Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions - Ziener, Christian H.; Kurz, Felix T.; Buschle, Lukas R.; Kampf, Thomas (SpringerPlus, 2015)
The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval \le r \le γ R$$R≤r≤γRwith Neumann boundary conditions are considered. The eigenfunctions are linear combinations of the Bessel function $$Φ _{n,ν }(r)=Y_{ν }^{′ }(λ _{n,ν }) J_{ν }(λ _{n,ν } r/R)-J_{ν }^{′ }(λ _{n,ν }) Y_{ν }(λ _{n,ν } r/R)$$Φn,ν(r)=Yν′(λn,ν)Jν(λn,νr/R)-Jν′(λn,ν)Yν(λn,νr/R)or linear combinations of the spherical Bessel functions $$ψ _{m,ν }(r)=y_{ν }^{′ }(λ _{m,ν }) j_{ν }(λ _{m,ν } r/R)-j_{ν }^{′ }(λ _{m,ν }) y_{ν }(λ _{m,ν } r/R)$$ψm,ν(r)=yν′(λm,ν)jν(λm,νr/R)-jν′(λm,ν)yν(λm,νr/R). The orthogonality relations with analytical expressions for the normalization constant are given. Explicit expressions for the Lommel integrals in terms of Lommel functions are derived. The cross product zeros $$Y_{ν }^{′ }(λ _{n,ν }) J_{ν }^{′ }(γ λ _{n,ν })-J_{ν }^{′ }(λ _{n,ν }) Y_{ν }^{′ }(γ λ _{n,ν }) = 0$$Yν′(λn,ν)Jν′(γλn,ν)-Jν′(λn,ν)Yν′(γλn,ν)=0and $$y_{ν }^{′ }(λ _{m,ν }) j_{ν }^{′ }(γ λ _{m,ν })-j_{ν }^{′ }(λ _{m,ν }) y_{ν }^{′ }(γ λ _{m,ν }) = 0$$yν′(λm,ν)jν′(γλm,ν)-jν′(λm,ν)yν′(γλm,ν)=0are considered in the complex plane for real as well as complex values of the index $$ν $$νand approximations for the exceptional zero $$λ _{1,ν }$$λ1,νare 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.
Gesehen am 15.09.2020