Cauchy Integral Formula for Jordan Domains

Jordan Domains

Jordan Path

A Jordan path is a simple closed path , that is, is a closed path which doesn’t cross itself, except at the common endpoints.

Jordan Curve, Jordan Arc

A (piecewise-) Jordan curve is the image of a Jordan path, i.e. is a Jordan curve if there exists a Jordan path such that . Such a is called a Jordan parametrization of . Specifically, A Jordan arc is the image of an injective path .

Jordan parametrisations are not unique

For example, and both parametrize the unit circle .

Jordan Domain

A Jordan domain is a domain such that is a disjoint union of finitely many Jordan curves.

Remark

Jordan domains can be either bounded or unbounded.

Intuitively, the integral of a holomorphic function around a Jordan curve/arc is independent of the parametrization of the curve up to the sign. This is because we can integrate the function around the curve clockwise or counterclockwise, and the result should be the same in terms of magnitude but opposite in sign.

Lemma

Let be a Jordan curve or arc. Let and be Jordan parametrisations of . Then exactly one of the following is true:

• for all functions holomorphic in a neighborhood of we have
• for all functions holomorphic in a neighborhood of we have

Proof It suffices to just prove the statement for Jordan arcs, as any Jordan curve can be cut into Jordan arcs. So let be a Jordan arc. Then either or . For case 1, and are PCE-homotopic via Notice that by injectivity, the inverse exists. And is continuous since is compact, so is continuous as well by the proposition, leading to the continuity of . Therefore, by the Homotopy Invariance Theorem, we have The proof for case 2 is by applying case 1 to and , and we will get a negative sign eventually.

Integral along a Jordan curve

Now if is a Jordan curve with a chosen orientation, and is holomorphic in a neighborhood of , we can now write without ambiguity.

Standard Orientation

The standard orientation of the boundary of a Jordan domain is the orientation such that the region is to the left of the curve.

Complex Green's Theorem

Cauchy Integral Theorem

Let be a bounded Jordan domain. If , then

A picture proof of the Cauchy Integral Theorem

One intuitive way to understand this theorem is to think of the Jordan domain as indicated in the following figure. We add paths that connect the boundary and holes, then the integral of around the boundary of the Jordan domain is the the sum of integrals around the blue loop and the red loops. Since is holomorphic in the region enclosed by the blue loop, the integral around the blue loop is zero. Same for the red loops. Therefore, the integral around the boundary of the Jordan domain is zero.

Proof We evaluate the the integral using Green’s theorem: The last step is from condition for complex differentiability, so .

A result in complex differential geometry

We actually achieve a result in complex geometry:

Cauchy Integral Formula for Jordan Domains

Let be a bounded Jordan domain and let . Then for all ,

Proof Let and choose such that . Let , which is a Jordan domain with , but importantly the standard orientation on is the opposite of that as boundary of . Then , so by Cauchy integral theorem, we have where the last equality follows from the Cauchy integral formula for disks.

A new insight on calculating integrals

This idea where we use the Cauchy integral theorem to reduce the integral around the boundary to an integral around a “problem point” has wide applications in mathematics and physics. Suppose is holomorphic in a a neighborhood of . Then choosing small enough, we can cut disks out of , and get As an application, if is a polynomial, and is a bounded Jordan domain such that none of the roots of lie on the boundary of , then we can evaluate where are the roots of , and the right hand side can be calculated by Cauchy integral formula.