Fundamental theorem of contour integrals

The fundamental theorem of contour integrals is the complex equivalent of the fundamental theorem of line integrals. It says that for a continuous complex-valued function $f(z)\in D$ and has an antiderivative $F(z)\in D$, i.e., $F^{\prime }(z)=f(z)$, then:

$${\int }_{C}f(z)dz=F(z_2)-F(z_1)$$

for every curve $C$ from a starting point $z_1$ to $z_2$. Note what this means! It’s another perspective for independence of path, i.e., when we evaluate the contour integral, it only matters where we start and end.

Note

Suppose I have a circle, and there are two possible curves to get from $z=1$ to $z=-1$ (upper and lower curves), $z=e^{i\pi t}$ and $z=e^{-i\pi t}$. The functions aren’t continuous at $(0,0)$, i.e., it’s not simply connected.