Now, before anyone grabs their torches and pitchforks, I know that not all integrals of closed contours are $0$.
However, the fundamental theorem of contour integrals tells us that a curve (over an analytic path) only depends on its start point and terminal point, and of course, a closed contour has the same points. Therefore, all closed analytic contours are $0$.
This is obviously wrong, as residues exist for a reason. But, I'm not sure where I'm misunderstanding. Can someone help me sort this out?
EDIT: I guess my question is, why does FToCI fail for some curves?