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

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

33.5.1		$F_{\mathrm{\ell }}(\eta ,\rho )$	$\sim C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }+1},$
		$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )$	$\sim (\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }}.$
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\sim$: asymptotic equality, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: 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		$G_{\mathrm{\ell }}(\eta ,\rho )$	$\sim {\displaystyle \frac{{\rho }^{-\mathrm{\ell }}}{(2\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right)}},$
	$\mathrm{\ell }=0,1,2,\mathrm{\dots }$,
		$G_{\mathrm{\ell }}^{\prime }(\eta ,\rho )$	$\sim -{\displaystyle \frac{\mathrm{\ell }{\rho }^{-\mathrm{\ell }-1}}{(2\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right)}},$
	$\mathrm{\ell }=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\sim$: asymptotic equality, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $\mathrm{\ell }$: 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

33.5.3		$F_{\mathrm{\ell }}(0,\rho )$	$=\rho {\mathsf{j}}_{\mathrm{\ell }}\left(\rho \right),$
		$G_{\mathrm{\ell }}(0,\rho )$	$=-\rho {\mathsf{y}}_{\mathrm{\ell }}\left(\rho \right).$
ⓘ Symbols: $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, ${\mathsf{y}}_n\left(z\right)$: spherical Bessel function of the second kind, $\mathrm{\ell }$: 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		$F_{\mathrm{\ell }}(0,\rho )$	$={(\pi \rho /2)}^{1/2}{J}_{\mathrm{\ell }+\frac{1}{2}}\left(\rho \right),$
		$G_{\mathrm{\ell }}(0,\rho )$	$=-{(\pi \rho /2)}^{1/2}{Y}_{\mathrm{\ell }+\frac{1}{2}}\left(\rho \right).$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: 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		$F_0(0,\rho )$	$=\sin \rho ,$
		$G_0(0,\rho )$	$=\cos \rho ,$
		$H_0^{\pm }(0,\rho )$	$={\mathrm{e}}^{\pm \mathrm{i}\rho }.$
ⓘ Symbols: $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\sin z$: 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		$$C_{\mathrm{\ell }}\left(0\right)=\frac{2^{\mathrm{\ell }}\mathrm{\ell }!}{(2\mathrm{\ell }+1)!}=\frac{1}{(2\mathrm{\ell }+1)!!}.$$
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!!$: double factorial (as in $n!!$), $!$: factorial (as in $n!$) and $\mathrm{\ell }$: 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

33.5.7		$${\sigma }_0\left(\eta \right)\sim -\gamma \eta ,$$
$\eta \to 0$,
ⓘ Symbols: ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: 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)).

As $\mathrm{\ell }\to \mathrm{\infty }$ with $\eta$ and $\rho$ ($\ne 0$) fixed,

33.5.8		$F_{\mathrm{\ell }}(\eta ,\rho )$	$\sim C_{\mathrm{\ell }}\left(\eta \right){\rho }^{\mathrm{\ell }+1},$
		$G_{\mathrm{\ell }}(\eta ,\rho )$	$\sim {\displaystyle \frac{{\rho }^{-\mathrm{\ell }}}{(2\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right)}},$
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\sim$: asymptotic equality, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: 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		$$C_{\mathrm{\ell }}\left(\eta \right)\sim \frac{{\mathrm{e}}^{-\pi \eta /2}}{(2\mathrm{\ell }+1)!!}\sim {\mathrm{e}}^{-\pi \eta /2}\frac{{\mathrm{e}}^{\mathrm{\ell }}}{\sqrt{2}{(2\mathrm{\ell })}^{\mathrm{\ell }+1}}.$$
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!!$: double factorial (as in $n!!$), $\mathrm{e}$: base of natural logarithm, $\mathrm{\ell }$: 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