User:Anghel/Sandbox

Theorem
Let $C$ be a simple closed contour in the complex plane $\C$ with parameterization $\gamma: \closedint a b \to \C$.

Let $t \in \openint a b$ such that $\gamma$ is complex-differentiable at $t$.

Let $S \in \set {-1,1}$ and $r \in \R_{>0}$ such that:


 * for all $\epsilon \in \openint 0 r$, we have $\map \gamma {t} + \epsilon i S \map {\gamma '}{ t } \in \Int C$

where $\Int C$ denotes the interior of $C$.

If $S = 1$, then $C$ is positively oriented.

If $S = -1$, then $C$ is negatively oriented.

Proof
For simplicity, set $\map v { t, \epsilon } := \map \gamma {t} + \epsilon i S \map {\gamma '}{ t }$.

We show that for all $t_1 \in \openint a b$ where $\gamma$ is complex-differentiable, there exists $r_1 \in \R_{>0}$ such that for all $\epsilon \in \openint 0 {r_1}$, we have $\map v { t, \epsilon} \in \Int C$.

The result then follows by the definitions of positively oriented contour and negatively oriented contour.

By definition of parameterization of contour, there exists $N \in \N$ and a subdivision $\set { c_0, \ldots , c_N }$ such that $\gamma$ is complex-differentiable at all $t \in \openint {c_k}{ c_{k+1} }$.

Find $k \in \set {0, \ldots, N-1}$ such that $t_0 \in \openint {c_k}{ c_{k+1} }$.

First, suppose $t_1 \in \openint {c_k}{ c_{k+1} }$.

Let $\Img C$ denote the image of $C$.

From Normal Vectors Form Space around Simple Complex Contour, it follows that there exists $r_1 \in \R_{>0}$ such that for all $\epsilon \in \openint 0 {r_1}$, we have $\map v { t_1, \epsilon} \notin \Img C$.

Note that for all $\epsilon_0, \epsilon_1 \in \openint 0 {r_1}$ with $\epsilon_0 < \epsilon_1$, there is a path $\sigma$ between $\map v { t_1, \epsilon_0}$ and $\map v { t_1, \epsilon_1}$ in $\C \setminus \Img C$, defined by the line segment:


 * for all $s \in \closedint {\epsilon_0}{\epsilon_1} : \map \sigma s = \map v {t_1, s}$

Complex Plane is Homeomorphic to Real Plane shows that we can identify the complex plane $\C$ with the real plane $\R^2$ by the homeomorphism $\map \phi {x, y} = x + i y$.

Interior of Simple Closed Contour is Well-Defined shows that $\Img C$ can be identified with the image of a Jordan curve $g: \R^2 \to \R^2$.

From the same theorem, it follows that $\Int C$ can be identified with the interior of $g$.

From the Jordan Curve Theorem, it follows that $\Int C$ is an open connected component of $\C \setminus \Img C$.

From Connected Open Subset of Euclidean Space is Path-Connected, it follows that $\Int C$ is a path components of $\C \setminus \Img C$.

So if if $\map v { t, \epsilon_0 } \in \Int C$ for one value of $\epsilon_0 \in \openint 0 {r_1}$, it follows that $\map v { t, \epsilon } \in \Int C$ for all $\epsilon \in \openint 0 {r_1}$.

Category:Orientation of Complex Contour]]