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A. Bessel function identities

In this appendix we list some standard results used in the analysis of chapter 5 [186].
    $\displaystyle j_{\ell + 1}(x) = {\ell \over x} j_{\ell}(x) - j_{\ell}'(x)$ (A.1)
    $\displaystyle j_{\ell - 1}(x) = {{\ell + 1} \over x} j_{\ell}(x) + j_{\ell}'(x)$ (A.2)
    $\displaystyle \exp[{\mathrm{i}} {\bf k} \cdot {\bf r}] = 4 \pi
...\ell}(k r)~{\bar Y}_{\ell m} (\Omega_{\bf k})~{\bar Y}_{\ell m}(\Omega_{\bf r})$ (A.3)
    $\displaystyle \int_{a}^{b} j_{\ell}(mx) j_{\ell}(nx) x^2 {\mathrm d}x$  
$\displaystyle *$   $\displaystyle \qquad
= \frac{1}{m^2 - n^2} \left[ x^2 \left\{ n j_{\ell}(m x) j_{\ell-1}(n
x) - m j_{\ell-1}(m x) j_{\ell}(n x) \right\} \right]_{a}^{b}$ (A.4)
    $\displaystyle \int_{a}^{b} j_{\ell}^2 (m x) x^2 {\mathrm d}x =
{\textstyle{1 \o...
...biggl[ x^2 \biggl\{ x j_{\ell}^2 (m x) +
x j_{\ell - 1}^2 (m x)
\biggr. \biggr.$  
$\displaystyle *$   $\displaystyle \biggl. \biggl. \qquad - {2\ell + 1
\over m} j_{\ell - 1} (m x) j_{\ell} (m x) \biggr\} \biggr]_{a}^{b}$ (A.5)

Peter Haynes