A scalar potential formulation and translation theory for the time-harmonic Maxwell equations
Journal of Computational Physics
High performance BLAS formulation of the multipole-to-local operator in the fast multipole method
Journal of Computational Physics
Short Note: A fast and stable method for rotating spherical harmonic expansions
Journal of Computational Physics
Lattice Sums for the Helmholtz Equation
SIAM Review
A kernel class allowing for fast computations in shape spaces induced by diffeomorphisms
Journal of Computational and Applied Mathematics
Efficient FMM accelerated vortex methods in three dimensions via the Lamb-Helmholtz decomposition
Journal of Computational Physics
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We develop exact expressions for the coefficients of series representations of translations and rotations of local and multipole fundamental solutions of the Helmholtz equation in spherical coordinates. These expressions are based on the derivation of recurrence relations, some of which, to our knowledge, are presented here for the first time. The symmetry and other properties of the coefficients are also examined and, based on these, efficient procedures for calculating them are presented. Our expressions are direct and do not use the Clebsch--Gordan coefficients or the Wigner 3-j symbols, although we compare our results with methods that use these to prove their accuracy. For evaluating an Nt term truncation of the translated series (involving $O(N_t^2)$ multipoles), our expressions require $O(N_t^3)$ evaluations, compared to previous exact expressions that require $O(N_t^5)$ operations.