Orthogonality Lommel integrals and cross product zeros of linear combinations of Bessel functions

摘要

The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval R ≤ r ≤ γR with Neumann boundary conditions are considered. The eigenfunctions are linear combinations of the Bessel function Φn,ν(r)=Yν′(λn,ν)Jν(λn,νr/R)-Jν′(λn,ν)Yν(λn,νr/R) or linear combinations of the spherical Bessel functions ψm,ν(r)=yν′(λm,ν)jν(λm,νr/R)-jν′(λm,ν)yν(λm,νr/R). The orthogonality relations with analytical expressions for the normalization constant are given. Explicit expressions for the Lommel integrals in terms of Lommel functions are derived. The cross product zeros Yν′(λn,ν)Jν′(γλn,ν)-Jν′(λn,ν)Yν′(γλn,ν)=0 and yν′(λm,ν)jν′(γλm,ν)-jν′(λm,ν)yν′(γλm,ν)=0 are considered in the complex plane for real as well as complex values of the index ν and approximations for the exceptional zero λ1,ν are obtained. A numerical scheme based on the discretization of the two-dimensional and three-dimensional Laplace operator with Neumann boundary conditions is presented. Explicit representations of the radial part of the Laplace operator in form of a tridiagonal matrix allow the simple computation of the cross product zeros.