Table of Integrals, Series, and Products, 8th Edition

8.51–8.52 Series of Bessel functions

8.511 Generating functions for Bessel functions:

1. 

$\begin{array}{r}\hfill \exp \frac{1}{2}\left(t-\frac{1}{t}\right)z=J_0(z)+{\displaystyle \sum _{k=1}^{\infty }[t^k+{(-t)}^{-k}]J_k(z)={\displaystyle \sum _{k=-\infty }^{\infty }{J}_k(z)t^k}}\\ \hfill \left[|z|<|t|\right]\end{array}$

  KU 119(12)

2. 

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

  WA 40

3. 

$\exp (\pm izsin\mathrm{\Phi })=J_0(z)+2{\displaystyle \sum _{k=1}^{\infty }{J}_{2k}(z)cos2k\mathrm{\Phi }\pm 2i}{\displaystyle \sum _{k=0}^{\infty }{j}_{2k+1}(z)sin(2k+1)\mathrm{\Phi }}$

  KU 120(13)

4. 

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

  WA 401(1)

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

  MO 27

$=J_0(z)+2{\displaystyle \sum _{k=1}^{\infty }{i}^k{J}_k(z)cosk\mathrm{\Phi }}$

  MO 27

5. 

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