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 $k$.
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 $k$.
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.