# Bessel functions

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##### 13: 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 …
$ng_{n-1}(z)+(n+1)g_{n+1}(z)=(2n+1)g_{n}^{\prime}(z),$ $n=1,2,\dotsc$,
##### 16: 30.10 Series and Integrals
For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).
##### 19: 10.18 Modulus and Phase Functions
###### §10.18(ii) Basic Properties
10.18.12 $M_{\nu}\left(x\right)N_{\nu}\left(x\right)\sin\left(\phi_{\nu}\left(x\right)-% \theta_{\nu}\left(x\right)\right)=\frac{2}{\pi x}.$
###### §10.18(iii) Asymptotic Expansions for Large Argument
The remainder after $k$ terms in (10.18.17) does not exceed the $(k+1)$th term in absolute value and is of the same sign, provided that $k>\nu-\tfrac{1}{2}$.