33 Coulomb Functions Variables

\rho,\eta

33.10 Limiting Forms for Large

\rho

or Large

\left|\eta\right|

33.12 Asymptotic Expansions for Large

\eta

§33.11 Asymptotic Expansions for Large $\rho$

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , asymptotic expansions, large $\rho$ , large $\rho$ , limiting forms
Notes:: See Fröberg (1955).
Referenced by:: §33.23(ii), §33.23(iii), §33.23(iii)
Permalink:: http://dlmf.nist.gov/33.11
Clarification (effective with 1.0.22):: The arguments of some of the functions in (33.11.2)–(33.11.7) were included to improve clarity of the presentation.
See also:: Annotations for Ch.33

For large $\rho$ , with $\ell$ and $\eta$ fixed,

33.11.1		${H^{\pm}_{\ell}}\left(\eta,\rho\right)\sim e^{\pm\mathrm{i}{\theta_{\ell}}% \left(\eta,\rho\right)}\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{\left(b% \right)_{k}}}{k!(\pm 2\mathrm{i}\rho)^{k}},$
ⓘ Symbols: ${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : phase of Coulomb functions, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : 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, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $a$ : coefficient and $b$ : coefficient A&S Ref: 14.5.9 Referenced by: Erratum (V1.0.19) for Equation (33.11.1), Erratum (V1.0.22) for Equation (33.11.1) Permalink: http://dlmf.nist.gov/33.11.E1 Encodings: TeX, pMML, png Errata (effective with 1.0.22): Previously this formula was expressed as an equality. Since this formula expresses an asymptotic expansion, it has been corrected by using instead an $\sim$ relation. Reported 2019-01-29 by Gergő Nemes Errata (effective with 1.0.19): Originally the factor in the denominator on the right-hand side was written incorrectly as $(\mp 2\mathrm{i}\rho)^{k}$ . This has been corrected to $(\pm 2\mathrm{i}\rho)^{k}$ . Reported 2018-05-15 by Ian Thompson See also: Annotations for §33.11 and Ch.33

where ${\theta_{\ell}}\left(\eta,\rho\right)$ is defined by (33.2.9), and $a$ and $b$ are defined by (33.8.3).

An equivalent formulation is given by

33.11.2	$\displaystyle F_{\ell}\left(\eta,\rho\right)$	$\displaystyle=g(\eta,\rho)\cos{\theta_{\ell}}+f(\eta,\rho)\sin{\theta_{\ell}},$
33.11.2	$\displaystyle G_{\ell}\left(\eta,\rho\right)$	$\displaystyle=f(\eta,\rho)\cos{\theta_{\ell}}-g(\eta,\rho)\sin{\theta_{\ell}},$
ⓘ Defines: $f(\eta,\rho)$ : function (locally) and $g(\eta,\rho)$ : function (locally) Symbols: ${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : phase of Coulomb functions, $\cos\NVar{z}$ : cosine function, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\sin\NVar{z}$ : sine function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter A&S Ref: 14.5.1 14.5.2 Referenced by: §33.11, Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7), Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7) Permalink: http://dlmf.nist.gov/33.11.E2 Encodings: TeX, TeX, pMML, pMML, png, png Clarification (effective with 1.0.22): The arguments of some of the functions in these equations were included to improve clarity of the presentation. See also: Annotations for §33.11 and Ch.33

33.11.3	$\displaystyle F_{\ell}'\left(\eta,\rho\right)$	$\displaystyle=\widehat{g}(\eta,\rho)\cos{\theta_{\ell}}+\widehat{f}(\eta,\rho)% \sin{\theta_{\ell}},$
33.11.3	$\displaystyle G_{\ell}'\left(\eta,\rho\right)$	$\displaystyle=\widehat{f}(\eta,\rho)\cos{\theta_{\ell}}-\widehat{g}(\eta,\rho)% \sin{\theta_{\ell}},$
ⓘ Defines: $\widehat{f}(\eta,\rho)$ : function (locally) and $\widehat{g}(\eta,\rho)$ : function (locally) Symbols: ${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : phase of Coulomb functions, $\cos\NVar{z}$ : cosine function, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\sin\NVar{z}$ : sine function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter A&S Ref: 14.5.3 14.5.4 Permalink: http://dlmf.nist.gov/33.11.E3 Encodings: TeX, TeX, pMML, pMML, png, png Clarification (effective with 1.0.22): The arguments of some of the functions in these equations were included to improve clarity of the presentation. See also: Annotations for §33.11 and Ch.33

