# Integration against a certain Fourier transform

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?

By Fourier inversion formula, $$\int_{-\infty}^\infty e^{-2\pi i u x} \hat f(u) du = f(x)$$ for all $$x$$ (up to some constant factors that depend on normalisation of the Fourier transform). By Fubini's theorem, $$\int_0^1 e^{h x} f(x) dx = \int_{-\infty}^\infty \frac{e^{h - 2\pi i u} - 1}{h - 2 \pi i u} \hat f(u) du .$$