Hi I think this may help you a bit to understand it.
$\overline{G(\bar{z};u)} $
$=\overline{e^{(u/2)(\bar{z}-\frac{1}{\bar{z}})}}$
$=e^{(\frac{u}{2})(z-\frac{1}{\bar{z}})}$
$=G(z;u)$ if u is real.(to prove Hint)
Therefore if u is real,then $J_n(u)$ is real
Then given by (9) with $s=1:$
$J_n(u) = Re(J_n(u))$
$=Re(\int_{0}^{2\pi}e^{i(usin\theta - n\theta)}d\theta \frac{1}{2\pi})$
$=\int_{0}^{2\pi}cos(usin\theta - n\theta)d\theta\frac{1}{2\pi}$
