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

§33.1 Special Notation

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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

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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

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