I'm trying to calculate this integral using residue theorem: $$ lim_{\epsilon \rightarrow 0}\int_{-\infty}^{\infty}\text{d}\omega\frac{\omega^2-k^2v_F^2}{(\omega+kv+i\epsilon)(\omega-kv-i\epsilon)}\frac{1}{(\omega-kv_q+i\epsilon)(\omega-kv_q-i\epsilon)} $$
It seems to me that the integrand is proportional to $\frac{1}{\omega^2}$, so it should be convergent at infinity in both upper complex plane and lower complex plane. But if I close the contour in the upper half plane, I got $$ \frac{i\pi(v_F^2-v^2)}{kv(v-v_q)^2}+\frac{v_F^2-v_q^2}{v_q^2-v^2}\frac{\pi}{\epsilon} $$ and when I close the contour in the lower half plane, I got $$ \frac{i\pi(v_F^2-v^2)}{kv(v+v_q)^2}+\frac{v_F^2-v_q^2}{v_q^2-v^2}\frac{\pi}{\epsilon} $$ My question is why these two results do not coincide?