§33.9 Expansions in Series of Bessel Functions

33.9.1		$$F_{\mathrm{\ell }}(\eta ,\rho )=\rho \sum _{k=0}^{\mathrm{\infty }}{a}_k{\mathsf{j}}_{\mathrm{\ell }+k}\left(\rho \right),$$
ⓘ Symbols: $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, ${\mathsf{j}}_n\left(z\right)$: spherical Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: 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), $a_{-1}=0$, $a_0=(2\mathrm{\ell }+1)!!C_{\mathrm{\ell }}\left(\eta \right)$, and

33.9.2		$$\frac{k(k+2\mathrm{\ell }+1)}{2k+2\mathrm{\ell }+1}{a}_k-2\eta a_{k-1}+\frac{(k-2)(k+2\mathrm{\ell }-1)}{2k+2\mathrm{\ell }-3}{a}_{k-2}=0,$$
$k=1,2,\mathrm{\dots }$.
ⓘ Symbols: $k$: nonnegative integer, $\mathrm{\ell }$: 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 $\rho$.

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		$$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\frac{(2\mathrm{\ell }+1)!}{{(2\eta )}^{2\mathrm{\ell }+1}}{\rho }^{-\mathrm{\ell }}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{b}_k{t}^{k/2}{I}_k\left(2\sqrt{t}\right),$$
$\eta >0$,
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: 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		$$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\frac{(2\mathrm{\ell }+1)!}{{(2\left|\eta \right|)}^{2\mathrm{\ell }+1}}{\rho }^{-\mathrm{\ell }}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{b}_k{t}^{k/2}{J}_k\left(2\sqrt{t}\right),$$
.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $!$: factorial (as in $n!$), $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: 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 $b_{2\mathrm{\ell }}=b_{2\mathrm{\ell }+2}=0$, $b_{2\mathrm{\ell }+1}=1$, and

33.9.5		$$4{\eta }^2(k-2\mathrm{\ell })b_{k+1}+k{b}_{k-1}+b_{k-2}=0,$$
$k=2\mathrm{\ell }+2,2\mathrm{\ell }+3,\mathrm{\dots }$.
ⓘ Symbols: $k$: nonnegative integer, $\mathrm{\ell }$: 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 $\eta \to +\mathrm{\infty }$ with $\rho$ ($>0$) fixed,

33.9.6		$$G_{\mathrm{\ell }}(\eta ,\rho )\sim \frac{{\rho }^{-\mathrm{\ell }}}{(\mathrm{\ell }+\frac{1}{2}){\lambda }_{\mathrm{\ell }}(\eta )C_{\mathrm{\ell }}\left(\eta \right)}\sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{(-1)}^k{b}_k{t}^{k/2}{K}_k\left(2\sqrt{t}\right),$$
ⓘ 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, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $t$: parameter, $b_k$: coefficient and ${\lambda }_{\mathrm{\ell }}(\eta )$: 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		$${\lambda }_{\mathrm{\ell }}(\eta )\sim \sum _{k=2\mathrm{\ell }+1}^{\mathrm{\infty }}{(-1)}^k(k-1)!b_k.$$
ⓘ Defines: ${\lambda }_{\mathrm{\ell }}(\eta )$: coefficient (locally) Symbols: $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $k$: nonnegative integer, $\mathrm{\ell }$: 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 $G_{\mathrm{\ell }}(\eta ,\rho )$ see Fröberg (1955, §8) and Humblet (1985).