§33.21 Asymptotic Approximations for Large $|r|$

Referenced by:: §33.23(ii), §33.23(iii), §33.23(iii)
Permalink:: http://dlmf.nist.gov/33.21

Contents

§33.21(i) Limiting Forms
§33.21(ii) Asymptotic Expansions

§33.21(i) Limiting Forms

Notes:: See Seaton (2002a, Eqs. 104, 107), or apply (13.14.21) to (33.16.9).
Keywords:: Coulomb functions: variables $r,\epsilon$
Permalink:: http://dlmf.nist.gov/33.21.i

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

1

When $r\to\pm\infty$ with $\epsilon>0$ , Equations (33.16.4)–(33.16.7) are combined with (33.10.1).
2

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

33.21.1
$\zeta _{\ell}(\nu,r)\sim e^{{-r/\nu}}(2r/\nu)^{\nu},$

$\xi _{\ell}(\nu,r)\sim e^{{r/\nu}}(2r/\nu)^{{-\nu}}$ , $r\to\infty$ ,

Symbols:

$e$ : base of exponential function, $\sim$ : asymptotic equality, $\ell$ : nonnegative integer, $r$ : real variable, $\nu$ : parameter, $\zeta _{\ell}(\nu,r)$ : function and $\xi _{\ell}(\nu,r)$ : function

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Encodings:

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33.21.2
$\zeta _{\ell}(-\nu,r)\sim e^{{r/\nu}}(-2r/\nu)^{{-\nu}},$

$\xi _{\ell}(-\nu,r)\sim e^{{-r/\nu}}(-2r/\nu)^{\nu},$ $r\to-\infty$ .

Symbols:

$e$ : base of exponential function, $\sim$ : asymptotic equality, $\ell$ : nonnegative integer, $r$ : real variable, $\nu$ : parameter, $\zeta _{\ell}(\nu,r)$ : function and $\xi _{\ell}(\nu,r)$ : function

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Encodings:

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Corresponding approximations for $\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ and $\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$ as $r\to\infty$ can be obtained via (33.16.17), and as $r\to-\infty$ via (33.16.18).
3

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

Keywords:: Coulomb functions: variables $r,\epsilon$
Permalink:: http://dlmf.nist.gov/33.21.ii

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