# Identity concerning $e^{ia\sin{x}}$ as a series of bessel functions

Prove the following identity:

$$e^{ia\sin{x}}=\sum_{-\infty}^{+\infty}J_k(a) e^{ikx},$$

where $a$ is a real constant and $J_k$ is the Bessel function of the first type of order $k$.

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This was already bumped by Community, so I took the opportunity to make anotherwise way too minor a correction and upvoted vesszabo'z solution along the way :-) –  Jyrki Lahtonen Sep 15 '13 at 16:23

We know that $$\exp\left[\frac{z}{2}\left( t-\frac{1}{t}\right)\right]=\sum_{k=-\infty}^{\infty}J_k(z)t^k,$$ using this your identity follows. See e.g. Properties in Wikipedia and references therein.