# Bessel functions and spherical Bessel functions

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##### 3: 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 $\mathsf{k}_{n}\left(z\right)\sim\tfrac{1}{2}\pi z^{-1}e^{-z}.$
##### 4: 10.1 Special Notation
The main functions treated in this chapter are the Bessel functions $J_{\nu}\left(z\right)$, $Y_{\nu}\left(z\right)$; Hankel functions ${H^{(1)}_{\nu}}\left(z\right)$, ${H^{(2)}_{\nu}}\left(z\right)$; modified Bessel functions $I_{\nu}\left(z\right)$, $K_{\nu}\left(z\right)$; spherical Bessel functions $\mathsf{j}_{n}\left(z\right)$, $\mathsf{y}_{n}\left(z\right)$, ${\mathsf{h}^{(1)}_{n}}\left(z\right)$, ${\mathsf{h}^{(2)}_{n}}\left(z\right)$; modified spherical Bessel functions ${\mathsf{i}^{(1)}_{n}}\left(z\right)$, ${\mathsf{i}^{(2)}_{n}}\left(z\right)$, $\mathsf{k}_{n}\left(z\right)$; Kelvin functions $\operatorname{ber}_{\nu}\left(x\right)$, $\operatorname{bei}_{\nu}\left(x\right)$, $\operatorname{ker}_{\nu}\left(x\right)$, $\operatorname{kei}_{\nu}\left(x\right)$. For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. … For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
##### 5: 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 …
$ng_{n-1}(z)+(n+1)g_{n+1}(z)=(2n+1)g_{n}^{\prime}(z),$ $n=1,2,\dotsc$,
##### 6: 10.50 Wronskians and Cross-Products
###### §10.50 Wronskians and Cross-Products
10.50.4 $\mathsf{j}_{0}\left(z\right)\mathsf{j}_{n}\left(z\right)+\mathsf{y}_{0}\left(z% \right)\mathsf{y}_{n}\left(z\right)=\cos\left(\tfrac{1}{2}n\pi\right)\sum_{k=0% }^{\left\lfloor n/2\right\rfloor}(-1)^{k}\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+2% }}+\sin\left(\tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor(n-1)/2\right% \rfloor}(-1)^{k}\frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+3}},$
##### 8: 10.73 Physical Applications
###### §10.73(ii) SphericalBesselFunctions
The functions $\mathsf{j}_{n}\left(x\right)$, $\mathsf{y}_{n}\left(x\right)$, ${\mathsf{h}^{(1)}_{n}}\left(x\right)$, and ${\mathsf{h}^{(2)}_{n}}\left(x\right)$ arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates $\rho,\theta,\phi$1.5(ii)): …Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation. …In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential. …