Jacobi-Anger identity

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The Jacobi-Anger identity is an expansion of exponentials of trigonometric functions in the basis of their harmonics. It is useful in physics (for example, to convert between plane waves and cylindrical waves), and in signal processing (to describe FM signals).

The most general identity is given by:

e^{i z \cos \theta}=\sum_{n=-\infty}^{\infty} i^n J_n(z) e^{i n \theta}.

where Jn(z) is the n-th Bessel function.

The following real-valued variations are often useful as well:

\cos(z \cos \theta) = J_0(z)+2 \sum_{n=1}^{\infty}(-1)^n J_{2n}(z) \cos(2n \theta)
\sin(z \cos \theta) = -2 \sum_{n=1}^{\infty}(-1)^n J_{2n-1}(z) \cos\left[\left(2n-1\right) \theta\right]
\cos(z \sin \theta) = J_0(z)+2 \sum_{n=1}^{\infty} J_{2n}(z) \cos(2n \theta)
\sin(z \sin \theta) = 2 \sum_{n=1}^{\infty} J_{2n-1}(z) \sin\left[\left(2n-1\right) \theta\right]