I would like to compute
$$ f(E,q):=\lim_{\epsilon\to 0}\text{Im}\left[ \frac{(E^2-q^2)^2-i\epsilon}{(E^2-q^2)^4+\epsilon^2}\left( E^2 + \alpha(m+in) \right) \right] $$
with an infinitesimal constant $\epsilon >0$, $\alpha \in \mathcal{R}$ and $m,n$ are real functions of $q$. If I haven't made a mistake, this evaluates to
$$\begin{align*} f(E,q) &= \lim_{\epsilon\to0}\frac{\alpha n(E^2-q^2)^2-\epsilon(E^2+\alpha m)}{(E^2-q^2)^4+\epsilon^2} \\ &= \frac{\alpha n}{(E^2-q^2)^2}-\pi \delta\left( (E^2-q^2)^2 \right)(E^2+\alpha m) \end{align*}$$
where $\delta(.)$ is the Dirac delta distribution defined by
$$ \pi\delta(x) = \lim_{\epsilon\to0}\frac{\epsilon}{x^2+\epsilon^2}. $$
My question is how I deal with the poles in the first term of the expression. When $E^2 = q^2$ the term $\frac{\alpha n}{(E^2-q^2)^2}$ diverges when the function $n=n(q)$ does not somehow prevent it from diverging.