Theorem: Let f be a holomorphic function on a disc U, then there exist a function g such that g is the primitive of f and $\displaystyle \int _ {\gamma} f = 0 $ for every closed path gamma.

Proof.

Define $\displaystyle g(z_1)= \int ^{z} _{z_0}f \ \ \ \ \ \forall z \in U $

Then define the path from $\displaystyle z_0$ to $\displaystyle z$ by t and from z to $\displaystyle z_0$ by $\displaystyle k^{-1}$, so that t and k forms a circle.

Then the integral over t and k are equal and the integration is independent of path.

Now, I don't really understand why they have to be equal, and what does "independent of path" means, would someone please explain?

Thank you!