I have two related questions:
Is there a loophole in the following argument: Suppose I want to calculate the Fourier transform of a function $f(x)$: $$ \mathcal{F}f(p) = \int d\omega e^{ipx} f(x) dx $$ suppose that I am only interested in $p>0$, in that case I can carry out the integral by contour integration on a semicircle and closing the contour on the upper half plane, provided that f(x) either decays in the upper $x$ plane or at least grows slower than the exponential so that the circular part of the contour can be ignored. In this case the integral will be given by the sum of the residues. Now, I am interested in the asymptotic behavior at large $p$. The $p$ dependence in the sum of the residues only comes from the exponential factors from the poles $X$: $e^{ipx} = e^{-p\Im X}e^{+i p \Re X}$. This means that at large $p$ the function will exponentially decay as: $exp(-kp)$, where $k$ is the imaginary part of the position of the pole closest to the real axis.
If there is no loophole in my previous argument, will a similar statement be true if the function also has branch cuts but they are farther away from the real axis compared to the pole. Say the function has a pole with imaginary part $1$ and a line shaped branch cut with imaginary part $2$. Will it still be true that the Fourier tranform at large $p$ decays like $\exp(-p)$? The reason I think this might be true is that I imagine a sequence of approximating functions, that approximates the branch cut by a series of poles. In that case at all finite order of this approximation case 1 applies.