A. Bessel function identities

In this appendix we list some standard results used in the analysis of chapter 5 [186].

$$j_{\ell + 1}(x) = {\ell \over x} j_{\ell}(x) - j_{\ell}'(x)$$ (A.1)

$$j_{\ell - 1}(x) = {{\ell + 1} \over x} j_{\ell}(x) + j_{\ell}'(x)$$ (A.2)

$$\exp[{\mathrm{i}} {\bf k} \cdot {\bf r}] = 4 \pi \sum_{\ell=0}^{\... ...\ell}(k r)~{\bar Y}_{\ell m} (\Omega_{\bf k})~{\bar Y}_{\ell m}(\Omega_{\bf r})$$ (A.3)

$$\int_{a}^{b} j_{\ell}(mx) j_{\ell}(nx) x^2 {\mathrm d}x$$

$$* \qquad = \frac{1}{m^2 - n^2} \left[ x^2 \left\{ n j_{\ell}(m x) j_{\ell-1}(n x) - m j_{\ell-1}(m x) j_{\ell}(n x) \right\} \right]_{a}^{b}$$ (A.4)

$$\int_{a}^{b} j_{\ell}^2 (m x) x^2 {\mathrm d}x = {\textstyle{1 \o... ...biggl[ x^2 \biggl\{ x j_{\ell}^2 (m x) + x j_{\ell - 1}^2 (m x) \biggr. \biggr.$$

$$* \biggl. \biggl. \qquad - {2\ell + 1 \over m} j_{\ell - 1} (m x) j_{\ell} (m x) \biggr\} \biggr]_{a}^{b}$$ (A.5)