# 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

## Contents

## Proofs

### Proof 1

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.

### Proof 2

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.

## Meaning

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