t(e^{\frac{xt}{2}}\right) \left(e^{\frac{-x}{2t}}\right) = \left(\sum_{0}^{\infty} \frac{x^{r} t^{r}}{2^{r} r!}\right) \left(\sum_{0}^{\infty} \frac{(-1)^{s} x^{s} t^{-s}}{2^{s} s!}\right) = \sum \sum \frac{(-1)^{s}x^{r+s} t^{r-s}}{2^{r+s}\, r! s!} = \sum_{-\infty}^{\infty} J_{n}(x) t^{n} \quad
\mathrm{and}\ \mathrm{let}\  t = e^{i\phi} \\
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Then there's the perennial question "why Bessel functions?". 
Most explanations start with <img src="pix/fmeq45.png" alt="jacobi 