  In this paper, we prove the exponential convergence of the multipole and
local expansions, shifting and translation operators used in fast multipole
methods (FMMs) for 3-dimensional Laplace equations in layered media. These
theoretical results ensure the exponential convergence of the FMM which has
been shown by the numerical results recently reported in [9]. As the free space
components are calculated by the classic FMM, this paper will focus on the
analysis for the reaction components of the Green's function for the Laplace
equation in layered media. We first prove that the density functions in the
integral representations of the reaction components are analytic and bounded in
the right half complex plane. Then, using the Cagniard-de Hoop transform and
contour deformations, estimate for the remainder terms of the truncated
expansions is given, and, as a result, the exponential convergence for the
expansions and translation operators is proven.
