I've seen this (page 112) Wick rotation from several QFT source and all of them explain really bad at what is going on.
From Complex analysis I know that for instance if we have an integral from $-\infty$ to $\infty$, you can add a semicirle (or some other type) to close the contour and then count the residues inside this simple closed contour (very rough explanation I know). But what about the Wick contour in the link? how can they just go from $$\int_{-\infty}^\infty$$ to $$\int_{-i \infty}^{i\infty}$$. I mean they must add some contour to the original in order to close it and then count the residues inside (there are none btw), but I have a hard time spotting this simple closed curve, I think I have misunderstood the whole point?
Can someone please explain what the link is talking about, especially with the figure where he says the contour simply can be rotated?
Thanks for your efforts!
I'll not attempt any general comment about Wick rotations. However, the picture you mention is simple enough to understand:
I added the green part, I suspect you can argue it vanishes ( I haven't checked this yet) then the integrals around either the loop in quadrant I or III vanishes since there are no poles within the contour. (the red circles denote the poles). But, if those loops vanish then the sum of the sides vanishes showing the black and blue parts are equal which is what is claimed in the pdf you link.