User:Anghel/Sandbox

Theorem
Let $f: D \to \C$ be a holomorphic function, where $D \subseteq \C$ is a simply connected domain.

Let $C$ be a closed contour in $D$.

Then:


 * $\ds \oint_C \map f z \rd z = 0$

Proof
Suppose that $C$ is a simple closed staircase contour.

Then $C$ is a concatenation of $n$ directed smooth curves that can be parameterized as line segments, where $n \in \N_{ \ge 4}$.

The image of $C$ is equal to the boundary of a polygon embedded in the complex plane.

Denote this polygon as $P_n$, where $n$ will be equal to the number of sides of $P_n$, and denote the boundary of $P_n$ as $\partial P_n$.

Boundary of Polygon is Topological Boundary shows that $\partial P_n$ is the boundary of $\Int C$, where $\Int C$ denotes the interior of $C$.

Complex Plane is Homeomorphic to Real Plane shows that there exists a homeomorphism $\phi :\R^2 \to \C$.

Interior of Simple Closed Contour is Well-Defined shows existence of a Jordan curve $f: \closedint 0 1 \to \R^2$ with $\Img f = \phi^{-1} \sqbrk { \Img C }$.

Simple Connectedness is Preserved under Homeomorphism shows that $\phi^{-1} \sqbrk D$ is simply connected.

Jordan Curve Characterization of Simply Connected Set shows that $\Int f \subseteq \phi^{-1} \sqbrk D$.

It follows that $\Int C = \phi \sqbrk { \Int f } \subseteq D$.

Two Ears Theorem shows that $P_n$ has an ear, a triangle $\triangle_n$ with two sides in common with $P_n$.

The two sides will be represented by the directed smooth curves $C_k, C_{k+1}$ for some $k \in \set { 1, \ldots, n }$, or possibly $C_n , C_1$.

The third side is a chord of $P_n$ that dissects $P_n$ into $\triangle_n$ and another polygon $P_{n-1}$.

The polygon $P_{n-1}$ has $n-1$ sides.

Let $\tilde C$ be the directed smooth curve that can be parameterized as the line segment equal to the chord, with start point equal to the start point of $C_k$, and end point equal to the end point of $C_{k+1}$.

We define two contours with images equal to the boundaries of $\triangle_n$, respectively $P_{n-1}$, as follows:

As $\Int { \triangle_n } \subseteq \Int C \subseteq D$, we can use the Goursat Triangle Lemma to get:

We continue by finding an ear $\triangle_{n-1}$ of $P_{n-1}$, which results in a dissection of $P_{n-1}$ into $\triangle_{n-1}$ and $P_{n-2}$, which is a polygon with $n-2$ sides.

Zero Simple Staircase Integral Condition for Primitive shows that $f$ has a primitive defined in $D$.

Primitive of Function on Connected Domain shows that in the general case where $C$ is a closed contour, we have:


 * $\ds \oint_C \map f z \rd z = 0$