# §33.21 Asymptotic Approximations for Large $|r|$

## §33.21(i) Limiting Forms

We indicate here how to obtain the limiting forms of $f\left(\epsilon,\ell;r\right)$, $h\left(\epsilon,\ell;r\right)$, $s\left(\epsilon,\ell;r\right)$, and $c\left(\epsilon,\ell;r\right)$ as $r\to\pm\infty$, with $\epsilon$ and $\ell$ fixed, in the following cases:

1. (a)

When $r\to\pm\infty$ with $\epsilon>0$, Equations (33.16.4)–(33.16.7) are combined with (33.10.1).

2. (b)

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

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

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).

3. (c)

When $r\to\pm\infty$ with $\epsilon=0$, combine (33.20.1), (33.20.2) with §§10.7(ii), 10.30(ii).

## §33.21(ii) Asymptotic Expansions

For asymptotic expansions of $f\left(\epsilon,\ell;r\right)$ and $h\left(\epsilon,\ell;r\right)$ as $r\to\pm\infty$ with $\epsilon$ and $\ell$ fixed, see Curtis (1964a, §6).