Let $R(x) = P (x)/Q(x)$ be a rational function with $(\text{degree}\: Q)≥ (\text{degree}\: P )+2$ and $Q(x) \not= 0$ on the real axis. Then I want to prove that if $α_1 , . . . , α_k$ are the roots of $Q$ in the upper half-plane, then there exists polynomials $P_j(ξ)$ of degree less than the multiplicity of $α_j$ so that
$$\int_{-\infty}^{\infty}R(x) e^{-2 \pi i x ξ}dx= \sum_{j=1}^k P_j(ξ)e^{-2 \pi i \alpha_j ξ}$$
But the thing is Can I suppose that the poles of $R(x)$ are simple? and How Can I compute the residue if not?
I was thinking in doing the following, we write
$$Q(z)=(z-\alpha_{1})^{j_1}...(z-\alpha_{k})^{j_k}H(z)$$
where $H(z)$ encodes the roots of $Q$ but in the lower-half plane,and then compute each residue, but I think that there has to be a better way to write $Q$ so that I only have to compute one derivative. I want to take as my contour a semicircle with height the maximum of the distances of each root ( Am I right here ? )
I have read this reference but is not too clear (page 8), since they assume simple poles.
Thanks in advance.