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?
Best Answer
Just a heads up: I'm just a student taking a class in complex analysis, so forgive me if I'm wrong as I am not an expert, but I can give a quick simple answer to try to help.
Closed contour integrals of analytical functions are always = 0
When closed contour integrals do not always =0 is when the are not analytical. They still can be 0, but can also equal other things like 2*pi*i, but these you figure out by other methods.
To answer your question I guess my question is, why does FToCI fail for some curves? - Because they are not analytical
I'll update this if I find anything wrong or anything new