r,\epsilon

\ell

§33.17 Recurrence Relations and Derivatives

ⓘ

Keywords:: Coulomb functions: variables $r,\epsilon$ , derivatives, recurrence relations
Notes:: See Seaton (2002a, Eqs. 77, 78, 82).
Referenced by:: §33.23(iv), §33.23(iv)
Permalink:: http://dlmf.nist.gov/33.17
See also:: Annotations for Ch.33

33.17.1		$(\ell+1)rf\left(\epsilon,\ell-1;r\right)-(2\ell+1)\left(\ell(\ell+1)-r\right)f% \left(\epsilon,\ell;r\right)+\ell\left(1+(\ell+1)^{2}\epsilon\right)rf\left(% \epsilon,\ell+1;r\right)=0,$
ⓘ Symbols: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\ell$ : nonnegative integer, $r$ : real variable and $\epsilon$ : real parameter Permalink: http://dlmf.nist.gov/33.17.E1 Encodings: TeX, pMML, png See also: Annotations for §33.17 and Ch.33

33.17.2		$(\ell+1)\left(1+\ell^{2}\epsilon\right)rh\left(\epsilon,\ell-1;r\right)-(2\ell% +1)\left(\ell(\ell+1)-r\right)h\left(\epsilon,\ell;r\right)+\ell rh\left(% \epsilon,\ell+1;r\right)=0,$
ⓘ Symbols: $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $\ell$ : nonnegative integer, $r$ : real variable and $\epsilon$ : real parameter Permalink: http://dlmf.nist.gov/33.17.E2 Encodings: TeX, pMML, png See also: Annotations for §33.17 and Ch.33

33.17.3	$\displaystyle(\ell+1)rf'\left(\epsilon,\ell;r\right)$	$\displaystyle=\left((\ell+1)^{2}-r\right)f\left(\epsilon,\ell;r\right)-\left(1% +(\ell+1)^{2}\epsilon\right)rf\left(\epsilon,\ell+1;r\right),$
ⓘ Symbols: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\ell$ : nonnegative integer, $r$ : real variable and $\epsilon$ : real parameter Permalink: http://dlmf.nist.gov/33.17.E3 Encodings: TeX, pMML, png See also: Annotations for §33.17 and Ch.33
33.17.4	$\displaystyle(\ell+1)rh'\left(\epsilon,\ell;r\right)$	$\displaystyle=\left((\ell+1)^{2}-r\right)h\left(\epsilon,\ell;r\right)-rh\left% (\epsilon,\ell+1;r\right).$
ⓘ Symbols: $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $\ell$ : nonnegative integer, $r$ : real variable and $\epsilon$ : real parameter Permalink: http://dlmf.nist.gov/33.17.E4 Encodings: TeX, pMML, png See also: Annotations for §33.17 and Ch.33

\ell

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