Question on Gamelin's Proof of the FTOC for Analytic Functions (Complex Analysis) On Page 107 of Complex Analysis, Gamelin gives a proof of the Fundamental Theorem of Calculus for Analytic functions (Part II). This proof is all fine, but I have some questions on the presentation.
My overall concern with the proof is that it's absolutely not the first one that would come to mind. I would have just mimicked the proof in a standard Calc 1 text, where you prove $F'(x)=f(x)$ by the limit definition of the derivative. Why don't we do it this way? Another concern with the proof is that we require $D$ to be star-shaped. Doesn't the FTOC still hold for non-star shaped domains? (I could be wrong about this, but I believe the domain must be simply connected.)
Can someone provide some insight as to why we don't mimic the standard proof of FTOC II for analytic functions?
 A: The proof of the fundamental theorem for analytic functions with more generality goes along those lines :

*

*Define integral along rectifiable (or some other class of) paths

*Prove that the integral of an analytic function along homotopic paths rel end points are equal

*Prove some theorem that allows you approximate the paths as piecewise smooth

*Apply the fundamental theorem for real functions to the composite of your primitive with the path on each piece.

The only step where it could be useful to mimic the standard proof for the fundamental theorem is the last but we just use the fundamental theorem since it is already proved by then.
Then for the star shape it not as much important as simple connectedness is. The proof of your book wants to compute explicitly the integral along a straight path so it needs that any segment beginning at a certain point is contained inside the domain of your function hence the star shape restriction.
Now a more general version of this theorem states (see John Conway's Fucntions of one Complex variable, 6.16)

Let $G\subseteq \mathbb{C}$ be a simply connected region,
$f : G \rightarrow \mathbb{C}$ an analytic function, then $f$ has a primitive
in $G$

The proof then shows that for any point $a,z\in G$, a primitive $F$ of $f$ is given by  $F(z) - F(a) = \int_\gamma f$ with $\gamma$ any path between $a$ and $z$. The crux of the proof is the homotopical cauchy theorem that proves that integrals along homotopic curves rel end points are equal.
