Cauchy's integral theorem

The Cauchy integral theorem (after Augustin Louis Cauchy ) is one of the most important theorems of function theory. It deals with curve integrals for holomorphic (on an open set complex - differentiable ) functions. In essence, it says that two same points connecting paths have the same path integral if the function is holomorphic everywhere between the two paths. The set draws its meaning among other things from the fact that it's used to prove the Cauchy integral formula and the residue theorem.

The first formulation of the theorem comes from 1814, when Cauchy proved him for rectangular areas. This he generalized in the next few years, but he sat there between the Jordan curve theorem for granted. Modern evidence comes through the Lemma of Goursat without this profound statement from the topology.

• 2.1 turns of the path of integration
• 2.2 Example

• 3.1 Cauchy 's integral theorem with Wirtinger calculus and Stokes' theorem

The set

The integral theorem was formulated in numerous versions.

Cauchy 's integral theorem for elementary regions

Be an elementary domain, ie, a field in which every holomorphic function has an antiderivative. Star of application include elementary regions. The Cauchy integral theorem states that,

For each closed curve vanishes (where and ). For the integral sign with circle see notation for line integrals of closed curves.

If no elementary region, so the statement is false. For example, holomorphic in the art, however, does not disappear on each closed curve. For example, applies

For just traversed edge curve of a disk around with positive radius.

Cauchy 's integral theorem ( homotopy version)

Is open and are two mutually homotopic curves in D, then

For any holomorphic function.

If a simply connected domain, then the integral vanishes by the homotopy version for each closed curve, ie is an elementary region.

In Returning to the example above you will notice that it is not simply connected.

Cauchy 's integral theorem ( homology version)

Where an area and a cycle, then disappears

Iff for every holomorphic function if null homologous in.

Isolated singularities

Number of turns of the path of integration

It is an area, an interior point and holomorphic. Be a punctured neighborhood on which is holomorphic. Further, let a fully running closed curve which rotates exactly once positively oriented, ie for the winding number applies ( not on ). With the integral theorem now applies

By generalization to any number of turns of obtained

From the definition of the residue obtained even

The residue theorem is a generalization of this approach to several isolated singularities and cycles.

Example

It is the integral with determined below. Choose the path of integration a circle with radius, so

Makes used:

Since you every function that is holomorphic on an annulus around, can develop into a Laurent series, resulting in the integration to:

Now lets apply the above result:

Where the expansion coefficient was called residual.

Derivation

The following derivation, however, presupposes the continuous differentiability complex, the complex integral leads back to real two-dimensional integrals.

Be with and with. Then, the integral along the curve in the complex plane, or for the equivalent line integral along the curve

In the real plane

Thus, the complex contour integral was expressed by two real line integrals.

For a closed curve, which bordered a simply connected region S, the set of Gauss can (in this case the continuity of the partial derivatives is used) apply

Or alternatively, the set of Stokes

If the function is complex differentiable in S, have the Cauchy- Riemann equations there

Apply, so the above integrand (whether in the Gaussian or Stokes Version) vanish:

Thus, the Cauchy integral theorem for holomorphic functions on simply connected domains is proved.

Cauchy 's integral theorem with Wirtinger calculus and Stokes' theorem

The Cauchy integral theorem follows as an easy consequence of the theorem of Stokes, when you put the Wirtinger calculus used. In this case, the proof of the integral theorem, the computation of the line integral as an integration for the complex-valued differential form

Over the closed curve that surrounds the simply connected and bounded region, interpreted.

The Wirtinger calculus now states that the differential representation

Has, resulting directly

Followed.

Now there is the fundamental

Furthermore does the prerequisite for Holomorphiebedingung after Wirtinger calculus nothing more than

What immediately

Moves to be.

The overall result is therefore:

And ultimately by means of Stokes' theorem:

If you want to show that from the complex differentiability (without a priori assumption of the continuity of the derivatives ) is already the Cauchy integral theorem (and thus a posteriori the existence of all higher derivatives ) follows, so you have to use the Integrallemma of Goursat. This also has the didactic advantage that the set of Gauss (or Stokes ) is not required.