33.11.4		${H^{\pm}_{\ell}}\left(\eta,\rho\right)=e^{\pm\mathrm{i}{\theta_{\ell}}}(f(\eta% ,\rho)\pm\mathrm{i}g(\eta,\rho)),$
ⓘ Symbols: ${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : phase of Coulomb functions, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $f(\eta,\rho)$ : function and $g(\eta,\rho)$ : function Permalink: http://dlmf.nist.gov/33.11.E4 Encodings: TeX, pMML, png Clarification (effective with 1.0.22): The arguments of some of the functions in these equations were included to improve clarity of the presentation. See also: Annotations for §33.11 and Ch.33

where

33.11.5	$\displaystyle f(\eta,\rho)$	$\displaystyle\sim\sum_{k=0}^{\infty}f_{k},$
33.11.5	$\displaystyle g(\eta,\rho)$	$\displaystyle\sim\sum_{k=0}^{\infty}g_{k},$
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $k$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $f(\eta,\rho)$ : function, $g(\eta,\rho)$ : function, $f_{k}$ : coefficient and $g_{k}$ : coefficient Permalink: http://dlmf.nist.gov/33.11.E5 Encodings: TeX, TeX, pMML, pMML, png, png Clarification (effective with 1.0.22): The arguments of some of the functions in these asymptotic expansions were included to improve clarity of the presentation. See also: Annotations for §33.11 and Ch.33

33.11.6	$\displaystyle\widehat{f}(\eta,\rho)$	$\displaystyle\sim\sum_{k=0}^{\infty}\widehat{f}_{k},$
33.11.6	$\displaystyle\widehat{g}(\eta,\rho)$	$\displaystyle\sim\sum_{k=0}^{\infty}\widehat{g}_{k},$
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $k$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $\widehat{f}(\eta,\rho)$ : function and $\widehat{g}(\eta,\rho)$ : function Permalink: http://dlmf.nist.gov/33.11.E6 Encodings: TeX, TeX, pMML, pMML, png, png Clarification (effective with 1.0.22): The arguments of some of the functions in these asymptotic expansions were included to improve clarity of the presentation. See also: Annotations for §33.11 and Ch.33

33.11.7		$g(\eta,\rho)\widehat{f}(\eta,\rho)-f(\eta,\rho)\widehat{g}(\eta,\rho)=1.$
ⓘ Symbols: $\rho$ : nonnegative real variable, $\eta$ : real parameter, $f(\eta,\rho)$ : function, $g(\eta,\rho)$ : function, $\widehat{f}(\eta,\rho)$ : function and $\widehat{g}(\eta,\rho)$ : function A&S Ref: 14.5.8 Referenced by: §33.11, Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7), Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7) Permalink: http://dlmf.nist.gov/33.11.E7 Encodings: TeX, pMML, png Clarification (effective with 1.0.22): The arguments of some of the functions in this equation were included to improve clarity of the presentation. See also: Annotations for §33.11 and Ch.33

Here $f_{0}=1$ , $g_{0}=0$ , $\widehat{f}_{0}=0$ , $\widehat{g}_{0}=1-(\eta/\rho)$ , and for $k=0,1,2,\dots$ ,

33.11.8	$\displaystyle f_{k+1}$	$\displaystyle=\lambda_{k}f_{k}-\mu_{k}g_{k},$
	$\displaystyle g_{k+1}$	$\displaystyle=\lambda_{k}g_{k}+\mu_{k}f_{k},$
	$\displaystyle\widehat{f}_{k+1}$	$\displaystyle=\lambda_{k}\widehat{f}_{k}-\mu_{k}\widehat{g}_{k}-(f_{k+1}/\rho),$
	$\displaystyle\widehat{g}_{k+1}$	$\displaystyle=\lambda_{k}\widehat{g}_{k}+\mu_{k}\widehat{f}_{k}-(g_{k+1}/\rho),$
ⓘ Defines: $f_{k}$ : coefficient (locally) and $g_{k}$ : coefficient (locally) Symbols: $k$ : nonnegative integer, $\rho$ : nonnegative real variable, $\widehat{f}(\eta,\rho)$ : function, $\widehat{g}(\eta,\rho)$ : function, $\lambda_{k}$ : coefficient and $\mu_{k}$ : coefficient Permalink: http://dlmf.nist.gov/33.11.E8 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §33.11 and Ch.33

where

33.11.9	$\displaystyle\lambda_{k}$	$\displaystyle=\frac{(2k+1)\eta}{(2k+2)\rho},$
33.11.9	$\displaystyle\mu_{k}$	$\displaystyle=\frac{\ell(\ell+1)-k(k+1)+\eta^{2}}{(2k+2)\rho}.$
ⓘ Defines: $\lambda_{k}$ : coefficient (locally) and $\mu_{k}$ : coefficient (locally) Symbols: $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter Permalink: http://dlmf.nist.gov/33.11.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.11 and Ch.33

33.10 Limiting Forms for Large

\rho

or Large

\left|\eta\right|

33.12 Asymptotic Expansions for Large

\eta

NIST

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§33.11 Asymptotic Expansions for Large ρ

§33.11 Asymptotic Expansions for Large $\rho$