User:Anghel/Sandbox

Theorem
Let $f: D \to \C$ be a complex-differentiable function, where $D\subseteq \C$ is a connected domain.

Let $u, v: \R^2 \to \R$ be defined by:


 * $\map f { x + iy } = \map u {x,y} + i \map v {x,y}$

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

Then there exists a piecewiese continuously differentiable function $\gamma: \closedint 0 1 \to \R^2$ such that


 * $\ds \int_{ C } \map f z \rd z = \ds \int_\gamma \tuple{ u, -v } \cdot \rd \mathbf l + i \ds \int_\gamma \tuple{ v, u } \cdot \rd \mathbf l $

where the left integral is a contour integral, and the two integrals on the right are line integrals.

Proof
First, suppose that $C$ consists of one directed smooth curve $C_1$.

Reparameterization of Directed Smooth Curve with Given Domain shows that we can find a parameterization $\gamma_1 : \closedint 0 1 \to D$ of $C_1$.

Define $x, y: \R \to \R$ by:


 * $\map { \gamma_1 }{ t } = \map x t + i \map y t$

Then:

where $\gamma : \R \to \R^2$ is defined by


 * $\map \gamma t = \tuple{ \map x t, \map y t } $

By definition of smooth path, $\gamma$ is continuously differentiable.

In the general case, $C$ is a concatenation of $n$ directed smooth curves $C_1, \ldots, C_n$.

Each directed smooth curves $C_i$ has a parameterization $\gamma_i : \closedint { \dfrac { i - 1 }{ n } }{ \dfrac i n } \to D$ for all $i \in \set { 1, \ldots, n }$.

Then:

Sum of Integrals on Adjacent Intervals for Continuous Functions