I'm trying find the Fourier transform $F(u)$ for the utility function for the entropic risk measure, which is given below. However my knowledge on Fourier analysis is fairly poor.
$f(x)=\dfrac{e^{\gamma x}-1}{\gamma} $
I'm running into problems because when I try to do the integration I end up evaluating $\dfrac{e^{x(\gamma +iu)}}{\gamma (\gamma +iu)}-\dfrac{e^{iux}}{iu\gamma}$ at infinity. Could anyone shed some light on what I'm messing up?