### (JEC) 7/10/15 complete the documentation for tau-factor and Fourier-Bessel transform.

parent 0ceb5d88
 ... ... @@ -15,28 +15,28 @@ Mathematics in summary Any 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) \label{eq:FLagfulla}\\ f_{lmn} &=& \int_{\mathrm{B}^3} dr d\Omega\ r^2\ f(r,\Omega)\ K^\ast_{lmn}(r,\Omega) 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}\\ 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}\$\$ based on the orthogonality of the set of functions \$\$\begin{equation} K_{lmn}(r,\Omega) \equiv Y_{l,m}(\Omega)\times {\cal L}_n(r) 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 L}_n\$ using the spherical harmonic functions \$Y_{l,m}\$ and the genralized Laguerre functions \${\cal K}_n\$: \$\$\begin{eqnarray} 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 L}_n(r) &=& \sqrt{\frac{n!}{(n+\alpha)!}} e^{-r/2} L_n^{(\alpha)}(r) {\cal K}_n(r,\tau) &=& \sqrt{\frac{n!}{(n+\alpha)!}} \frac{e^{-r/2\tau} L_n^{(\alpha)}(r/\tau)}{\tau^{3/2}} \end{eqnarray}\$\$ with \$ L_n^{(\alpha)}(r)\$ the generalized Laguerre polynomials (notice that by default \$ \alpha = 2\$). with \$ L_n^{(\alpha)}(r)\$ the generalized Laguerre polynomials (notice that by default \$ \alpha = 2\$). 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 bnad-limited functions the discret sums are truncated with \$ l
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