§33.1 Special Notation

Referenced by:: §33.24, §33.5(iv)
Permalink:: http://dlmf.nist.gov/33.1

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

$k,\ell$	nonnegative integers.
$r,x$	real variables.
$\rho$	nonnegative real variable.
$\epsilon,\eta$	real parameters.
$\mathop{\psi\/}\nolimits\!\left(x\right)$	logarithmic derivative of $\mathop{\Gamma\/}\nolimits\!\left(x\right)$ ; see §5.2(i).
$\mathop{\delta\/}\nolimits\!\left(x\right)$	Dirac delta; see §1.17.
primes	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

Keywords:: Coulomb radial functions, Coulomb wave functions

Curtis (1964a):
$P_{\ell}(\epsilon,r)=(2\ell+1)!\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)/2^{{\ell+1}}$ , $Q_{\ell}(\epsilon,r)=-(2\ell+1)!\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)/(2^{{\ell+1}}A(\epsilon,\ell))$ .
Greene et al. (1979):
$f^{{(0)}}(\epsilon,\ell;r)=\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ , $f(\epsilon,\ell;r)=\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ , $g(\epsilon,\ell;r)=\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$ .