I asked the following question on mathstack but didn't receive any answers. I suspect that this question has a simple answer but I haven't thought about Fourier transforms in a while so am being sluggish at figuring it out. Any help/comments/suggestions would be appreciated.

Consider a function $f$ that has smooth Fourier transform $\widehat{f}$ with compact support. In a particular problem I am considering, it would be useful to be able to compute the integral

$$\int_{\mathbb{R}} \widehat{f}(u) \ \frac{(e^{h-2\pi i u}-1)}{h-2\pi i u} \ du$$

where $h >0$. Is anything known about this integral? In particular can we compute this integral for general $f$ as above?