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Concept
Cauchy's integral theorem
Summary
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if is holomorphic in a simply connected domain Ω, then for any simply closed contour in Ω, that contour integral is zero.
If f(z) is a holomorphic function on an open region U, and is a curve in U from to then,
Also, when f(z) has a single-valued antiderivative in an open region U, then the path integral is path independent for all paths in U.
Let be a simply connected open set, and let be a holomorphic function. Let be a smooth closed curve. Then:
(The condition that be simply connected means that has no "holes", or in other words, that the fundamental group of is trivial.)
Let be an open set, and let be a holomorphic function. Let be a smooth closed curve. If is homotopic to a constant curve, then:
(Recall that a curve is homotopic to a constant curve if there exists a smooth homotopy (within ) from the curve to the constant curve. Intuitively, this means that one can shrink the curve into a point without exiting the space.) The first version is a special case of this because on a simply connected set, every closed curve is homotopic to a constant curve.
In both cases, it is important to remember that the curve does not surround any "holes" in the domain, or else the theorem does not apply. A famous example is the following curve:
which traces out the unit circle. Here the following integral:
is nonzero. The Cauchy integral theorem does not apply here since is not defined at . Intuitively, surrounds a "hole" in the domain of , so cannot be shrunk to a point without exiting the space. Thus, the theorem does not apply.
As Édouard Goursat showed, Cauchy's integral theorem can be proven assuming only that the complex derivative exists everywhere in . This is significant because one can then prove Cauchy's integral formula for these functions, and from that deduce these functions are infinitely differentiable.
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