Proving the generalized Cauchy integral theorem using Green's theorem

Question

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?

Answer 1

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?

The standard direct proof of Green's theorem works for open sets that can be divided into finitely many type I regions and into finitely many type II regions. These sets can have pieces of their boundary that are merely continuous.

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.