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On Goursat's proof of Cauchy's integral theorem

By Harald Hanche-Olsen. Appeared in *Amer. Math. Monthly* **115** (2008), 648–652 (full text/pdf, amended on 2008-10-17 with a note acknowledging Výborný's paper).

From the introduction: One standard proof for the Cauchy integral theorem goes something like this: First one proves it for triangular paths, and uses this to establish the existence of an antiderivative on star shaped regions. The Cauchy integral theorem follows on such regions. Next, the homotopy version of the theorem is derived from this, typically with some difficulty of a didactic nature.

The purpose of this note is to point out that the homotopy version is easily derived directly, by the simple expedient of employing Goursat's trick in the domain of the homotopy.

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A missed reference

(Note added 2008-10-15)

It has come to my attention that the main idea of this note has been noted earlier:

R. Výborný, On the use of a differentiable homotopy in the proof of the Cauchy theorem, *Amer. Math. Monthly* **86** (1979), 380–382 (accessible at JSTOR; subscription required).

I apologise to Výborný for having missed his paper in my literature search. (The idea seemed so natural I was very surprised not to find any papers using it. This is yet another example of how easy it is to miss relevant references, even using modern search tools.)

Here is a scan from Editor's Endnotes in the April 2009 issue of the *Monthly* (pp. 381–382):

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Discussion and further references

To appear. I have already received some feedback on the paper, including a number of interesting references. I'll have a look and post my findings here when I have the time. Write me an email if you wish to be alerted whan this happens.