Fourier series expansion of $\exp[-j2kA\sin(\omega t)]$ I've been going through this paper where we exploit the well known Fourier expansion of the model signal as shown in the image below.



I've never come across this well known fourier expansion before. Can someone point me towards right direction to fill in the missing steps in above image and also the use of Bessel functions got me confused a bit. Any references would be of great help. Thanks.
They're talking about the Fourier series expansion of the periodic function

$$f(\tau)=e^{-jx\sin(\tau)}=\sum_{n=-\infty}^{\infty}c_ne^{jn\tau}\tag{1}$$

which sometimes appears in the literature, e.g., when computing the spectrum of a frequency modulated sine wave. The complex Fourier coefficients in $(1)$ are given by

$$c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\tau)e^{-jn\tau}d\tau=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{j(-n\tau-x\sin(\tau))}d\tau=J_{-n}(x)\tag{2}$$

where the last equality follows from the integral representation of the Bessel function of the first kind of order $n$.

Combining $(1)$ and $(2)$ we get

$$f(\tau)=\sum_{n=-\infty}^{\infty}J_{-n}(x)e^{jn\tau}\tag{3}$$

With $x=2k_0A$ and $\tau=\omega_Dt$ you obtain the expansion used in the paper you quoted.