Β§33.9 Expansions in Series of Bessel Functions

Β§33.9(i) Spherical Bessel Functions

 33.9.1 $F_{\ell}\left(\eta,\rho\right)=\rho\sum_{k=0}^{\infty}a_{k}\mathsf{j}_{\ell+k}% \left(\rho\right),$

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

 33.9.2 $\frac{k(k+2\ell+1)}{2k+2\ell+1}a_{k}-2\eta a_{k-1}+\frac{(k-2)(k+2\ell-1)}{2k+% 2\ell-3}a_{k-2}=0,$ $k=1,2,\dots$. β 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 Β§33.9(i), Β§33.9 and Ch.33

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

Β§33.9(ii) Bessel Functions and Modified Bessel Functions

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

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

 33.9.3 $F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\frac{(2\ell+1)!}{(2% \eta)^{2\ell+1}}\rho^{-\ell}\*\sum_{k=2\ell+1}^{\infty}b_{k}t^{k/2}I_{k}\left(% \textstyle 2\sqrt{t}\right),$ $\eta>0$,
 33.9.4 $F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\frac{(2\ell+1)!}{(2% \left|\eta\right|)^{2\ell+1}}\rho^{-\ell}\*\sum_{k=2\ell+1}^{\infty}\!\!b_{k}t% ^{k/2}J_{k}\left(\textstyle 2\sqrt{t}\right),$ $\eta<0$.

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

 33.9.5 ${4\eta^{2}(k-2\ell)b_{k+1}+kb_{k-1}+b_{k-2}=0},$ $k=2\ell+2,2\ell+3,\dots$. β 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 Β§33.9(ii), Β§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 $G_{\ell}\left(\eta,\rho\right)\sim\frac{\rho^{-\ell}}{(\ell+\frac{1}{2})% \lambda_{\ell}(\eta)C_{\ell}\left(\eta\right)}\*\sum_{k=2\ell+1}^{\infty}(-1)^% {k}b_{k}t^{k/2}K_{k}\left(\textstyle 2\sqrt{t}\right),$

where

 33.9.7 $\lambda_{\ell}(\eta)\sim\sum_{k=2\ell+1}^{\infty}(-1)^{k}(k-1)!b_{k}.$ β Defines: $\lambda_{\ell}(\eta)$: coefficient (locally) Symbols: $\sim$: PoincarΓ© 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 Β§33.9(ii), Β§33.9 and Ch.33

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