For the purposes of complex integration with the residue theorem, what happens if one or more of the poles are on the contour, rather than within it? Is the residue theorem still applicable?
It depends. If it's a straight line contour running through simple pole, I believe it turns out that the contribution is just half the residue (as you could infinitesimally move the pole inside the contour or go around the pole with an infinitesimal semi-circle).
If you have a non-simple pole, I believe it is more complicated. I once found a post online on a different forum where someone proved that straight-line contour that runs through an odd-degree pole still contributes half the residue, but even degree poles or contours which only touch an odd-degree non-simple pole tangentially result in a divergent integral. I'll try to see if I can find that post again.
Edit: Ah, there was a previous thread on Physicsforums in which someone asked the question, and someone linked to this other forum in which a poster proved the statement that the half residue theorem worked for odd-degree poles.