  In this paper, we develop fast multipole methods for 3D Helmholtz kernel in
layered media. Two algorithms based on different forms of Taylor expansion of
layered media Green's function are developed. A key component of the first
algorithm is an efficient algorithm based on discrete complex image
approximation and recurrence formula for the calculation of the layered media
Green's function and its derivatives, which are given in terms of Sommerfeld
integrals. The second algorithm uses symmetric derivatives in the Taylor
expansion to reduce the size of precomputed tables for the derivatives of
layered media Green's function. Numerical tests in layered media have validated
the accuracy and O(N) complexity of the proposed algorithms.
