§33.11 Asymptotic Expansions for Large $\rho$

For large $\rho$, with $\mathrm{\ell }$ and $\eta$ fixed,

33.11.1		$$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\sim {\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k{\left(b\right)}_k}{k!{(\pm 2\mathrm{i}\rho )}^k},$$
ⓘ Symbols: ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $k$: nonnegative integer, $\mathrm{\ell }$: 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 ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$ 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		$F_{\mathrm{\ell }}(\eta ,\rho )$	$=g(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}+f(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$
		$G_{\mathrm{\ell }}(\eta ,\rho )$	$=f(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}-g(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$
ⓘ Defines: $f(\eta ,\rho )$: function (locally) and $g(\eta ,\rho )$: function (locally) Symbols: ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\cos z$: cosine function, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\sin z$: sine function, $\mathrm{\ell }$: 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		$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )$	$=\widehat{g}(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}+\widehat{f}(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$
		$G_{\mathrm{\ell }}^{\prime }(\eta ,\rho )$	$=\widehat{f}(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}-\widehat{g}(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$
ⓘ Defines: $\widehat{f}(\eta ,\rho )$: function (locally) and $\widehat{g}(\eta ,\rho )$: function (locally) Symbols: ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\cos z$: cosine function, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\sin z$: sine function, $\mathrm{\ell }$: 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		$$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}}(f(\eta ,\rho )\pm \mathrm{i}g(\eta ,\rho )),$$
ⓘ Symbols: ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: 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		$f(\eta ,\rho )$	$\sim {\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{f}_k,$
		$g(\eta ,\rho )$	$\sim {\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{g}_k,$
ⓘ 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		$\widehat{f}(\eta ,\rho )$	$\sim {\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\widehat{f}}_k,$
		$\widehat{g}(\eta ,\rho )$	$\sim {\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\widehat{g}}_k,$
ⓘ 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 $k=0,1,2,\mathrm{\dots }$,

33.11.8		$f_{k+1}$	$={\lambda }_k{f}_k-{\mu }_k{g}_k,$
		$g_{k+1}$	$={\lambda }_k{g}_k+{\mu }_k{f}_k,$
		${\widehat{f}}_{k+1}$	$={\lambda }_k{\widehat{f}}_k-{\mu }_k{\widehat{g}}_k-(f_{k+1}/\rho ),$
		${\widehat{g}}_{k+1}$	$={\lambda }_k{\widehat{g}}_k+{\mu }_k{\widehat{f}}_k-(g_{k+1}/\rho ),$
ⓘ 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		${\lambda }_k$	$={\displaystyle \frac{(2k+1)\eta }{(2k+2)\rho }},$
		${\mu }_k$	$={\displaystyle \frac{\mathrm{\ell }(\mathrm{\ell }+1)-k(k+1)+{\eta }^2}{(2k+2)\rho }}.$
ⓘ Defines: ${\lambda }_k$: coefficient (locally) and ${\mu }_k$: coefficient (locally) Symbols: $k$: nonnegative integer, $\mathrm{\ell }$: 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