Proof of Cauchy integral theorem on any simply connected set

Question

I am trying to find a reference for a proof of Cauchy integral theorem, ie the fact that given a simply connected open subset $U$ of the complex plane, a rectifiable loop $\gamma$ contained in $U$ and a function $f$ holomorphic on $U$, then :

$$\int_\gamma f(z)dz=0$$

Is it only true for specific (convex, starred) simply-connected subsets ? I've found proof for a disk or a keyhole from "Goursat's Lemma" in Elias M. Stein and Rami Shakarchi book on complex analysis, but I've struggled to find one that applies to any (simply connected) set. Does it require advanced knowledge on other topics, regarding homotopy for example ?

Answer 1

You will find a proof in Richard A. Silverman's Introductory Complex Analysis, §36.