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 ... ... @@ -16,40 +16,45 @@ Mathematics in summary Any real function $f(r,\Omega)$ square-integrable on $\mathrm{B}^3 = \mathrm{R}^+ \times [0,\pi] \times [0,2\pi)$ can be decomposed as: $$\begin{eqnarray} f(r,\Omega) &=& \sum_{n=0}^\infty \sum_{l=0}^{\infty}\sum_{m=-l}^{l}\ f_{lmn}\ K_{lmn}(r,\Omega; \tau) \label{eq:FLagfulla}\\ math \begin{array}{rcl} f(r,\Omega) &=& \sum_{n=0}^\infty \sum_{l=0}^{\infty}\sum_{m=-l}^{l}\ f_{lmn}\ K_{lmn}(r,\Omega; \tau)\\ f_{lmn} &=& \int_{\mathrm{B}^3} dr d\Omega\ r^2\ f(r,\Omega)\ K^\ast_{lmn}(r,\Omega; \tau) \label{eq:FLagfullb} \end{eqnarray}$$ \end{array}  based on the orthogonality of the set of functions $$\begin{equation} K_{lmn}(r,\Omega; \tau) \equiv Y_{l,m}(\Omega)\times {\cal K}_n(r,\tau) \label{eq:FLagOrthoFunc} \end{equation}$$ using the spherical harmonic functions $Y_{l,m}$ and the genralized Laguerre functions ${\cal K}_n$: math K_{lmn}(r,\Omega; \tau) \equiv Y_{l,m}(\Omega)\times \mathcal{K}_n(r,\tau)  using the spherical harmonic functions $Y_{l,m}(\Omega)$ and the genralized Laguerre functions $\mathcal{K}_n$: $$\begin{eqnarray} math \begin{array}{rcl} Y_{l,m}(\Omega) &=& \lambda_{lm}(\theta) e^{i m\phi} = \sqrt{\frac{2 l +1}{4 \pi}\frac{(l-m)!}{(l+m)!}}P_l^m({\cos \theta}) e^{i m\phi} \\ {\cal K}_n(r,\tau) &=& \tau^{-3/2} \sqrt{\frac{n!}{(n+\alpha)!}} e^{-r/2\tau} \left(\frac{r}{\tau}\right)^{\frac{\alpha}{2}-1} L_n^{(\alpha)}(r/\tau) = \tau^{-3/2} {\cal K}_n(r/\tau,1) \end{eqnarray}$$ \mathcal{K}_n(r,\tau) &=& \tau^{-3/2} \sqrt{\frac{n!}{(n+\alpha)!}} e^{-r/2\tau} \left(\frac{r}{\tau}\right)^{\frac{\alpha}{2}-1} L_n^{(\alpha)}(r/\tau) = \tau^{-3/2} \mathcal{K}_n(r/\tau,1) \end{array} ` with $L_n^{(\alpha)}(r)$ the generalized Laguerre polynomials (notice that by default $\alpha = 2$ but one is free to use real numbers). The $\tau$-scaling parameter is a free parameter but can be nicely set to $\tau = R_{max}/r_{N-1}$ with $R_{max}$ the limit in radial direction of the function $f(r,\Omega)$, and $r_{N-1}$ the largest node of the Gauss-Laguerre quadrature (see below). For band-limited functions the discret sums are truncated with \$ l
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