r,\epsilon

\ell

|\epsilon|

§33.19 Power-Series Expansions in $r$

ⓘ

Keywords:: Coulomb functions: variables $r,\epsilon$ , power-series expansions in $r$
Notes:: See Seaton (2002a, Eqs. 15–17, 31–48).
Referenced by:: §33.23(ii), §33.23(iii), §33.23(iii)
Permalink:: http://dlmf.nist.gov/33.19
See also:: Annotations for Ch.33

33.19.1		$f\left(\epsilon,\ell;r\right)=r^{\ell+1}\sum_{k=0}^{\infty}\alpha_{k}r^{k},$
ⓘ 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 $\alpha_{k}$ : term Referenced by: §33.18, §33.19 Permalink: http://dlmf.nist.gov/33.19.E1 Encodings: TeX, pMML, png See also: Annotations for §33.19 and Ch.33

where

33.19.2		$\displaystyle\alpha_{0}$	$\displaystyle=2^{\ell+1}/(2\ell+1)!,$
		$\displaystyle\alpha_{1}$	$\displaystyle=-\alpha_{0}/(\ell+1),$
		$\displaystyle k(k+2\ell+1)\alpha_{k}+2\alpha_{k-1}+\epsilon\alpha_{k-2}=0,$
	$k=2,3,\dots$ .
ⓘ Defines: $\alpha_{k}$ : term (locally) Symbols: $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer, $\ell$ : nonnegative integer and $\epsilon$ : real parameter Permalink: http://dlmf.nist.gov/33.19.E2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §33.19 and Ch.33

33.19.3		$2\pi h\left(\epsilon,\ell;r\right)={\sum_{k=0}^{2\ell}\frac{(2\ell-k)!\gamma_{% k}}{k!}(2r)^{k-\ell}-\sum_{k=0}^{\infty}\delta_{k}r^{k+\ell+1}}-A(\epsilon,% \ell)\left(2\ln\|2r/\kappa\|+\Re\psi\left(\ell+1+\kappa\right)+\Re\psi\left(-% \ell+\kappa\right)\right){f\left(\epsilon,\ell;r\right),}$
$r\neq 0$ .
ⓘ 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, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $\Re$ : real part, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $\kappa$ : quantity, $A(\epsilon,\ell)$ : function, $\gamma_{k}$ : term and $\delta_{k}$ : term Referenced by: §33.19 Permalink: http://dlmf.nist.gov/33.19.E3 Encodings: TeX, pMML, png See also: Annotations for §33.19 and Ch.33

Here $\kappa$ is defined by (33.14.6), $A(\epsilon,\ell)$ is defined by (33.14.11) or (33.14.12), $\gamma_{0}=1$ , $\gamma_{1}=1$ , and

33.19.4		$\gamma_{k}-\gamma_{k-1}+\tfrac{1}{4}(k-1)(k-2\ell-2)\epsilon\gamma_{k-2}=0,$
$k=2,3,\dots$ .
ⓘ Symbols: $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\epsilon$ : real parameter and $\gamma_{k}$ : term Permalink: http://dlmf.nist.gov/33.19.E4 Encodings: TeX, pMML, png See also: Annotations for §33.19 and Ch.33

Also,

33.19.5	$\displaystyle\delta_{0}$	$\displaystyle=\left(\beta_{2\ell+1}-2(\psi\left(2\ell+2\right)+\psi\left(1% \right))A(\epsilon,\ell)\right)\alpha_{0},$
33.19.5	$\displaystyle\delta_{1}$	$\displaystyle=\left(\beta_{2\ell+2}-2(\psi\left(2\ell+3\right)+\psi\left(2% \right))A(\epsilon,\ell)\right)\alpha_{1},$
ⓘ Defines: $\delta_{k}$ : term (locally) Symbols: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\epsilon$ : real parameter, $A(\epsilon,\ell)$ : function, $\alpha_{k}$ : term and $\beta_{k}$ : term Permalink: http://dlmf.nist.gov/33.19.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.19 and Ch.33

33.19.6		$k(k+2\ell+1)\delta_{k}+2\delta_{k-1}+\epsilon\delta_{k-2}+2(2k+2\ell+1)A(% \epsilon,\ell)\alpha_{k}=0,$
$k=2,3,\dots$ ,
ⓘ Symbols: $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\epsilon$ : real parameter, $A(\epsilon,\ell)$ : function, $\alpha_{k}$ : term and $\delta_{k}$ : term Permalink: http://dlmf.nist.gov/33.19.E6 Encodings: TeX, pMML, png See also: Annotations for §33.19 and Ch.33

with $\beta_{0}=\beta_{1}=0$ , and

33.19.7		$\beta_{k}-\beta_{k-1}+\tfrac{1}{4}(k-1)(k-2\ell-2)\epsilon\beta_{k-2}+\tfrac{1% }{2}(k-1)\epsilon\gamma_{k-2}=0,$
$k=2,3,\dots$ .
ⓘ Symbols: $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\epsilon$ : real parameter, $\gamma_{k}$ : term and $\beta_{k}$ : term Permalink: http://dlmf.nist.gov/33.19.E7 Encodings: TeX, pMML, png See also: Annotations for §33.19 and Ch.33

The expansions (33.19.1) and (33.19.3) converge for all finite values of $r$ , except $r=0$ in the case of (33.19.3).

\ell

|\epsilon|

§33.19 Power-Series Expansions in r