For a physics problem I need to calculate the following integral:
$$I = \int \frac{(\alpha^2/k^2)}{\frac{\vec{p}^2}{2}-\frac{(\vec{p} - \vec{k})^2}{2} -1 +i\varepsilon} \frac{d^3 \vec{k}}{(2\pi)^3} $$
First I rewrite as
$$I = \int_{0}^{2\pi}\int_{0}^{\pi}\int_{-\infty}^{\infty} \frac{\alpha^2 \sin(\theta) }{\frac{k^2}{2} - pk\cos\theta -1 + i\varepsilon}dkd\theta d\phi$$
Where $k = |\vec{k}|$ and $\varepsilon$ is a positive infinitesimal. I suspect that to calculate this integral I need a contour integral but I'm lost on how to calculate it. $\alpha$ is positive and I may assume $\alpha <<1$ But I would like to calculate exactly if possible.