title: Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions.
abstract: The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval [Formula: see text] with Neumann boundary conditions are considered. The eigenfunctions are linear combinations of the Bessel function [Formula: see text] or linear combinations of the spherical Bessel functions [Formula: see text]. 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 [Formula: see text] and [Formula: see text] are considered in the complex plane for real as well as complex values of the index [Formula: see text] and approximations for the exceptional zero [Formula: see text] 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.
authors.affiliation: Department of Radiology, German Cancer Research Center-DKFZ, Im Neuenheimer Feld 280, 69120 Heidelberg, Germany ; Division of Neuroradiology, Heidelberg University, Im Neuenheimer Feld 400, 69120 Heidelberg, Germany.
authors.forename: Christian H
authors.initials: CH
authors.lastname: Ziener
authors.affiliation: Department of Radiology, German Cancer Research Center-DKFZ, Im Neuenheimer Feld 280, 69120 Heidelberg, Germany ; Division of Neuroradiology, Heidelberg University, Im Neuenheimer Feld 400, 69120 Heidelberg, Germany.
authors.forename: Felix T
authors.initials: FT
authors.lastname: Kurz
authors.affiliation: Department of Radiology, German Cancer Research Center-DKFZ, Im Neuenheimer Feld 280, 69120 Heidelberg, Germany.
authors.forename: Lukas R
authors.initials: LR
authors.lastname: Buschle
authors.affiliation: Department of Experimental Physics 5, University of Würzburg, Am Hubland, 97074 Würzburg, Germany.
authors.forename: Thomas
authors.initials: T
authors.lastname: Kampf
keywords: Bessel function; Integral; Linear combination