Highly accurate approximations of Green's and Neumann functions on rectangular domains

Green's and Neumann functions of -Δ, where Δ is the Laplacian operator, on a rectangular domain are approximated to any desired degree of accuracy by finite series. Many applications require only a modest number of terms. Upper bounds for the errors in these approximations are also derived. The approximating functions reveal the structural similarities and differences in Green's and Neumann functions.