assuming that the integral exists
$ I(u)= \int_{-\infty}^{\infty}dxe^{iux}e^{ax}f(x) $
using the shift properties of Fourier function is that integral equal to
$ I(u)= \frac{F(u+ia)+F(u-ia)}{2} $
with $ F(u)=\int_{-\infty}^{\infty}dxe^{iux}f(x) $ ??
or it is just equal to $ I(u)= F(u+ia) $
what should be the correct solution ? here $ f(x) $ real or complex
if $ f(x)=f(-x) $ is even then its Fourier Transform must be real but how about in other cases ??