Well-defined function in a disk

Question

Say $f$ is holomorphic in the disk $D$. If we denote $F(z) = \int_{\gamma_z} f(w)dw$, $\gamma_z \subset D$, how can I prove that $F$ is well-defined, for $z \in D$? ($\gamma_z$ being the curve from the center of the disk to $z$)

Answer 1

Actually, in the case of the disk, there does not have to be any ambiguity because we could take $\gamma_z =[0,z]$, the straight line segment from $0$ to $z$. This gives a definitive curve. But, if the domain was more complicated (not star convex or something), then you can use Cauchy's integral theorem, provided that $D$ is nice enough (disk is sufficient, but really simple connectedness).

To see this, if $\gamma_z$ and $\sigma_z$ were two such paths, then the path $\gamma_z$ followed by the reverse of $\sigma_z$ is a closed path. The integral over the holomorphic $f$ is thus zero by Cauchy. But this is the same thing as the integrals being equal.

Answer 2

You proof that there exits a curve for which $F$ is defined, and then you prove that if you take any other curve, you'll get the same answer. Hint: what do you know about integrating analytic functions over closed loops?