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Hansen-Bessel Formula

$\displaystyle J_n(z)$ $\textstyle =$ $\displaystyle {1\over 2\pi}\int_{-\pi}^\pi e^{iz\cos t}e^{in(t-\pi/2)}\,dt$  
  $\textstyle =$ $\displaystyle {i^{-n}\over\pi} \int_0^\pi e^{iz\cos t}cos(nt)\,dt$  
  $\textstyle =$ $\displaystyle {1\over\pi}\int_0^\pi \cos(z\sin t-nt)\,dt$  

for $n=0$, 1, 2, ..., where $J_n(z)$ is a Bessel Function of the First Kind.


Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1472, 1980.

© 1996-9 Eric W. Weisstein