# Coulomb radial functions

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##### 2: 33.5 Limiting Forms for Small $\rho$, Small $|\eta|$, or Large $\ell$
33.5.6 $C_{\ell}\left(0\right)=\frac{2^{\ell}\ell!}{(2\ell+1)!}=\frac{1}{(2\ell+1)!!}.$
33.5.9 $C_{\ell}\left(\eta\right)\sim\dfrac{e^{-\pi\eta/2}}{(2\ell+1)!!}\sim e^{-\pi% \eta/2}\dfrac{e^{\ell}}{\sqrt{2}(2\ell)^{\ell+1}}.$
##### 3: 33.1 Special Notation
The main functions treated in this chapter are first the Coulomb radial functions $F_{\ell}\left(\eta,\rho\right)$, $G_{\ell}\left(\eta,\rho\right)$, ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$ (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions $f\left(\epsilon,\ell;r\right)$, $h\left(\epsilon,\ell;r\right)$, $s\left(\epsilon,\ell;r\right)$, $c\left(\epsilon,\ell;r\right)$ (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions. …
• Greene et al. (1979):

$f^{(0)}(\epsilon,\ell;r)=f\left(\epsilon,\ell;r\right)$, $f(\epsilon,\ell;r)=s\left(\epsilon,\ell;r\right)$, $g(\epsilon,\ell;r)=c\left(\epsilon,\ell;r\right)$.

• ##### 4: 33.2 Definitions and Basic Properties
33.2.2 $\rho_{\operatorname{tp}}\left(\eta,\ell\right)=\eta+(\eta^{2}+\ell(\ell+1))^{1% /2}.$
33.2.3 $F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)2^{-\ell-1}(\mp\mathrm% {i})^{\ell+1}M_{\pm\mathrm{i}\eta,\ell+\frac{1}{2}}\left(\pm 2\mathrm{i}\rho% \right),$
33.2.12 $\mathscr{W}\left\{G_{\ell},F_{\ell}\right\}=\mathscr{W}\left\{{H^{\pm}_{\ell}}% ,F_{\ell}\right\}=1.$
##### 5: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$
33.10.2 ${H^{\pm}_{\ell}}\left(\eta,\rho\right)\sim\exp\left(\pm\mathrm{i}{\theta_{\ell% }}\left(\eta,\rho\right)\right),$
##### 6: 33.13 Complex Variable and Parameters
33.13.1 $C_{\ell}\left(\eta\right)=2^{\ell}e^{\mathrm{i}{\sigma_{\ell}}\left(\eta\right% )-(\pi\eta/2)}\Gamma\left(\ell+1-\mathrm{i}\eta\right)/\Gamma\left(2\ell+2% \right),$
##### 7: 33.8 Continued Fractions
33.8.2 $\frac{{H^{\pm}_{\ell}}'}{{H^{\pm}_{\ell}}}=c\pm\frac{\mathrm{i}}{\rho}\cfrac{% ab}{2(\rho-\eta\pm\mathrm{i})+\cfrac{(a+1)(b+1)}{2(\rho-\eta\pm 2\mathrm{i})+% \cdots}},$
##### 8: 33.6 Power-Series Expansions in $\rho$
33.6.1 $F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\sum_{k=\ell+1}^{% \infty}A_{k}^{\ell}(\eta)\rho^{k},$
33.6.2 $F_{\ell}'\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\sum_{k=\ell+1}^{% \infty}kA_{k}^{\ell}(\eta)\rho^{k-1},$
33.6.5 ${H^{\pm}_{\ell}}\left(\eta,\rho\right)=\frac{e^{\pm\mathrm{i}{\theta_{\ell}}% \left(\eta,\rho\right)}}{(2\ell+1)!\Gamma\left(-\ell\pm\mathrm{i}\eta\right)}% \left(\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}}{{\left(2\ell+2\right)_{k}% }k!}(\mp 2\mathrm{i}\rho)^{a+k}\left(\ln\left(\mp 2\mathrm{i}\rho\right)+\psi% \left(a+k\right)-\psi\left(1+k\right)-\psi\left(2\ell+2+k\right)\right)-\sum_{% k=1}^{2\ell+1}\frac{(2\ell+1)!(k-1)!}{(2\ell+1-k)!{\left(1-a\right)_{k}}}(\mp 2% \mathrm{i}\rho)^{a-k}\right),$
##### 9: 33.7 Integral Representations
33.7.1 $F_{\ell}\left(\eta,\rho\right)=\frac{\rho^{\ell+1}2^{\ell}e^{\mathrm{i}\rho-(% \pi\eta/2)}}{|\Gamma\left(\ell+1+\mathrm{i}\eta\right)|}\int_{0}^{1}e^{-2% \mathrm{i}\rho t}t^{\ell+\mathrm{i}\eta}(1-t)^{\ell-\mathrm{i}\eta}\,\mathrm{d% }t,$
33.7.2 ${H^{-}_{\ell}}\left(\eta,\rho\right)=\frac{e^{-\mathrm{i}\rho}\rho^{-\ell}}{(2% \ell+1)!C_{\ell}\left(\eta\right)}\int_{0}^{\infty}e^{-t}t^{\ell-\mathrm{i}% \eta}(t+2\mathrm{i}\rho)^{\ell+\mathrm{i}\eta}\,\mathrm{d}t,$
33.7.3 ${H^{-}_{\ell}}\left(\eta,\rho\right)=\frac{-\mathrm{i}e^{-\pi\eta}\rho^{\ell+1% }}{(2\ell+1)!C_{\ell}\left(\eta\right)}\int_{0}^{\infty}\left(\frac{\exp\left(% -\mathrm{i}(\rho\tanh t-2\eta t)\right)}{(\cosh t)^{2\ell+2}}+\mathrm{i}(1+t^{% 2})^{\ell}\exp\left(-\rho t+2\eta\operatorname{arctan}t\right)\right)\,\mathrm% {d}t,$
33.7.4 ${H^{+}_{\ell}}\left(\eta,\rho\right)=\frac{\mathrm{i}e^{-\pi\eta}\rho^{\ell+1}% }{(2\ell+1)!C_{\ell}\left(\eta\right)}\int_{-1}^{-\mathrm{i}\infty}e^{-\mathrm% {i}\rho t}(1-t)^{\ell-\mathrm{i}\eta}(1+t)^{\ell+\mathrm{i}\eta}\,\mathrm{d}t.$
##### 10: 33.11 Asymptotic Expansions for Large $\rho$
33.11.1 ${H^{\pm}_{\ell}}\left(\eta,\rho\right)\sim e^{\pm\mathrm{i}{\theta_{\ell}}% \left(\eta,\rho\right)}\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{\left(b% \right)_{k}}}{k!(\pm 2\mathrm{i}\rho)^{k}},$