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In this paper, we will first give a derivation of the multipole expansion(ME) and local expansion (LE) for the far field from sources in general 2-Dlayered media and the multipole-to-local translation (M2L) operator by usingthe generating function for Bessel functions. Then, we present a rigorous proofof the exponential convergence of the ME, LE, and M2L for 2-D Helmholtzequations in layered media. It is shown that the convergence of ME, LE, and M2Lfor the reaction field component of the Green's function depends on a polarizeddistance between the target and a polarized image of the source.

Exponential convergence for multipole and local expansions and their translations for sources in layered media: 2-D acoustic wave