§33.21 Asymptotic Approximations for Large $|r|$

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

1. (a)

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

2. (b)

   When $r\to \pm \mathrm{\infty }$ with , Equations (33.16.10)–(33.16.13) are combined with

   33.21.1		${\zeta }_{\mathrm{\ell }}(\nu ,r)$	$\sim {\mathrm{e}}^{-r/\nu }{(2r/\nu )}^{\nu },$
   		${\xi }_{\mathrm{\ell }}(\nu ,r)$	$\sim {\mathrm{e}}^{r/\nu }{(2r/\nu )}^{-\nu }$,
   	$r\to \mathrm{\infty }$,
   ⓘ Symbols: $\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function Permalink: http://dlmf.nist.gov/33.21.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.21(i), §33.21 and Ch.33

   33.21.2		${\zeta }_{\mathrm{\ell }}(-\nu ,r)$	$\sim {\mathrm{e}}^{r/\nu }{(-2r/\nu )}^{-\nu },$
   		${\xi }_{\mathrm{\ell }}(-\nu ,r)$	$\sim {\mathrm{e}}^{-r/\nu }{(-2r/\nu )}^{\nu },$
   	$r\to -\mathrm{\infty }$.
   ⓘ Symbols: $\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $\nu$: parameter, ${\zeta }_{\mathrm{\ell }}(\nu ,r)$: function and ${\xi }_{\mathrm{\ell }}(\nu ,r)$: function Permalink: http://dlmf.nist.gov/33.21.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.21(i), §33.21 and Ch.33

   Corresponding approximations for $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ as $r\to \mathrm{\infty }$ can be obtained via (33.16.17), and as $r\to -\mathrm{\infty }$ via (33.16.18).

3. (c)

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

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