# Proving the generalized Cauchy integral theorem using Green's theorem

I was looking for proofs of the generalized Cauchy integral theorem:

Theorem (Generalized Cauchy integral theorem): Let $$\Omega\subset\mathbb{C}$$ be an open set and $$\gamma:[a,b]\to\Omega$$ be a piecewise $$C^1$$ closed curve such that $$\text{ind}_\gamma(z)=0$$ for all $$z\in\mathbb{C}\backslash\Omega$$. Let $$f:\Omega\to\mathbb{C}$$ be an holomorphic function. Then $$\int_\gamma f(z)dz=0$$.

Now all the proofs that I found not using topological arguments were based on proving Cauchy's generalized integral formula using Liouville's theorem. However, after we have shown that a function holomorphic on an open set is analytic on that set, isn't it possible to prove Cauchy's theorem by dividing $$\mathbb{C}$$ into its connected components induced by $$\gamma$$, and then using green's theorem on each simple closed piecewise $$C^1$$ component of $$\gamma$$? The only problem that I see is that looking at the conditions for Green's theorem, I saw that it requires $$\gamma$$ to be piecewise smooth instead of piecewise $$C^1$$. So is there a way around this condition on the curve and do you think that otherwise the proof is valid?

Let $$\Omega \subset \mathbb{R}^2$$ be a bounded and open set, and let $$f$$ be holomorphic on $$\Omega$$ and $$C^1$$ on a neighborhood of $$\overline{\Omega}$$. The proof of Cauchy's theorem through Green's theorem uses $$d(f\,dz) = 0$$ to deduce $$\int_{\partial \Omega}f\,dz = \int_{\Omega}d(f\,dz) = 0.$$ The only question is how regular does $$\partial \Omega$$ need to be for Green's theorem to be true?
A version of Green's theorem that follows directly from the Stokes theorem for manifolds with corners is that if $$\Omega$$ is a bounded open set such that $$\overline{\Omega}$$ is a $$C^2$$ manifold with corners and $$f, g$$ are $$C^1$$ on a neighborhood of $$\overline{\Omega}$$, then $$\int_{\Omega}(g_x - f_y) \,dx\,dy = \int_{\partial \Omega}(f\,dx + g\,dy).$$
According to "Introduction to Analysis in Several Variables" here, the regularity assumption can be relaxed to $$\overline{\Omega}$$ only being $$C^1$$ with corners. It can even be relaxed to $$\Omega$$ having Lipschitz boundary. A direct proof of this version of Green's theorem can be found in appendix I of "Measure Theory and Integration" by Taylor. The book refers to "Geometric Measure Theory" by Federer for "substantially more sophisticated generalizations" of the Stoke's theorem.