Cauchy's integral theorem

Cauchy's integral theorem
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.
Statement
Fundamental theorem for complex line integrals
If is a holomorphic function on an open region , and is a curve in from to then,
Also, when has a single-valued antiderivative in an open region , then the path integral is path independent for all paths in .
Formulation on simply connected regions
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.)
General formulation
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.
Main example
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.
Discussion
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.
The condition that be simply connected means that has no "holes" or, in homotopy terms, that the fundamental group of is trivial; for instance, every open disk , for , qualifies. The condition is crucial; consider
which traces out the unit circle, and then the path integral
is nonzero; the Cauchy integral theorem does not apply here since is not defined (and is 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 calculus: let be a simply connected open subset of , let be a holomorphic function, and let be a piecewise continuously differentiable path in with start point and end point . If is a complex antiderivative of , then
The Cauchy integral theorem is valid with a weaker hypothesis than given above, e.g. given , a simply connected open subset of , we can weaken the assumptions to being holomorphic on and continuous on and a rectifiable simple loop in .
The Cauchy integral theorem leads to Cauchy's integral formula and the residue theorem.
Proof
If one assumes that the partial derivatives of a holomorphic function are continuous, the Cauchy integral theorem can be proven as a direct consequence of Green's theorem and the fact that the real and imaginary parts of must satisfy the Cauchy–Riemann equations in the region bounded by and moreover in the open neighborhood of this region. Cauchy provided this proof, but it was later proven by Goursat without requiring techniques from vector calculus, or the continuity of partial derivatives.
We can break the integrand as well as the differential into their real and imaginary components:
In this case we have
By Green's theorem, we may then replace the integrals around the closed contour with an area integral throughout the domain that is enclosed by as follows:
But as the real and imaginary parts of a function holomorphic in the domain and must satisfy the Cauchy–Riemann equations there:
We therefore find that both integrands (and hence their integrals) are zero
This gives the desired result
See also
Morera's theorem
Methods of contour integration
Star domain
References
External links
Jeremy Orloff, 18.04 Complex Variables with Applications Spring 2018 Massachusetts Institute of Technology: MIT OpenCourseWare Creative Commons.
Augustin-Louis Cauchy
Theorems in complex analysis