I have to calculate this integral :
$\int_{-\infty}^{+\infty} \hat G(\omega)e^{i\frac{\pi}{2}\omega}d\omega$
($\hat G(\omega)$ is the Fourier trasform) with:
$G:x\in \Bbb{R}\to \begin{cases} g(x), & \text{if |x| $\leq$ $\pi$} \\ 0, & \text{if |x| $\gt$$\pi$} \end{cases}$
and: $g:x \in[-\pi,\pi[\to \begin{cases} 0, & \text{if |x| $\ge$ $\frac{\pi}{2}$} \\ e^x, & \text{if |x| $\lt$$\frac{\pi}{2}$} \end{cases}$