Cauchy's Integral Theorem
Cauchy's Integral Theorem is one of two fundamental results in complex analysis due to Augustin Louis Cauchy. It states that if is a complex-differentiable function in some simply connected region , and is a path in of finite length whose endpoints are identical, then The other result, which is arbitrarily distinguished from this one as Cauchy's Integral Formula, says that under the same premises if is a circle of center with counteclockwise direction, then
We will prove the theorem for the case when is a triangle. We leave it as an exercise to verify that all other paths can be sufficiently approximated with triangles.
Lemma. Cauchy's Integral Theorem holds when is a constant function.
Proof. Let be a triangle; let , , and be its vertices, Let a mapping of onto such that , , , and such that is linear on each of these subintervals. Let be the constant value of . Then
Now we prove the main result.
We construct a sequence parths , recursively, as follows. Let be the vertices of , and let , , be the respective midpoints of , , . Then We choose from the paths , , , so that the quantity is maximal. Then so that
Let denote the longest side length in the triangle ; let denote the perimeter of . Then and .
Let denote the closed region bounded by . Since is a descending chain of nonempty closed sets, the set is not empty, so let us choose some that is an element of , for all .
The function is differentiable at the point , so for every , there exists an such that for all for which , If we pick such that , then this inequality holds for all . By the lemma, Now, let be a parameterization of such that for all . Then Therefore Since can be arbitrarily small, it follows that whence as desired.
We use Green's Theorem.
Let denote the real numbers such that . Let Let and be the functions mapping into such that . Then Now, since is complex-differentiable, Let be the region bounded by . Then by Green's theorem, and similarly, Thus Cauchy's theorem holds.
The Cauchy Integral Theorem guarantees that the integral of a function over a path depends only on the endpoints of a path, provided the function in question is complex-differentiable in all the areas bounded by the paths. Indeed, if and are two paths from to , then