This question is in reference to eq. (2.52) on the bottom of page 27 in Peskin and Schroeder.
To evaluate the Klein-Gordon field propagator along a spacelike interval we wrap the contour around the branch cut at $p=+im$. My question is (possibly a mathematical one) why it is necessary to move the contour in the first place? The function is analytic along the real axis, which is what we are originally integrating along.