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

Question

Prove the following identity:

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

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

Answer 1

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.