§33.9 Expansions in Series of Bessel Functions

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§33.9(i) Spherical Bessel Functions

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Keywords:: Coulomb functions: variables $\rho,\eta$ , expansions in spherical Bessel functions
Notes:: The convergence of (33.9.1) follows from the asymptotic forms, for large $k$ , of $a_{k}$ (obtained by application of §2.9(i)) and $\mathsf{j}_{\ell+k}\left(\rho\right)$ (obtained from (10.19.1) and (10.47.3)).
Permalink:: http://dlmf.nist.gov/33.9.i
See also:: Annotations for §33.9 and Ch.33

33.9.1		$F_{\ell}\left(\eta,\rho\right)=\rho\sum_{k=0}^{\infty}a_{k}\mathsf{j}_{\ell+k}% \left(\rho\right),$
ⓘ Symbols: $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $a_{k}$ : coefficients A&S Ref: 14.4.5 Referenced by: §33.9(i), §33.9(i) Permalink: http://dlmf.nist.gov/33.9.E1 Encodings: TeX, pMML, png See also: Annotations for §33.9(i), §33.9 and Ch.33

where the function $\mathsf{j}$ is as in §10.47(ii), $a_{-1}=0$ , $a_{0}=(2\ell+1)!!C_{\ell}\left(\eta\right)$ , and

33.9.2		$\frac{k(k+2\ell+1)}{2k+2\ell+1}a_{k}-2\eta a_{k-1}+\frac{(k-2)(k+2\ell-1)}{2k+% 2\ell-3}a_{k-2}=0,$
$k=1,2,\dots$ .
ⓘ Symbols: $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $a_{k}$ : coefficients A&S Ref: 14.4.6 Permalink: http://dlmf.nist.gov/33.9.E2 Encodings: TeX, pMML, png See also: Annotations for §33.9(i), §33.9 and Ch.33

The series (33.9.1) converges for all finite values of $\eta$ and $\rho$ .

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Keywords:: Coulomb functions: variables $\rho,\eta$ , expansions in Bessel functions, expansions in modified Bessel functions
Notes:: For (33.9.3) see Yost et al. (1936), Abramowitz (1954), and Humblet (1985). For (33.9.4) see Curtis (1964a, §5.1). For (33.9.6) see Yost et al. (1936) and Abramowitz (1954).
Permalink:: http://dlmf.nist.gov/33.9.ii
See also:: Annotations for §33.9 and Ch.33

In this subsection the functions $J$ , $I$ , and $K$ are as in §§10.2(ii) and 10.25(ii).

With $t=2\left|\eta\right|\rho$ ,

33.9.3		$F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\frac{(2\ell+1)!}{(2% \eta)^{2\ell+1}}\rho^{-\ell}\*\sum_{k=2\ell+1}^{\infty}b_{k}t^{k/2}I_{k}\left(% \textstyle 2\sqrt{t}\right),$
$\eta>0$ ,
ⓘ Symbols: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $!$ : factorial (as in $n!$ ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $t$ : parameter and $b_{k}$ : coefficient A&S Ref: 14.4.1 Referenced by: §33.9(ii), §33.9(ii) Permalink: http://dlmf.nist.gov/33.9.E3 Encodings: TeX, pMML, png See also: Annotations for §33.9(ii), §33.9 and Ch.33

33.9.4		$F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\frac{(2\ell+1)!}{(2% \left\|\eta\right\|)^{2\ell+1}}\rho^{-\ell}\*\sum_{k=2\ell+1}^{\infty}\!\!b_{k}t% ^{k/2}J_{k}\left(\textstyle 2\sqrt{t}\right),$
$\eta<0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $!$ : factorial (as in $n!$ ), $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $t$ : parameter and $b_{k}$ : coefficient Referenced by: §33.9(ii), §33.9(ii) Permalink: http://dlmf.nist.gov/33.9.E4 Encodings: TeX, pMML, png See also: Annotations for §33.9(ii), §33.9 and Ch.33

Here $b_{2\ell}=b_{2\ell+2}=0$ , $b_{2\ell+1}=1$ , and

33.9.5		${4\eta^{2}(k-2\ell)b_{k+1}+kb_{k-1}+b_{k-2}=0},$
$k=2\ell+2,2\ell+3,\dots$ .
ⓘ Symbols: $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $b_{k}$ : coefficient A&S Ref: 14.4.3 ((Early printings contained an error. All editions miss $b_{2L}=0$ .)) Permalink: http://dlmf.nist.gov/33.9.E5 Encodings: TeX, pMML, png See also: Annotations for §33.9(ii), §33.9 and Ch.33

The series (33.9.3) and (33.9.4) converge for all finite positive values of $\left|\eta\right|$ and $\rho$ .

Next, as $\eta\to+\infty$ with $\rho$ ( $>0$ ) fixed,

33.9.6		$G_{\ell}\left(\eta,\rho\right)\sim\frac{\rho^{-\ell}}{(\ell+\frac{1}{2})% \lambda_{\ell}(\eta)C_{\ell}\left(\eta\right)}\*\sum_{k=2\ell+1}^{\infty}(-1)^% {k}b_{k}t^{k/2}K_{k}\left(\textstyle 2\sqrt{t}\right),$
ⓘ 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, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $t$ : parameter, $b_{k}$ : coefficient and $\lambda_{\ell}(\eta)$ : coefficient A&S Ref: 14.4.2 Referenced by: §33.9(ii) Permalink: http://dlmf.nist.gov/33.9.E6 Encodings: TeX, pMML, png See also: Annotations for §33.9(ii), §33.9 and Ch.33

where

33.9.7		$\lambda_{\ell}(\eta)\sim\sum_{k=2\ell+1}^{\infty}(-1)^{k}(k-1)!b_{k}.$
ⓘ Defines: $\lambda_{\ell}(\eta)$ : coefficient (locally) Symbols: $\sim$ : Poincaré asymptotic expansion, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $b_{k}$ : coefficient A&S Ref: 14.4.4 Permalink: http://dlmf.nist.gov/33.9.E7 Encodings: TeX, pMML, png See also: Annotations for §33.9(ii), §33.9 and Ch.33

For other asymptotic expansions of $G_{\ell}\left(\eta,\rho\right)$ see Fröberg (1955, §8) and Humblet (1985).