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Jacobi-Anger Expansion


\begin{displaymath}
e^{iz\cos\theta} = \sum_{n=-\infty}^\infty i^nJ_n(z)e^{in\theta},
\end{displaymath}

where $J_n(z)$ is a Bessel Function of the First Kind. The identity can also be written

\begin{displaymath}
e^{iz\cos\theta}=J_0(z)+2\sum_{n=1}^\infty i^n J_n(z)\cos(n\theta).
\end{displaymath}

This expansion represents an expansion of plane waves into a series of cylindrical waves.

See also Bessel Function of the First Kind




© 1996-9 Eric W. Weisstein
1999-05-25