Talk:Cauchy-Goursat Theorem

I don't think that's the problem with the proof: once you have the existence of an antiderivative; you can apply the fundamental theorem of calculus as is done here.

The problem is that to show that $F$ is well defined we want that:
 * $\displaystyle\int_{\gamma_1} f = \int_{\gamma_2} f$

for any two paths going between two points in the domain; that is, we want that the integral around $\gamma_1$ follows by $\gamma_2$ traversed backwards is zero; and this is precisely Cauchy's theorem for $\gamma_1 \cdot\gamma_2^{-1}$. So the statement:
 * "Let $F$ be an antiderivative of $f$"

needs proving; and is most naturally proved as a minor corollary of Cauchy's theorem. Unless the proof of existence of antiderivatives contains a proof of Cauchy's theorem to show well-definedness, the argument on this page becomes circular. --Linus44 (talk) 19:20, 2 May 2013 (UTC)


 * Moreover, the example given isn't an example of Cauchy's theorem: there is no simply connected subset of $\C$ that contains $\gamma : t \mapsto \exp(it)$ in which $1/z^2$ is holomorphic.


 * The fact that the integral vanishes here is an example of the residue theorem. --Linus44 (talk) 22:13, 2 May 2013 (UTC)