Coulomb functions: variables r,ϵ

(0.012 seconds)

1—10 of 29 matching pages

1: 33.20 Expansions for Small $|\epsilon|$
§33.20(iii) Asymptotic Expansion for the Irregular Solution
where $A(\epsilon,\ell)$ is given by (33.14.11), (33.14.12), and …
2: 33.17 Recurrence Relations and Derivatives
§33.17 Recurrence Relations and Derivatives
33.17.1 $(\ell+1)rf\left(\epsilon,\ell-1;r\right)-(2\ell+1)\left(\ell(\ell+1)-r\right)f% \left(\epsilon,\ell;r\right)+\ell\left(1+(\ell+1)^{2}\epsilon\right)rf\left(% \epsilon,\ell+1;r\right)=0,$
33.17.2 $(\ell+1)\left(1+\ell^{2}\epsilon\right)rh\left(\epsilon,\ell-1;r\right)-(2\ell% +1)\left(\ell(\ell+1)-r\right)h\left(\epsilon,\ell;r\right)+\ell rh\left(% \epsilon,\ell+1;r\right)=0,$
33.17.3 $(\ell+1)rf'\left(\epsilon,\ell;r\right)=\left((\ell+1)^{2}-r\right)f\left(% \epsilon,\ell;r\right)-\left(1+(\ell+1)^{2}\epsilon\right)rf\left(\epsilon,% \ell+1;r\right),$
33.17.4 $(\ell+1)rh'\left(\epsilon,\ell;r\right)=\left((\ell+1)^{2}-r\right)h\left(% \epsilon,\ell;r\right)-rh\left(\epsilon,\ell+1;r\right).$
6: 33.21 Asymptotic Approximations for Large $|r|$
§33.21(i) Limiting Forms
• (b)

When $r\to\pm\infty$ with $\epsilon<0$, Equations (33.16.10)–(33.16.13) are combined with

33.21.1
$\zeta_{\ell}(\nu,r)\sim e^{-r/\nu}(2r/\nu)^{\nu},$
$\xi_{\ell}(\nu,r)\sim e^{r/\nu}(2r/\nu)^{-\nu}$ , $r\to\infty$,
33.21.2
$\zeta_{\ell}(-\nu,r)\sim e^{r/\nu}(-2r/\nu)^{-\nu},$
$\xi_{\ell}(-\nu,r)\sim e^{-r/\nu}(-2r/\nu)^{\nu},$ $r\to-\infty$.

Corresponding approximations for $s\left(\epsilon,\ell;r\right)$ and $c\left(\epsilon,\ell;r\right)$ as $r\to\infty$ can be obtained via (33.16.17), and as $r\to-\infty$ via (33.16.18).