# §33.11 Asymptotic Expansions for Large $\rho$

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 $\displaystyle F_{\ell}\left(\eta,\rho\right)$ $\displaystyle=g(\eta,\rho)\cos{\theta_{\ell}}+f(\eta,\rho)\sin{\theta_{\ell}},$ $\displaystyle G_{\ell}\left(\eta,\rho\right)$ $\displaystyle=f(\eta,\rho)\cos{\theta_{\ell}}-g(\eta,\rho)\sin{\theta_{\ell}},$ ⓘ 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 $\displaystyle F_{\ell}'\left(\eta,\rho\right)$ $\displaystyle=\widehat{g}(\eta,\rho)\cos{\theta_{\ell}}+\widehat{f}(\eta,\rho)% \sin{\theta_{\ell}},$ $\displaystyle G_{\ell}'\left(\eta,\rho\right)$ $\displaystyle=\widehat{f}(\eta,\rho)\cos{\theta_{\ell}}-\widehat{g}(\eta,\rho)% \sin{\theta_{\ell}},$ ⓘ 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 $\displaystyle f(\eta,\rho)$ $\displaystyle\sim\sum_{k=0}^{\infty}f_{k},$ $\displaystyle g(\eta,\rho)$ $\displaystyle\sim\sum_{k=0}^{\infty}g_{k},$ ⓘ 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 $\displaystyle\widehat{f}(\eta,\rho)$ $\displaystyle\sim\sum_{k=0}^{\infty}\widehat{f}_{k},$ $\displaystyle\widehat{g}(\eta,\rho)$ $\displaystyle\sim\sum_{k=0}^{\infty}\widehat{g}_{k},$ ⓘ 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 $\displaystyle\lambda_{k}$ $\displaystyle=\frac{(2k+1)\eta}{(2k+2)\rho},$ $\displaystyle\mu_{k}$ $\displaystyle=\frac{\ell(\ell+1)-k(k+1)+\eta^{2}}{(2k+2)\rho}.$ ⓘ 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