Talk:Cauchy's integral theorem

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Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 20:07, 21 May 2007 (UTC)

It might be a good idea to distinguish between Cauchy´s integral Theorem and Cauchy Integrals.

[edit] Missing information

The last few lines that speak about the book by Marsden-Tromba are unclear; I have the latest edition (5th) and this is not cited on the given page. More specific reference should be given. It would be better to mention the applications of the topics along with its theory to make them more interesting. Its always a welcome feeling if you know why you are learning a particular thing.

[edit] "general Caucy theorem"

I guess that Tetvesdugo was actually thinking of a slight variation on what he/she wrote: if the two paths have the same start points and the same end points, but are not necessarily closed, then the integrals along them are the same. Of course it is still equivalent to the closed-contour version, but the equivalence is a tiny bit less trivial. McKay 07:30, 1 February 2007 (UTC)

OK, I see. But keeping with the standard (and a little less general form) is I think better. As to the more general version, I think it can be proved by making a closed path aout of those two paths with the same starting and ending points (by reversing one of the paths). Oleg Alexandrov (talk) 15:41, 1 February 2007 (UTC)

there is a somewhat different Cauchy integral theorem: if f is holomorphic on an open connected star-shaped region D, then f has a complex anti-derivative. (so the simple-connectedness of the domain is replaced by star-shapedness and continuity on its closure is not needed.) slightly more generally, if D has star center c and f is holomorphic on the punctured set D - c and continuous on D, the same is true.

instead of borrowing Green's from real variable calculus, it can be shown using Goursat's integral lemma. (Cauchy-Riemann gives the continuity of the partial derivatives? if not invoking Green's as the article does might be a problem.) someone correct me if i'm wrong but i think this reflects the historical developement of the integral theorem better than the version in article. Mct mht 20:41, 3 August 2007 (UTC)

[edit] The proof is wrong

The proof presented here is wrong. It relies on Green's theorem, but the latter requires the partial derivatives to be continuous - and this continuity has not been established. Of course, it will follow once we have shown that the derivative of a complex function is differentiable, but the proof of this fact typically relies on Cauchy's integral theorem to begin with. This circular reasoning is not unique to Wikipedia, but I think we should be a little better than that and give a valid proof. -- Meni Rosenfeld (talk) 13:04, 29 October 2007 (UTC)

Quite right. I am going to delete it. Also there is a fashion emerging to move proofs to sub pages.Billlion 09:39, 31 October 2007 (UTC)