33 Coulomb Functions Variables $r,\epsilon$ 33.16 Connection Formulas 33.18 Limiting Forms for Large $\ell$

§33.17 Recurrence Relations and Derivatives

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

33.17.1 $(\ell+1)r\mathop{f\/}\nolimits\!\left(\epsilon,\ell-1;r\right)-(2\ell+1)\left(% \ell(\ell+1)-r\right)\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)+\ell% \left(1+(\ell+1)^{2}\epsilon\right)r\mathop{f\/}\nolimits\!\left(\epsilon,\ell% +1;r\right)=0,$

Symbols:: $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\ell$ : nonnegative integer, : real variable and $\epsilon$ : real parameter
Permalink:: http://dlmf.nist.gov/33.17.E1
Encodings:: TeX, pMML, png

33.17.2 $(\ell+1)\left(1+\ell^{2}\epsilon\right)r\mathop{h\/}\nolimits\!\left(\epsilon,% \ell-1;r\right)-(2\ell+1)\left(\ell(\ell+1)-r\right)\mathop{h\/}\nolimits\!% \left(\epsilon,\ell;r\right)+\ell r\mathop{h\/}\nolimits\!\left(\epsilon,\ell+% 1;r\right)=0,$

Symbols:: $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $\ell$ : nonnegative integer, : real variable and $\epsilon$ : real parameter
Permalink:: http://dlmf.nist.gov/33.17.E2
Encodings:: TeX, pMML, png

33.17.3 $(\ell+1)r{\mathop{f\/}\nolimits^{{\prime}}}\!\left(\epsilon,\ell;r\right)=% \left((\ell+1)^{2}-r\right)\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)% -{\left(1+(\ell+1)^{2}\epsilon\right)r\mathop{f\/}\nolimits\!\left(\epsilon,% \ell+1;r\right),}$

Symbols:: $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\ell$ : nonnegative integer, : real variable and $\epsilon$ : real parameter
Permalink:: http://dlmf.nist.gov/33.17.E3
Encodings:: TeX, pMML, png

33.17.4 $(\ell+1)r{\mathop{h\/}\nolimits^{{\prime}}}\!\left(\epsilon,\ell;r\right)=% \left((\ell+1)^{2}-r\right)\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)% -r\mathop{h\/}\nolimits\!\left(\epsilon,\ell+1;r\right).$

Symbols:: $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $\ell$ : nonnegative integer, : real variable and $\epsilon$ : real parameter
Permalink:: http://dlmf.nist.gov/33.17.E4
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 33.16 Connection Formulas 33.18 Limiting Forms for Large $\ell$