33 Coulomb Functions Variables

\rho,\eta

33.3 Graphics 33.5 Limiting Forms for Small

\rho

, Small

|\eta|

, or Large

\ell

§33.4 Recurrence Relations and Derivatives

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , derivatives, recurrence relations
Notes:: See Powell (1947).
Referenced by:: §33.23(iv), §33.23(iv)
Permalink:: http://dlmf.nist.gov/33.4
See also:: Annotations for Ch.33

For $\ell=1,2,3,\dots$ , let

33.4.1	$\displaystyle R_{\ell}$	$\displaystyle=\sqrt{1+\dfrac{\eta^{2}}{\ell^{2}}},$
	$\displaystyle S_{\ell}$	$\displaystyle=\dfrac{\ell}{\rho}+\dfrac{\eta}{\ell},$
	$\displaystyle T_{\ell}$	$\displaystyle=S_{\ell}+S_{\ell+1}.$
ⓘ Defines: $R_{\ell}$ : factor (locally), $S_{\ell}$ : factor (locally) and $T_{\ell}$ : factor (locally) Symbols: $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter Referenced by: §33.13, §33.8 Permalink: http://dlmf.nist.gov/33.4.E1 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §33.4 and Ch.33

Then, with $X_{\ell}$ denoting any of $F_{\ell}\left(\eta,\rho\right)$ , $G_{\ell}\left(\eta,\rho\right)$ , or ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$ ,

33.4.2		$R_{\ell}X_{\ell-1}-T_{\ell}X_{\ell}+R_{\ell+1}X_{\ell+1}=0,$
$\ell\geq 1$ ,
ⓘ Symbols: $\ell$ : nonnegative integer, $R_{\ell}$ : factor, $T_{\ell}$ : factor and $X_{\ell}$ : any Coulomb function A&S Ref: 14.2.3 Permalink: http://dlmf.nist.gov/33.4.E2 Encodings: TeX, pMML, png See also: Annotations for §33.4 and Ch.33

33.4.3		$X_{\ell}^{\prime}=R_{\ell}X_{\ell-1}-S_{\ell}X_{\ell},$
$\ell\geq 1$ ,
ⓘ Symbols: $\ell$ : nonnegative integer, $R_{\ell}$ : factor, $S_{\ell}$ : factor and $X_{\ell}$ : any Coulomb function A&S Ref: 14.2.1 Referenced by: §33.6 Permalink: http://dlmf.nist.gov/33.4.E3 Encodings: TeX, pMML, png See also: Annotations for §33.4 and Ch.33

33.4.4		$X_{\ell}^{\prime}=S_{\ell+1}X_{\ell}-R_{\ell+1}X_{\ell+1},$
$\ell\geq 0$ .
ⓘ Symbols: $\ell$ : nonnegative integer, $R_{\ell}$ : factor, $S_{\ell}$ : factor and $X_{\ell}$ : any Coulomb function A&S Ref: 14.2.2 Referenced by: §33.6 Permalink: http://dlmf.nist.gov/33.4.E4 Encodings: TeX, pMML, png See also: Annotations for §33.4 and Ch.33

33.3 Graphics 33.5 Limiting Forms for Small

\rho

, Small

|\eta|

, or Large

\ell

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§33.4 Recurrence Relations and Derivatives

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