\rho,\eta

33.4 Recurrence Relations and Derivatives 33.6 Power-Series Expansions in

\rho

§33.5 Limiting Forms for Small $\rho$ , Small $|\eta|$ , or Large $\ell$

ⓘ

Permalink:: http://dlmf.nist.gov/33.5
See also:: Annotations for Ch.33

§33.5(i) Small $\rho$
§33.5(ii) $\eta=0$
§33.5(iii) Small $|\eta|$
§33.5(iv) Large $\ell$

§33.5(i) Small $\rho$

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , limiting forms, small $\rho$
Notes:: See Yost et al. (1936).
Permalink:: http://dlmf.nist.gov/33.5.i
See also:: Annotations for §33.5 and Ch.33

As $\rho\to 0$ with $\eta$ fixed,

33.5.1	$\displaystyle F_{\ell}\left(\eta,\rho\right)$	$\displaystyle\sim C_{\ell}\left(\eta\right)\rho^{\ell+1},$
33.5.1	$\displaystyle F_{\ell}'\left(\eta,\rho\right)$	$\displaystyle\sim(\ell+1)C_{\ell}\left(\eta\right)\rho^{\ell}.$
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $\sim$ : asymptotic equality, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter A&S Ref: 14.6.3 Permalink: http://dlmf.nist.gov/33.5.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.5(i), §33.5 and Ch.33

33.5.2		$\displaystyle G_{\ell}\left(\eta,\rho\right)$	$\displaystyle\sim\frac{\rho^{-\ell}}{(2\ell+1)C_{\ell}\left(\eta\right)},$
	$\ell=0,1,2,\dots$ ,
		$\displaystyle G_{\ell}'\left(\eta,\rho\right)$	$\displaystyle\sim-\frac{\ell\rho^{-\ell-1}}{(2\ell+1)C_{\ell}\left(\eta\right)},$
	$\ell=1,2,3,\dots$ .
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $\sim$ : asymptotic equality, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter A&S Ref: 14.6.3 Permalink: http://dlmf.nist.gov/33.5.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.5(i), §33.5 and Ch.33

§33.5(ii) $\eta=0$

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , case $\eta=0$
Notes:: See Hull and Breit (1959, pp. 435–436); also Wheeler (1937).
Permalink:: http://dlmf.nist.gov/33.5.ii
See also:: Annotations for §33.5 and Ch.33

33.5.3	$\displaystyle F_{\ell}\left(0,\rho\right)$	$\displaystyle=\rho\mathsf{j}_{\ell}\left(\rho\right),$
33.5.3	$\displaystyle G_{\ell}\left(0,\rho\right)$	$\displaystyle=-\rho\mathsf{y}_{\ell}\left(\rho\right).$
ⓘ Symbols: $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the second kind, $\ell$ : nonnegative integer and $\rho$ : nonnegative real variable Permalink: http://dlmf.nist.gov/33.5.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.5(ii), §33.5 and Ch.33

Equivalently,

33.5.4	$\displaystyle F_{\ell}\left(0,\rho\right)$	$\displaystyle=(\pi\rho/2)^{1/2}J_{\ell+\frac{1}{2}}\left(\rho\right),$
33.5.4	$\displaystyle G_{\ell}\left(0,\rho\right)$	$\displaystyle=-(\pi\rho/2)^{1/2}Y_{\ell+\frac{1}{2}}\left(\rho\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer and $\rho$ : nonnegative real variable A&S Ref: 14.6.6 Permalink: http://dlmf.nist.gov/33.5.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.5(ii), §33.5 and Ch.33

For the functions $\mathsf{j}$ , $\mathsf{y}$ , $J$ , $Y$ see §§10.47(ii), 10.2(ii).

