im trying to sort it out in my head
might sound convicing but im gonna give it a shot...
basically the FTC is ok to use iff there are no branch cuts when doing the integral (i.e. Log), it is simpley connected.. if that is alll tru we can do the calculus way of doing integrals... if not we have to use the contour integral...
back to cauchy gousat... if the function is analytic AND is inside a simple closed contour and there are no trouble spots..then the integral of fdz is 0
however if there is some kind of problem like 1/(z+3)(z-3)
and 3 is in the closed countour we use the Cauchy Integral Formula..
hope thats clear