r,\epsilon

r

33.21 Asymptotic Approximations for Large

|r|

§33.20 Expansions for Small $|\epsilon|$

ⓘ

Permalink:: http://dlmf.nist.gov/33.20
See also:: Annotations for Ch.33

§33.20(i) Case $\epsilon=0$
§33.20(ii) Power-Series in $\epsilon$ for the Regular Solution
§33.20(iii) Asymptotic Expansion for the Irregular Solution
§33.20(iv) Uniform Asymptotic Expansions

§33.20(i) Case $\epsilon=0$

ⓘ

Keywords:: Coulomb functions: variables $r,\epsilon$ , case $\epsilon=0$ , expansions in Bessel functions, expansions in modified Bessel functions
Notes:: See Seaton (2002a, Eqs. 96, 98, 100, 102(corrected)).
Permalink:: http://dlmf.nist.gov/33.20.i
See also:: Annotations for §33.20 and Ch.33

33.20.1		$\displaystyle f\left(0,\ell;r\right)$	$\displaystyle=(2r)^{1/2}J_{2\ell+1}\left(\sqrt{8r}\right),$
		$\displaystyle h\left(0,\ell;r\right)$	$\displaystyle=-(2r)^{1/2}Y_{2\ell+1}\left(\sqrt{8r}\right)$ ,
	$r>0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $\ell$ : nonnegative integer and $r$ : real variable Referenced by: item 3 Permalink: http://dlmf.nist.gov/33.20.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.20(i), §33.20 and Ch.33

33.20.2		$\displaystyle f\left(0,\ell;r\right)$	$\displaystyle=(-1)^{\ell+1}(2\|r\|)^{1/2}I_{2\ell+1}\left(\sqrt{8\|r\|}\right),$
		$\displaystyle h\left(0,\ell;r\right)$	$\displaystyle=(-1)^{\ell}(2/\pi)(2\|r\|)^{1/2}K_{2\ell+1}\left(\sqrt{8\|r\|}\right)$ ,
	$r<0$ .
ⓘ Symbols: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $\pi$ : the ratio of the circumference of a circle to its diameter, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\ell$ : nonnegative integer and $r$ : real variable Referenced by: item 3 Permalink: http://dlmf.nist.gov/33.20.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.20(i), §33.20 and Ch.33

For the functions $J$ , $Y$ , $I$ , and $K$ see §§10.2(ii), 10.25(ii).

§33.20(ii) Power-Series in $\epsilon$ for the Regular Solution

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Keywords:: Coulomb functions: variables $r,\epsilon$ , power-series expansions in $\epsilon$
Notes:: See Seaton (2002a, Eqs. 58, 64, 67, 68).
Referenced by:: §33.14(ii)
Permalink:: http://dlmf.nist.gov/33.20.ii
See also:: Annotations for §33.20 and Ch.33

33.20.3		$f\left(\epsilon,\ell;r\right)=\sum_{k=0}^{\infty}\epsilon^{k}{\sf F}_{k}(\ell;% r),$
ⓘ Symbols: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter and $\mathsf{F}_{k}(\ell;r)$ : coefficient Referenced by: §33.20(ii) Permalink: http://dlmf.nist.gov/33.20.E3 Encodings: TeX, pMML, png See also: Annotations for §33.20(ii), §33.20 and Ch.33

where

33.20.4		${\sf F}_{k}(\ell;r)=\sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}J_{2\ell+1+p}\left(% \sqrt{8r}\right),$
$r>0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\mathsf{F}_{k}(\ell;r)$ : coefficient and $C_{k,p}$ : coefficients Permalink: http://dlmf.nist.gov/33.20.E4 Encodings: TeX, pMML, png See also: Annotations for §33.20(ii), §33.20 and Ch.33

33.20.5		${\sf F}_{k}(\ell;r)=\sum_{p=2k}^{3k}(-1)^{\ell+1+p}(2\|r\|)^{(p+1)/2}C_{k,p}I_{2% \ell+1+p}\left(\sqrt{8\|r\|}\right),$
$r<0$ .
ⓘ Symbols: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\mathsf{F}_{k}(\ell;r)$ : coefficient and $C_{k,p}$ : coefficients Permalink: http://dlmf.nist.gov/33.20.E5 Encodings: TeX, pMML, png See also: Annotations for §33.20(ii), §33.20 and Ch.33

The functions $J$ and $I$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by $C_{0,0}=1$ , $C_{1,0}=0$ , and

