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I am looking at a paper that has the following expression

$$J_\mu = J \exp[i \alpha \sin(\omega t-\phi_\mu)]\exp[-i \omega t]$$ It then says "parameters whose Fourier series read" $$J_\mu(t)=\sum_{s=-\infty}^\infty\mathscr{J}_{1+s}(\alpha)e^{-i (1+s)\phi_\mu}e^{is \omega t}$$

I am failing to see where this comes from. In order to get the bessel functions, do I have to compute the coefficients of the fourier series using orthogonality?

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From the generating function for the Bessel functions \begin{equation} e^{\frac{1}{2}z(t-t^{-1})}=\sum_{m=-\infty}^{\infty}t^{m}J_{m}\left(z\right) \end{equation} with $z=\alpha$ and $t=\exp\left[ i\left( \omega t-\phi_\mu \right) \right]$ one obtains \begin{equation} \exp[i \alpha \sin(\omega t-\phi_\mu)]=\sum_m \exp\left[ im\left( \omega t-\phi_\mu \right) \right] \end{equation} \begin{align} J_\mu &= J \exp[i \alpha \sin(\omega t-\phi_\mu)]\exp[-i \omega t]\\ &=J\sum_m J_m(\alpha)\exp\left[ im\left( \omega t-\phi_\mu \right) \right]e^{-i\omega t}\\ &=J\sum_mJ_m(\alpha) \exp\left( i(m-1) \omega t \right)e^{-im\phi_\mu}\\ &=J\sum_sJ_{s+1}(\alpha) e^{-i(s+1)\phi_\mu }e^{is \omega t} \end{align} as expected.

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