Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions
The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval
Gespeichert in:
| Hauptverfasser: | , , , |
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| Dokumenttyp: | Article (Journal) |
| Sprache: | Englisch |
| Veröffentlicht: |
04 August 2015
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| In: |
SpringerPlus
Year: 2015, Jahrgang: 4, Heft: 1, Pages: 1-26 |
| ISSN: | 2193-1801 |
| DOI: | 10.1186/s40064-015-1142-0 |
| Online-Zugang: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1186/s40064-015-1142-0
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| Verfasserangaben: | Christian H. Ziener, Felix T. Kurz, Lukas R. Buschle and Thomas Kampf |
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| 245 | 1 | 0 | |a Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions |c Christian H. Ziener, Felix T. Kurz, Lukas R. Buschle and Thomas Kampf |
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| 520 | |a The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval |r \le r \le \gamma R$$R≤r≤γRwith Neumann boundary conditions are considered. The eigenfunctions are linear combinations of the Bessel function $$\Phi _{n,\nu }(r)=Y_{\nu }^{\prime }(\lambda _{n,\nu }) J_{\nu }(\lambda _{n,\nu } r/R)-J_{\nu }^{\prime }(\lambda _{n,\nu }) Y_{\nu }(\lambda _{n,\nu } r/R)$$Φn,ν(r)=Yν′(λn,ν)Jν(λn,νr/R)-Jν′(λn,ν)Yν(λn,νr/R)or linear combinations of the spherical Bessel functions $$\psi _{m,\nu }(r)=y_{\nu }^{\prime }(\lambda _{m,\nu }) j_{\nu }(\lambda _{m,\nu } r/R)-j_{\nu }^{\prime }(\lambda _{m,\nu }) y_{\nu }(\lambda _{m,\nu } 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_{\nu }^{\prime }(\lambda _{n,\nu }) J_{\nu }^{\prime }(\gamma \lambda _{n,\nu })-J_{\nu }^{\prime }(\lambda _{n,\nu }) Y_{\nu }^{\prime }(\gamma \lambda _{n,\nu }) = 0$$Yν′(λn,ν)Jν′(γλn,ν)-Jν′(λn,ν)Yν′(γλn,ν)=0and $$y_{\nu }^{\prime }(\lambda _{m,\nu }) j_{\nu }^{\prime }(\gamma \lambda _{m,\nu })-j_{\nu }^{\prime }(\lambda _{m,\nu }) y_{\nu }^{\prime }(\gamma \lambda _{m,\nu }) = 0$$yν′(λm,ν)jν′(γλm,ν)-jν′(λm,ν)yν′(γλm,ν)=0are considered in the complex plane for real as well as complex values of the index $$\nu $$νand approximations for the exceptional zero $$\lambda _{1,\nu }$$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. | ||
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