# spherical Bessel functions

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##### 2: 10.52 Limiting Forms
###### §10.52 Limiting Forms
10.52.1 $\mathsf{j}_{n}\left(z\right),{\mathsf{i}^{(1)}_{n}}\left(z\right)\sim z^{n}/(2% n+1)!!,$
10.52.2 $-\mathsf{y}_{n}\left(z\right),i{\mathsf{h}^{(1)}_{n}}\left(z\right),-i{\mathsf% {h}^{(2)}_{n}}\left(z\right),(-1)^{n}{\mathsf{i}^{(2)}_{n}}\left(z\right),(2/% \pi)\mathsf{k}_{n}\left(z\right)\sim(2n-1)!!/z^{n+1}.$
10.52.5 ${\mathsf{i}^{(1)}_{n}}\left(z\right)\sim{\mathsf{i}^{(2)}_{n}}\left(z\right)% \sim\tfrac{1}{2}z^{-1}e^{z},$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta(<\tfrac{1}{2}\pi)$,
10.52.6 $\mathsf{k}_{n}\left(z\right)\sim\tfrac{1}{2}\pi z^{-1}e^{-z}.$
##### 3: 10.50 Wronskians and Cross-Products
###### §10.50 Wronskians and Cross-Products
10.50.4 $\mathsf{j}_{0}\left(z\right)\mathsf{j}_{n}\left(z\right)+\mathsf{y}_{0}\left(z% \right)\mathsf{y}_{n}\left(z\right)=\cos\left(\tfrac{1}{2}n\pi\right)\sum_{k=0% }^{\left\lfloor n/2\right\rfloor}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+2% }}+\sin\left(\tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor(n-1)/2\right% \rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+3}},$
##### 4: 10.51 Recurrence Relations and Derivatives
Let $f_{n}(z)$ denote any of $\mathsf{j}_{n}\left(z\right)$, $\mathsf{y}_{n}\left(z\right)$, ${\mathsf{h}^{(1)}_{n}}\left(z\right)$, or ${\mathsf{h}^{(2)}_{n}}\left(z\right)$. …
$nf_{n-1}(z)-(n+1)f_{n+1}(z)=(2n+1)f_{n}^{\prime}(z),$ $n=1,2,\dots$,
$f_{n}^{\prime}(z)=-f_{n+1}(z)+(n/z)f_{n}(z),$ $n=0,1,\dots.$
Then …
$ng_{n-1}(z)+(n+1)g_{n+1}(z)=(2n+1)g_{n}^{\prime}(z),$ $n=1,2,\dotsc$,
##### 8: 30.10 Series and Integrals
For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).
##### 9: 10.60 Sums
###### §10.60 Sums
10.56.3 $\frac{\cosh\sqrt{z^{2}+2izt}}{z}=\frac{\cosh z}{z}+\sum_{n=1}^{\infty}\frac{(% it)^{n}}{n!}{\mathsf{i}^{(1)}_{n-1}}\left(z\right),$
10.56.4 $\frac{\sinh\sqrt{z^{2}+2izt}}{z}=\frac{\sinh z}{z}+\sum_{n=1}^{\infty}\frac{(% it)^{n}}{n!}{\mathsf{i}^{(2)}_{n-1}}\left(z\right),$