# §33.1 Special Notation

(For other notation see Notation for the Special Functions.)

$k,\ell$ nonnegative integers. real variables. nonnegative real variable. real parameters. logarithmic derivative of $\mathop{\Gamma\/}\nolimits\!\left(x\right)$; see §5.2(i). Dirac delta; see §1.17. derivatives with respect to the variable.

The main functions treated in this chapter are first the Coulomb radial functions $\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$, $\mathop{G_{\ell}\/}\nolimits\!\left(\eta,\rho\right)$, $\mathop{{H^{\pm}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$, $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$, $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$, $\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$ (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions.

## Alternative Notations

1. Curtis (1964a):

$P_{\ell}(\epsilon,r)=(2\ell+1)!\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r% \right)/2^{\ell+1}$, $Q_{\ell}(\epsilon,r)=-(2\ell+1)!\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r% \right)/(2^{\ell+1}A(\epsilon,\ell))$.

2. Greene et al. (1979):

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