Discussion
As was shown by Goursat, Cauchy's integral theorem can be proven assuming only that the complex derivative f '(z) exists everywhere in U. This is significant, because one can then prove Cauchy's integral formula for these functions, and from that deduce these functions are in fact infinitely differentiable.
The condition that U be simply connected means that U has no "holes" or, in homotopy terms, that the fundamental group of U is trivial; for instance, every open disk qualifies. The condition is crucial; consider
which traces out the unit circle, and then the path integral
is non-zero; the Cauchy integral theorem does not apply here since is not defined (and certainly not holomorphic) at .
One important consequence of the theorem is that path integrals of holomorphic functions on simply connected domains can be computed in a manner familiar from the fundamental theorem of real calculus: let U be a simply connected open subset of C, let f : U → C be a holomorphic function, and let γ be a piecewise continuously differentiable path in U with start point a and end point b. If F is a complex antiderivative of f, then
The Cauchy integral theorem is valid in slightly stronger forms than given above. e.g. Let U be a simply connected open subset of C and f a function which is holomorphic on U and continuous on . Let be a loop in which is uniform limit of a sequence of rectifiable loops in U with bounded length. Then, applying the Cauchy theorem to the, and passing to the limit one has
See e.g. (Kodaira 2007, Theorem 2.3) for a more general result.
The Cauchy integral theorem leads to Cauchy's integral formula and the residue theorem.
Read more about this topic: Cauchy's Integral Theorem
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