§33.6 Power-Series Expansions in $\rho$

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , power-series expansions in $\rho$
Notes:: For (33.6.5) use the definition (33.2.8) with $U\left(a,b,z\right)$ expanded as in (13.2.9). For (33.6.4) use (33.2.4) with Eq. (1.12) of Buchholz (1969).
Referenced by:: §33.23(ii), §33.23(iii), §33.23(iii)
Permalink:: http://dlmf.nist.gov/33.6
See also:: Annotations for Ch.33

33.6.1		$F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\sum_{k=\ell+1}^{% \infty}A_{k}^{\ell}(\eta)\rho^{k},$
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $A_{\ell+1}^{\ell}$ : coefficient A&S Ref: 14.1.4, 14.1.5 Referenced by: §33.6 Permalink: http://dlmf.nist.gov/33.6.E1 Encodings: TeX, pMML, png See also: Annotations for §33.6 and Ch.33

33.6.2		$F_{\ell}'\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\sum_{k=\ell+1}^{% \infty}kA_{k}^{\ell}(\eta)\rho^{k-1},$
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $A_{\ell+1}^{\ell}$ : coefficient A&S Ref: 14.1.4, 14.1.5 Referenced by: §33.6 Permalink: http://dlmf.nist.gov/33.6.E2 Encodings: TeX, pMML, png See also: Annotations for §33.6 and Ch.33

where $A_{\ell+1}^{\ell}=1$ , $A_{\ell+2}^{\ell}=\eta/(\ell+1)$ , and

33.6.3		$(k+\ell)(k-\ell-1)A_{k}^{\ell}=2\eta A_{k-1}^{\ell}-A_{k-2}^{\ell},$
$k=\ell+3,\ell+4,\dots$ ,
ⓘ Symbols: $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $A_{\ell+1}^{\ell}$ : coefficient A&S Ref: 14.1.6 Permalink: http://dlmf.nist.gov/33.6.E3 Encodings: TeX, pMML, png See also: Annotations for §33.6 and Ch.33

or in terms of the hypergeometric function (§§15.1, 15.2(i)),

33.6.4		$A_{k}^{\ell}(\eta)=\dfrac{(-\mathrm{i})^{k-\ell-1}}{(k-\ell-1)!}\*{{}_{2}F_{1}% }\left(\ell+1-k,\ell+1-\mathrm{i}\eta;2\ell+2;2\right).$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $A_{\ell+1}^{\ell}$ : coefficient Referenced by: §33.6 Permalink: http://dlmf.nist.gov/33.6.E4 Encodings: TeX, pMML, png See also: Annotations for §33.6 and Ch.33

33.6.5		${H^{\pm}_{\ell}}\left(\eta,\rho\right)=\frac{e^{\pm\mathrm{i}{\theta_{\ell}}% \left(\eta,\rho\right)}}{(2\ell+1)!\Gamma\left(-\ell\pm\mathrm{i}\eta\right)}% \left(\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}}{{\left(2\ell+2\right)_{k}% }k!}(\mp 2\mathrm{i}\rho)^{a+k}\left(\ln\left(\mp 2\mathrm{i}\rho\right)+\psi% \left(a+k\right)-\psi\left(1+k\right)-\psi\left(2\ell+2+k\right)\right)-\sum_{% k=1}^{2\ell+1}\frac{(2\ell+1)!(k-1)!}{(2\ell+1-k)!{\left(1-a\right)_{k}}}(\mp 2% \mathrm{i}\rho)^{a-k}\right),$
ⓘ Symbols: ${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : phase of Coulomb functions, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $a$ : variable A&S Ref: 14.1.14 (equivalent) Referenced by: §33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5) Permalink: http://dlmf.nist.gov/33.6.E5 Encodings: TeX, pMML, png Errata (effective with 1.0.19): On the right-hand side the factor in the denominator was incorrectly written as $\Gamma\left(-\ell+\mathrm{i}\eta\right)$ . This has been corrected to $\Gamma\left(-\ell\pm\mathrm{i}\eta\right)$ . Reported 2018-05-15 by Ian Thompson See also: Annotations for §33.6 and Ch.33

where $a=1+\ell\pm\mathrm{i}\eta$ and $\psi\left(x\right)=\Gamma'\left(x\right)/\Gamma\left(x\right)$ (§5.2(i)).

The series (33.6.1), (33.6.2), and (33.6.5) converge for all finite values of $\rho$ . Corresponding expansions for ${H^{\pm}_{\ell}}'\left(\eta,\rho\right)$ can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).

§33.6 Power-Series Expansions in ρ

§33.6 Power-Series Expansions in $\rho$