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

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§33.11 Asymptotic Expansions for Large ρ

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!

§33.11 Asymptotic Expansions for Large $\rho$