# Coulomb functions

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##### 2: 33.22 Particle Scattering and Atomic and Molecular Spectra
###### §33.22(i) Schrödinger Equation
Attractive potentials: $Z_{1}Z_{2}<0$, $\eta<0$. …
Attractive potentials: $Z_{1}Z_{2}<0$, $\Im\kappa<0$. …
##### 3: 33.14 Definitions and Basic Properties
###### §33.14(iii) Irregular Solution $h\left(\epsilon,\ell;r\right)$
For nonzero values of $\epsilon$ and $r$ the function $h\left(\epsilon,\ell;r\right)$ is defined by … The function $s\left(\epsilon,\ell;r\right)$ has the following properties: …
##### 5: 33.2 Definitions and Basic Properties
###### §33.2(i) Coulomb Wave Equation
The function $F_{\ell}\left(\eta,\rho\right)$ is recessive (§2.7(iii)) at $\rho=0$, and is defined by … The functions ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$ are defined by … …
##### 6: 33.20 Expansions for Small $|\epsilon|$
###### §33.20(iii) Asymptotic Expansion for the Irregular Solution
where $A(\epsilon,\ell)$ is given by (33.14.11), (33.14.12), and …
##### 8: 33.17 Recurrence Relations and Derivatives
###### §33.17 Recurrence Relations and Derivatives
33.17.1 $(\ell+1)rf\left(\epsilon,\ell-1;r\right)-(2\ell+1)\left(\ell(\ell+1)-r\right)f% \left(\epsilon,\ell;r\right)+\ell\left(1+(\ell+1)^{2}\epsilon\right)rf\left(% \epsilon,\ell+1;r\right)=0,$
33.17.2 $(\ell+1)\left(1+\ell^{2}\epsilon\right)rh\left(\epsilon,\ell-1;r\right)-(2\ell% +1)\left(\ell(\ell+1)-r\right)h\left(\epsilon,\ell;r\right)+\ell rh\left(% \epsilon,\ell+1;r\right)=0,$
33.17.3 $(\ell+1)rf'\left(\epsilon,\ell;r\right)=\left((\ell+1)^{2}-r\right)f\left(% \epsilon,\ell;r\right)-\left(1+(\ell+1)^{2}\epsilon\right)rf\left(\epsilon,% \ell+1;r\right),$
33.17.4 $(\ell+1)rh'\left(\epsilon,\ell;r\right)=\left((\ell+1)^{2}-r\right)h\left(% \epsilon,\ell;r\right)-rh\left(\epsilon,\ell+1;r\right).$
##### 10: 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):

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