Fast Evaluation of the Azimuthal Fourier Modes of the 3D Helmholtz Green's Function and Their Derivatives

Abstract

We introduce an $O(M)$ algorithm for evaluating the azimuthal Fourier modes $G_{k,m}$, $m = 0, 1, ..., M$, of the three-dimensional Helmholtz Green's function with real wavenumber $k$, together with all their first- and second-order derivatives with respect to the cylindrical source and target coordinates. The cost is independent of both the wavenumber and the source-target separation, and high relative accuracy is retained even for modes whose magnitude is exponentially small. The method combines contour deformation at a few boundary modes with a boundary-value formulation of the five-term recurrence in the mode index. Derivative quantities are obtained from stable recurrences, adding only a small constant factor to the cost of $G_{k,m}$ alone. Numerical experiments demonstrate high relative accuracy, linear scaling in $M$, and applications to modal boundary integral equation solvers for axisymmetric acoustic scattering, where the $k$-independent kernel evaluator makes dense per-mode linear algebra the dominant cost.