Contour Integral with Multiple Poles

Question

I'm reading through a physics textbook and came across the integral in the image below. The author says we're using contour integration for this, but I'm not sure how they are getting 2 terms in the final answer.

Can someone show me how to derive equation (37.17) in the attached image? (Preferably through contour integration.)

Thanks in advance.

Image of the contour integral to be integrated.

Answer 1

There are two poles, one in the upper-half plane and the lower-half plane. Assume you close the contour in the upper-half plane. Then you only pick up the contribution from the pole in upper-half plane. Note that the pole is a second-order pole. Can you take it from here?