**1. I'm having some trouble with some of the contour integrals covered in Chapter 2 of**

**Peskin & Schroeder's**Intro to QTF. I'm not so much as looking for answers to the integral (in fact, the answers are given in the textbook), but I was hoping someone could point me to some resources to use to become more familiar and confident with integrals like this as so far my understanding of it is very "hand-wavy."**2. Relevant equations: In section 2.4 of P&S, the propagator for the Klein Gordon field is derived and found to have the form: $$D(x-y) = \frac{-i}{2(2\pi)^2 r} \int_{-\infty}^{\infty}{dp \frac{p e^{ipr}}{\sqrt{p^2+m^2}}}$$**

Now, I can readily see how this is a contour integral along the real p-axis with (isolated?) singular points at $$p=\pm im$$

But, I'm not familiar with the contour deformation done in the book:

3. My attempt: I'm not very clear on how pushing the contour up like that doesn't change the value of the integral, or why this is easier to do in practice than to do the integral along the real line. I suppose it makes sense that you'd want to wrap around one of the poles so that you can use the residue theorem to show that the part of the contour that wraps around the pole goes to zero, since the Residue at p=+/- im is zero (or I may be completely wrong about that) leaving only the nonzero part of the contour along the branch cut, but this only makes sense to me in a "hand-wavy" sort of way.

I did take complex analysis a long time ago as an undergrad, but either my professor wasn't that great or I just have a terrible memory because I don't remember doing anything like this. From what I recall, we used the book "Complex Variables and Applications" by Churchill but that book doesn't seem to prove entirely useful when trying to review integrals like this.

Any point in the right direction would be much appreciated!

Now, I can readily see how this is a contour integral along the real p-axis with (isolated?) singular points at $$p=\pm im$$

But, I'm not familiar with the contour deformation done in the book:

3. My attempt: I'm not very clear on how pushing the contour up like that doesn't change the value of the integral, or why this is easier to do in practice than to do the integral along the real line. I suppose it makes sense that you'd want to wrap around one of the poles so that you can use the residue theorem to show that the part of the contour that wraps around the pole goes to zero, since the Residue at p=+/- im is zero (or I may be completely wrong about that) leaving only the nonzero part of the contour along the branch cut, but this only makes sense to me in a "hand-wavy" sort of way.

I did take complex analysis a long time ago as an undergrad, but either my professor wasn't that great or I just have a terrible memory because I don't remember doing anything like this. From what I recall, we used the book "Complex Variables and Applications" by Churchill but that book doesn't seem to prove entirely useful when trying to review integrals like this.

Any point in the right direction would be much appreciated!