33 Coulomb Functions Notation 33 Coulomb Functions 33.2 Definitions and Basic Properties

§33.1 Special Notation

ⓘ

Referenced by:: 2nd item, §33.5(iv)
Permalink:: http://dlmf.nist.gov/33.1
See also:: Annotations for Ch.33

(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.
$\psi\left(x\right)$	logarithmic derivative of $\Gamma\left(x\right)$ ; see §5.2(i).
$\delta\left(x\right)$	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 $F_{\ell}\left(\eta,\rho\right)$ , $G_{\ell}\left(\eta,\rho\right)$ , ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$ (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.

Alternative Notations

ⓘ

Keywords:: Coulomb radial functions, Coulomb wave functions
See also:: Annotations for §33.1 and Ch.33

Curtis (1964a):

$P_{\ell}(\epsilon,r)=(2\ell+1)!f\left(\epsilon,\ell;r\right)/2^{\ell+1}$ , $Q_{\ell}(\epsilon,r)=-(2\ell+1)!h\left(\epsilon,\ell;r\right)/(2^{\ell+1}A(% \epsilon,\ell))$ .
Greene et al. (1979):

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

33 Coulomb Functions 33.2 Definitions and Basic Properties

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§33.1 Special Notation

Alternative Notations

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!