The discussion centers on the application of the residue theorem in complex integration when poles lie on the contour. It is established that for a straight line contour intersecting a simple pole, the contribution is half the residue. In contrast, for non-simple poles, the situation is more complex; odd-degree poles contribute half the residue, while even-degree poles or tangential contacts with odd-degree non-simple poles result in divergent integrals. References to previous discussions on Physicsforums and other forums provide additional context and validation for these findings.
PREREQUISITESMathematicians, physicists, and students of complex analysis who are dealing with contour integration and the residue theorem, particularly in scenarios involving poles on the contour.
Mute said: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.