# A boundary version of Cauchy's Integral Theorem

In Complex Analysis by Kodaira, a more powerful version of Cauchy's Integral Theorem (and consequently formula) was proven. The result generalizes the theorem to the boundary of an open set as follows

Let $D$ be a domain and $\overline{D}$ be it's closure. Suppose that $f:\overline{D} \rightarrow \mathbb{C}$ is holomorphic in $D$ and continuous on $\overline{D}$. If the boundary of $D$, denoted $\partial D$ is composed of piecewise $C^1$ curves then $$f(w) = \frac{1}{2\pi i}\oint_{\partial D}\frac{f(z)}{z-w}dz$$ for all $w\in D$.

This result generalizes to the boundary of a domain given that it's boundary is sufficiently "nice". In the book, it was proven through finding a "cellular decomposition" of a domain which essentially gives a homotopy from a loop inside the domain to the boundary. The limit is then passed through the continuity of $f$ on the boundary. The cellular decomposition found was long and tedious to follow so I'm wondering if there are alternatives.

I've heard of alternative methods where if we can find a sequence of curves $\left\{\gamma_n\right\}$ which uniformly converges to $\partial D$ then the result follows from that. So my question is, what are the general conditions for there to exist such a sequence? Or perhaps does someone have an alternative proof of this result?

• My opinion on this is that Cauchy's integral formula is best given in terms of cycles, and treat the question of when the boundary of a domain is homologous to a small circle around a point separately. – timur Aug 3 '12 at 20:18
• @timur Admittedly that probably would be simpler. But this stems mostly from curiosity. – EuYu Aug 3 '12 at 20:41