Bessel functions \(J_m\) and \(Y_m\) of integer order \(m\) are important for many problems. Let \({\mathcal B}_m\) denote \(J_m\), \(Y_m\) or any linear combination of these functions. Graf’s addition theorem represents \({\mathcal B}_m(|x - y|)\) for \(x, y \in {\mathbb R}^2\) with \(|y| < |x|\) as bi-infinite series. Analogously, Neumann’s addition theorem represents \({\mathcal B}_m(\xi \pm \eta)\) with \(0 < \eta < \xi\) as bi-infinite series. Graf’s and Neumann’s addition theorems are used to solve acoustic and electromagnetic scattering problems in 2 dimensions, especially for fast solution of 2-dimensional scattering and transmission problems using the fast multipole method.
Using truncation of the bi-infinite series, the authors analyze the truncation errors in the Graf’s and Neumann’s addition theorems and present explicit bounds as well as convergence rates of these truncation errors. Numerical experiments show that these bounds are sharp.