33.5.5	$\displaystyle F_{0}\left(0,\rho\right)$	$\displaystyle=\sin\rho,$
	$\displaystyle G_{0}\left(0,\rho\right)$	$\displaystyle=\cos\rho,$
	$\displaystyle{H^{\pm}_{0}}\left(0,\rho\right)$	$\displaystyle=e^{\pm\mathrm{i}\rho}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\sin\NVar{z}$ : sine function and $\rho$ : nonnegative real variable Permalink: http://dlmf.nist.gov/33.5.E5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §33.5(ii), §33.5 and Ch.33

33.5.6		$C_{\ell}\left(0\right)=\frac{2^{\ell}\ell!}{(2\ell+1)!}=\frac{1}{(2\ell+1)!!}.$
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $!!$ : double factorial, $!$ : factorial (as in $n!$ ) and $\ell$ : nonnegative integer Permalink: http://dlmf.nist.gov/33.5.E6 Encodings: TeX, pMML, png See also: Annotations for §33.5(ii), §33.5 and Ch.33

§33.5(iii) Small $|\eta|$

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , limiting forms, small $\left|\eta\right|$
Permalink:: http://dlmf.nist.gov/33.5.iii
See also:: Annotations for §33.5 and Ch.33

33.5.7		${\sigma_{0}}\left(\eta\right)\sim-\gamma\eta,$
$\eta\to 0$ ,
ⓘ Symbols: ${\sigma_{\NVar{\ell}}}\left(\NVar{\eta}\right)$ : Coulomb phase shift, $\gamma$ : Euler’s constant, $\sim$ : asymptotic equality and $\eta$ : real parameter Permalink: http://dlmf.nist.gov/33.5.E7 Encodings: TeX, pMML, png See also: Annotations for §33.5(iii), §33.5 and Ch.33

where $\gamma$ is Euler’s constant (§5.2(ii)).

§33.5(iv) Large $\ell$

ⓘ

Keywords:: Coulomb functions: variables $\rho,\eta$ , large $\ell$ , limiting forms
Notes:: See Yost et al. (1936) and Biedenharn et al. (1955). For (33.5.9) combine the second formula for $(2\ell+1)!!$ given in §33.1 with (5.11.7).
Permalink:: http://dlmf.nist.gov/33.5.iv
See also:: Annotations for §33.5 and Ch.33

As $\ell\to\infty$ with $\eta$ and $\rho$ ( $\neq 0$ ) fixed,

33.5.8	$\displaystyle F_{\ell}\left(\eta,\rho\right)$	$\displaystyle\sim C_{\ell}\left(\eta\right)\rho^{\ell+1},$
33.5.8	$\displaystyle G_{\ell}\left(\eta,\rho\right)$	$\displaystyle\sim\dfrac{\rho^{-\ell}}{(2\ell+1)C_{\ell}\left(\eta\right)},$
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $\sim$ : asymptotic equality, $G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial function, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable and $\eta$ : real parameter A&S Ref: 14.6.3 Referenced by: §33.18, §33.23(iv) Permalink: http://dlmf.nist.gov/33.5.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §33.5(iv), §33.5 and Ch.33

33.5.9		$C_{\ell}\left(\eta\right)\sim\dfrac{e^{-\pi\eta/2}}{(2\ell+1)!!}\sim e^{-\pi% \eta/2}\dfrac{e^{\ell}}{\sqrt{2}(2\ell)^{\ell+1}}.$
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $!!$ : double factorial, $\mathrm{e}$ : base of natural logarithm, $\ell$ : nonnegative integer and $\eta$ : real parameter Referenced by: §33.23(iv), §33.5(iv) Permalink: http://dlmf.nist.gov/33.5.E9 Encodings: TeX, pMML, png See also: Annotations for §33.5(iv), §33.5 and Ch.33

33.4 Recurrence Relations and Derivatives 33.6 Power-Series Expansions in

\rho

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§33.5 Limiting Forms for Small ρ, Small |η|, or Large ℓ

Contents

§33.5(i) Small ρ

§33.5(ii) η=0

§33.5(iii) Small |η|

§33.5(iv) Large ℓ

§33.5 Limiting Forms for Small $\rho$ , Small $|\eta|$ , or Large $\ell$

§33.5(i) Small $\rho$

§33.5(ii) $\eta=0$

§33.5(iii) Small $|\eta|$

§33.5(iv) Large $\ell$