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### 8.51–8.52 Series of Bessel functions

8.511 Generating functions for Bessel functions:

1.

$\begin{array}{r}\hfill \mathrm{exp}\frac{1}{2}\left(t-\frac{1}{t}\right)z={J}_{0}\left(z\right)+\sum _{k=1}^{\infty }\left[{t}^{k}+{\left(-t\right)}^{-k}\right]{J}_{k}\left(z\right)=\sum _{k=-\infty }^{\infty }{J}_{k}\left(z\right){t}^{k}\\ \hfill \left[|z|<|t|\right]\end{array}$

KU 119(12)

2.

$\mathrm{exp}\left(t-\frac{1}{t}\right)z=\left\{\sum _{k=-\infty }^{\infty }{t}^{k}{J}_{k}\left(z\right)\right\}\left\{\sum _{m=-\infty }^{\infty }{t}^{m}{J}_{m}\left(z\right)\right\}$

WA 40

3.

$\mathrm{exp}\left(±izsin\mathrm{\Phi }\right)={J}_{0}\left(z\right)+2\sum _{k=1}^{\infty }{J}_{2k}\left(z\right)cos2k\mathrm{\Phi }±2i\sum _{k=0}^{\infty }{j}_{2k+1}\left(z\right)sin\left(2k+1\right)\mathrm{\Phi }$

KU 120(13)

4.

$\mathrm{exp}\left(izcos\mathrm{\Phi }\right)=\sqrt{\frac{\pi }{2z}}\sum _{k=0}^{\infty }\left(2k+1\right){i}^{k}{J}_{k+\frac{1}{2}}\left(z\right){P}_{k}\left(cos\mathrm{\Phi }\right)$

WA 401(1)

$=\sum _{k=-\infty }^{\infty }{i}^{k}{J}_{k}\left(z\right){e}^{ik\mathrm{\Phi }}$

MO 27

$={J}_{0}\left(z\right)+2\sum _{k=1}^{\infty }{i}^{k}{J}_{k}\left(z\right)cosk\mathrm{\Phi }$

MO 27

5.

$\sqrt{\frac{i}{\pi }}{e}^{izcos2\mathrm{\Phi }}{\int }_{-}^{}$

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