§33.20 Expansions for Small $|ϵ|$

33.20.1		$f(0,\mathrm{\ell };r)$	$={(2r)}^{1/2}{J}_{2\mathrm{\ell }+1}\left(\sqrt{8r}\right),$
		$h(0,\mathrm{\ell };r)$	$=-{(2r)}^{1/2}{Y}_{2\mathrm{\ell }+1}\left(\sqrt{8r}\right)$,
	$r>0$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: 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		$f(0,\mathrm{\ell };r)$	$={(-1)}^{\mathrm{\ell }+1}{(2|r|)}^{1/2}{I}_{2\mathrm{\ell }+1}\left(\sqrt{8|r|}\right),$
		$h(0,\mathrm{\ell };r)$	$={(-1)}^{\mathrm{\ell }}(2/\pi ){(2|r|)}^{1/2}{K}_{2\mathrm{\ell }+1}\left(\sqrt{8|r|}\right)$,
	.
ⓘ Symbols: $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\ell }$: 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).

33.20.3		$$f(ϵ,\mathrm{\ell };r)=\sum _{k=0}^{\mathrm{\infty }}{ϵ}^k{\mathsf{F}}_k(\mathrm{\ell };r),$$
ⓘ Symbols: $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and ${\mathsf{F}}_k(\mathrm{\ell };r)$: 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		$${\mathsf{F}}_k(\mathrm{\ell };r)=\sum _{p=2k}^{3k}{(2r)}^{(p+1)/2}{C}_{k,p}{J}_{2\mathrm{\ell }+1+p}\left(\sqrt{8r}\right),$$
$r>0$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, ${\mathsf{F}}_k(\mathrm{\ell };r)$: 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		$${\mathsf{F}}_k(\mathrm{\ell };r)=\sum _{p=2k}^{3k}{(-1)}^{\mathrm{\ell }+1+p}{(2|r|)}^{(p+1)/2}{C}_{k,p}{I}_{2\mathrm{\ell }+1+p}\left(\sqrt{8|r|}\right),$$
.
ⓘ Symbols: $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, ${\mathsf{F}}_k(\mathrm{\ell };r)$: 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		$C_{k,p}$	$=0,$
	or $p>3k$,
		$C_{k,p}$	$=\left(-(2\mathrm{\ell }+p)C_{k-1,p-2}+C_{k-1,p-3}\right)/(4p),$
	$k>0$, $2k\le p\le 3k$.
ⓘ Symbols: $k$: nonnegative integer, $\mathrm{\ell }$: 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 $ϵ$.

As $ϵ\to 0$ with $\mathrm{\ell }$ and $r$ fixed,

33.20.7		$$h(ϵ,\mathrm{\ell };r)\sim -A(ϵ,\mathrm{\ell })\sum _{k=0}^{\mathrm{\infty }}{ϵ}^k{\mathsf{H}}_k(\mathrm{\ell };r),$$
ⓘ Symbols: $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function and ${\mathsf{H}}_k(\mathrm{\ell };r)$: 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 $A(ϵ,\mathrm{\ell })$ is given by (33.14.11), (33.14.12), and

33.20.8		$${\mathsf{H}}_k(\mathrm{\ell };r)=\sum _{p=2k}^{3k}{(2r)}^{(p+1)/2}{C}_{k,p}{Y}_{2\mathrm{\ell }+1+p}\left(\sqrt{8r}\right),$$
$r>0$,
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $C_{k,p}$: coefficients and ${\mathsf{H}}_k(\mathrm{\ell };r)$: 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		$${\mathsf{H}}_k(\mathrm{\ell };r)={(-1)}^{\mathrm{\ell }+1}\frac{2}{\pi }\sum _{p=2k}^{3k}{(2|r|)}^{(p+1)/2}{C}_{k,p}{K}_{2\mathrm{\ell }+1+p}\left(\sqrt{8|r|}\right),$$
.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $C_{k,p}$: coefficients and ${\mathsf{H}}_k(\mathrm{\ell };r)$: 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).

For a comprehensive collection of asymptotic expansions that cover $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$ as $ϵ\to 0\pm$ 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 $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.