Spherical Bessel functions

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 13.2, p. 274

Section A13.2.1.4. Spherical Bessel functions

J. Navaza^a ^*

^aLaboratoire de Génétique des Virus, CNRS-GIF, 1. Avenue de la Terrasse, 91198 Gif-sur-Yvette, France
Correspondence e-mail: jnavaza@pasteur.fr

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The $[j_{\ell}]$ 's, with $[0 \leq \ell \lt \infty]$ , constitute a complete set of functions having the following properties (Watson, 1958):

(1) Recurrence relation: $[j_{\ell-1}(x) - (2\ell+1) [\ j_{\ell}(x)/x] + j_{\ell+1}(x) = 0. \eqno\hbox{(A13.2.1.12)}]$
(2) Initial values: $[\eqalignno{j_{0}(x) &= \sin(x)/x\cr j_{1}(x) &= [\sin(x) - x \ \cos(x)]/x^{2}.&(\hbox{A}13.2.1.13)}]$
(3) Integral of a product of spherical Bessel functions: $[\eqalignno{U^{\ell}(\;p, q) &= {\textstyle\int\limits_{0}^{1}} j_{\ell}(\;px) j_{\ell}(qx) x^{2}\ \hbox{d}x\cr &= \cases{[\ j_{\ell}(p)j_{\ell - 1}(q)q - j_{\ell}(q)j_{\ell - 1}(\;p)p] / (\;p^{2} - q^{2})\;\; \hbox{if } {p} \neq {q} \cr {1 \over 2} [\ j_{\ell}(p)^{2} - j_{\ell - 1}(\;p)j_{\ell + 1}(\;p)] {\hbox to 5.3pc{}} \hbox{if } {\it p} = {\it q} \cr}\cr &= (2\ell + 3) [\ j_{\ell + 1}(\;p) j_{\ell + 1}(q)]/pq + U^{\ell + 2}(\;p, q)\cr &= {\textstyle\sum\limits_{n = 1}^{\infty}} [2(\ell + 2n)-1] [\ j_{\ell + 2n-1}(\;p)j_{\ell + 2n-1}(q)]/pq.\cr& &(\hbox{A}13.2.1.14)}]$

Watson, G. N. (1958). A treatise on the theory of Bessel functions, 2nd ed. Cambridge University Press.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 13.2, p. 274