Different values for same integral calculated via contour integration Perhaps this question should be asked on the Math community, but I think that, because of the specific example that I will cite, I might get a better insight on my doubt here. 
So, why can we get two different results (mathematically) according to whether we choose to circumvent or to contour a pole when calculating an integral? The most obvious example being the Feynman prescription on QFT and related calculations. 
I know that most integrals to which contour integration techniques are useful doesn't exist and, in these cases, what we do is 'just' a way of giving them a quantitative meaning (call it the Cauchy's Principal Value, if you want), however still can't understand intuitively or formally what causes the difference on the values calculated from different choices of contour.  
 A: This is simply a consequence of the Cauchy Goursat Theorem and Residue Theorem: the first tells you that an integral is invariant under an homotopy of a contour unless the the region between a contour and its homotopic image includes a new singularity; the second tells you that the integral changes by  $2\,\pi\,i$ times the residue of any new pole that becomes included under an homotopy as long as the contour does not contain branch points or cross brach cuts. 
The intermediate case, that leads to the Cauchy principal value, is where the contour goes exactly through the pole; this behavior can be understood by "dinting" the contour inwards by a semicircular dint of radius $\epsilon$ to avoid the pole and then evaluating the limit of the explicit calculation of the contour on this semicircle as $\epsilon\to0$.
As to the physical meaning of all these variations, one has to look in detail at the physical assumptions underlying the operations on a case-by-case basis to determine this meaning.
