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Table of Integrals, Series, and Products, 8th Edition by Daniel Zwillinger

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8.51–8.52 Series of Bessel functions

8.511 Generating functions for Bessel functions:

1. 

exp12(t1t)z=J0(z)+k=1[tk+(t)k]Jk(z)=k=Jk(z)tk[|z|<|t|]

si987_e  KU 119(12)

2. 

exp(t1t)z={k=tkJk(z)}{m=tmJm(z)}

si988_e  WA 40

3. 

exp(±izsinΦ)=J0(z)+2k=1J2k(z)cos2kΦ±2ik=0j2k+1(z)sin(2k+1)Φ

si989_e  KU 120(13)

4. 

exp(izcosΦ)=π2zk=0(2k+1)ikJk+12(z)Pk(cosΦ)

  WA 401(1)

=k=ikJk(z)eikΦ

  MO 27

=J0(z)+2k=1ikJk(z)coskΦ

  MO 27

5. 

iπeizcos2Φ

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