I have seen multiple proofs of Cauchy's integrals theorem, mostly using the Deformation Invariance theorem or a version of Green's Theorem. But why exacly is this necessary?

From my understanding of the indepence of paths, isn't it obvious that if you take 2 random point z1 and z2 on a closed contour, then the integral from z1 to z2 and back from z2 to z1 cancel out and thus results in zero?

For closed contours consisting of multiple smooth curves, simply taking the sum of the integrals from each start and end point should result in a telescoping sum that again cancels out to zero.

All of this assumes the function is analytic in the domain D. Is there some other condition I am missing? Or is this a circular reasoning? (no pun intended). For reference, my math education is at the level of an undergraduate, with mostly self taught complex analysis.


1 Answer 1


This reasoning is indeed circular, at least most of the time. Generally, Cauchy's integral theorem is used to prove the independence of paths for complex integration, so one can't use the independence of paths to prove Cauchy's integral theorem.


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