I'm now taking a course in complex analysis and in wikipedia it was said that Cauchy Integral formula is true also for a function which is "holomorphic in the open region enclosed by the path and continuous on its closure."

More precisely, How can I wonder how to prove the following theorem

given $f$ analytic in open set $\Omega\subset\mathbb{C}$ and continious in $\overline{\Omega}$, $$\forall z\in\Omega, f(z)=\frac 1 {2\pi i}\int_{\partial\Omega}\frac{f(\xi)}{\xi-z}d\xi$$

using only Cauchy integral formula for open subsets of the complex plane?

  • 1
    $\begingroup$ $\Omega$ needs to have a nice enough boundary for that. Then you can approximate the boundary by curves $\gamma_n$ in $\Omega$, and after parametrisation, the integrands $\dfrac{f(\gamma_n(t))}{\gamma_n(t) - z}\gamma_n'(t)$ converge uniformly to the boundary values. $\endgroup$ – Daniel Fischer Nov 8 '15 at 10:33
  • $\begingroup$ @DanielFischer thanks. What do you mean by "nice enough"? can you please explain your solution please? By the way, How do I use the continuity ? $\endgroup$ – Dirac Delta Nov 8 '15 at 10:37
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    $\begingroup$ I'm not sure how far one can push "nice enough". If the boundary consists of finitely many disjoint simple closed piecewise $C^1$ curves, that sure is "nice enough". With more work, you can extend it to rectifiable curves. But if we have for example a boundary with positive Lebesgue measure, it won't work. The continuity of $f$ on $\overline{\Omega}$ is needed to have good enough convergence of the integrands so that we can interchange taking the limit and integration. $\endgroup$ – Daniel Fischer Nov 8 '15 at 10:44

First of all, you need some assumption on $\partial\Omega$ so that the integral in the RHS makes sense. Say, $\partial D$ consists of finitely many smooth, or at least rectifiable curves. The difficulty of the proof depends on the exact assumption you make.

The complete proof is somewhat technical. (See, for example, Shabat, Complex Analysis, volume 1). The idea is the following. You consider a sequence $$D_1\subset D_2\subset\ldots$$ of regions with nice boundaries which exhaust $\Omega$ and whose boundaries tend to $\Omega$. Then you apply the usual Cauchy theorem to the $\Omega_n$ and pass to the limit. The difficult part is to justify this limit, and this depends on your exact assumptions about $\partial\Omega$.


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