33.20.6		$\displaystyle C_{k,p}$	$\displaystyle=0,$
	$p<2k$ or $p>3k$ ,
		$\displaystyle C_{k,p}$	$\displaystyle=\left(-(2\ell+p)C_{k-1,p-2}+C_{k-1,p-3}\right)/(4p),$
	$k>0$ , $2k\leq p\leq 3k$ .
ⓘ Symbols: $k$ : nonnegative integer, $\ell$ : nonnegative integer and $C_{k,p}$ : coefficients Referenced by: §33.20(iii) Permalink: http://dlmf.nist.gov/33.20.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.20(ii), §33.20 and Ch.33

The series (33.20.3) converges for all $r$ and $\epsilon$ .

§33.20(iii) Asymptotic Expansion for the Irregular Solution

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Keywords:: Coulomb functions: variables $r,\epsilon$ , asymptotic expansions as $\epsilon\rightarrow 0$ , expansions in Bessel functions, expansions in modified Bessel functions
Notes:: See Seaton (2002a, Eqs. 59, 69, 70).
Permalink:: http://dlmf.nist.gov/33.20.iii
See also:: Annotations for §33.20 and Ch.33

As $\epsilon\to 0$ with $\ell$ and $r$ fixed,

33.20.7		$h\left(\epsilon,\ell;r\right)\sim-A(\epsilon,\ell)\sum_{k=0}^{\infty}\epsilon^% {k}{\sf H}_{k}(\ell;r),$
ⓘ Symbols: $h\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : irregular Coulomb function, $\sim$ : Poincaré asymptotic expansion, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $A(\epsilon,\ell)$ : function and $\mathsf{H}_{k}(\ell;r)$ : coefficient Permalink: http://dlmf.nist.gov/33.20.E7 Encodings: TeX, pMML, png See also: Annotations for §33.20(iii), §33.20 and Ch.33

where $A(\epsilon,\ell)$ is given by (33.14.11), (33.14.12), and

33.20.8		${\sf H}_{k}(\ell;r)=\sum_{p=2k}^{3k}(2r)^{(p+1)/2}C_{k,p}Y_{2\ell+1+p}\left(% \sqrt{8r}\right),$
$r>0$ ,
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $C_{k,p}$ : coefficients and $\mathsf{H}_{k}(\ell;r)$ : coefficient Permalink: http://dlmf.nist.gov/33.20.E8 Encodings: TeX, pMML, png See also: Annotations for §33.20(iii), §33.20 and Ch.33

33.20.9		${\sf H}_{k}(\ell;r)=(-1)^{\ell+1}\frac{2}{\pi}\sum_{p=2k}^{3k}(2\|r\|)^{(p+1)/2}% C_{k,p}K_{2\ell+1+p}\left(\sqrt{8\|r\|}\right),$
$r<0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $C_{k,p}$ : coefficients and $\mathsf{H}_{k}(\ell;r)$ : coefficient Permalink: http://dlmf.nist.gov/33.20.E9 Encodings: TeX, pMML, png See also: Annotations for §33.20(iii), §33.20 and Ch.33

The functions $Y$ and $K$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by (33.20.6).

§33.20(iv) Uniform Asymptotic Expansions

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Keywords:: Coulomb functions: variables $r,\epsilon$ , asymptotic expansions as $\epsilon\rightarrow 0$ , expansions in Airy functions, expansions in Bessel functions, uniform
Referenced by:: §2.8(iii), §2.8(iv), §33.12(ii)
Permalink:: http://dlmf.nist.gov/33.20.iv
See also:: Annotations for §33.20 and Ch.33

For a comprehensive collection of asymptotic expansions that cover $f\left(\epsilon,\ell;r\right)$ and $h\left(\epsilon,\ell;r\right)$ as $\epsilon\to 0\pm$ and are uniform in $r$ , including unbounded values, see Curtis (1964a, §7). These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\ell+1$ and $2\ell+2$ .

33.19 Power-Series Expansions in

r

33.21 Asymptotic Approximations for Large

|r|

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§33.20 Expansions for Small |ϵ|

Contents

§33.20(i) Case ϵ=0

§33.20(ii) Power-Series in ϵ for the Regular Solution

§33.20(iii) Asymptotic Expansion for the Irregular Solution

§33.20(iv) Uniform Asymptotic Expansions

§33.20 Expansions for Small $|\epsilon|$

§33.20(i) Case $\epsilon=0$

§33.20(ii) Power-Series in $\epsilon$ for the Regular Solution