# §33.9 Expansions in Series of Bessel Functions

## §33.9(i) Spherical Bessel Functions

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

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

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

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

In this subsection the functions $\mathop{J\/}\nolimits$, $\mathop{I\/}\nolimits$, and $\mathop{K\/}\nolimits$ are as in §§10.2(ii) and 10.25(ii).

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

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

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

 33.9.5 ${4\eta^{2}(k-2\ell)b_{k+1}+kb_{k-1}+b_{k-2}=0},$ $k=2\ell+2,2\ell+3,\dots$. 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)

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 $\mathop{G_{\ell}\/}\nolimits\!\left(\eta,\rho\right)\sim\frac{\rho^{-\ell}}{(% \ell+\frac{1}{2})\lambda_{\ell}(\eta)\mathop{C_{\ell}\/}\nolimits\!\left(\eta% \right)}\*\sum_{k=2\ell+1}^{\infty}(-1)^{k}b_{k}t^{k/2}\mathop{K_{k}\/}% \nolimits\!\left(\textstyle 2\sqrt{t}\right),$

where

 33.9.7 $\lambda_{\ell}(\eta)\sim\sum_{k=2\ell+1}^{\infty}(-1)^{k}(k-1)!b_{k}.$ 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)

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