Fundamental Theorem of Calculus in complex analysis I've seen the proof of the FTC in complex analysis but it doesn't really make sense to me. How can it be path independent? Why can I not change the value of the function on a particular path while leaving another path alone and thus change the integral value, while keeping the function continuous/integrable? 

Remember that for the foundamental theorem of calcus in CA, is sensate take holomorphic functions in  simply connected domains. In a simply connected domain the integral of an holomorphic function is path indipendent. For istance:
Let $f: G\rightarrow \mathbb{C}$ and $\alpha , \beta$  two paths in $G$(a simply connected domain) with the same initial and final points , and with ounterclockwise orientation. We have  (Cauchy Theorem) $$\int_{\alpha - \beta}f(z)dz=0$$
then $0=\int_{\alpha -\beta}f(z)dz=\int_{\alpha}f(z)dz-\int_{\beta}f(z)dz \Rightarrow \int_{\alpha}f(z)dz=\int_{\beta}f(z)dz$ .
