristic is the screening term, which defines different regimes. When its
coefficient is large, we can accurately approximate the LGF with an
exponentially converging asymptotic expansion, and its convergence rate
monotonically increases with the coefficient of the screening term. To tabulate
the LGF when the coefficient is not large, we derive a one-dimensional integral
representation of the LGF. We show that the trapezoidal rule can approximate
this integral with exponential convergence, and we propose an efficient
algorithm for its evaluation via the Fast Fourier Transform. We discuss
applications including computing the LGF of the three-dimensional Poisson
equation with one periodic direction and the return probability of a
two-dimensional random walk with killing.
