On the azimuthal Fourier components of the Green’s function for the Helmholtz equation in three dimensions

Many algorithms that compute acoustic or electromagnetic fields scattered by surfaces of revolution require fast evaluation of the azimuthal Fourier components Gm of the Green’s function for the Helmholtz equation in three dimensions. In this article we derive a recurrence relation for the functions Gm and obtain explicit formulas for their partial derivatives. These observations significantly reduce the complexity of the computation of the scattered fields generated by axisymmetric scatterers.