 Fourier integral of a rational function with simple nonreal poles

For monic $q$ with distinct nonreal zeros and $\mathrm{deg}p\le \mathrm{deg}q-1$, close the contour in the upper half-plane. The rational factor is $O(R^{-1})$ on the large semicircle, whose exponential damping makes the arc integral $O(R^{-1})$. The two one-sided real integrals also converge by integration by parts, since the rational factor tends to zero and its derivative is $O(x^{-2})$. Thus the displayed formula is an ordinary improper integral, not merely a principal value.