In this document about quantum field theory in accelerated reference frames, the equation (8.21) introduces the function $F(\omega,\Omega)$
$$F(\omega,\Omega) = \int_{-\infty}^{\infty} \frac{\text{d}u}{2\pi} \exp\left[i\Omega u + i\frac{\omega}{a}e^{-au}\right]$$
which apparently satisfy this identity [equation (8.29) in the document]:
$$F(\omega,\Omega) = F(-\omega,\Omega) \exp\left(\frac{\pi \Omega}{a}\right), \qquad a>0, \omega>0, \tag{*}$$
My attempt was to prove this last identity using the obvious fact that $F(\omega,\Omega)$ is an inverse Fourier transform:
$$F(\omega,\Omega) = \mathcal{F}^{-1}\left[\exp\left(i\frac{\omega}{a}e^{-au}\right)\right]$$
where $u$ and $\Omega$ are the conjugate Fourier variables. But I do not see how to derive identity $(*)$ from this fact.