User:Anghel/Sandbox

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

Let $\triangle$ be a triangle embedded in the complex plane $\C$.

Let $\partial \triangle$ be the boundary of $\triangle$.

Let $\operatorname{Int} \left({\triangle}\right)$ be the interior of $\partial \triangle$, when $\partial \triangle$ is parameterized as a Jordan curve.

Suppose that $\partial \triangle \cup \operatorname{Int} \left({\triangle}\right) \subseteq D$.

If $C$ is a contour with image equal to $\partial \triangle$, then:


 * $\displaystyle \oint_C f \left({z}\right) \mathrm dz = 0$

Proof
We will create a sequence of triangles $\left \langle{ \triangle_n }\right \rangle_{n \in \N}$ by an inductive process.

Put $\triangle_0 = \triangle$ as the first element of the sequence.

Denote the vertices of $\triangle_n$ as $v_1, v_2, v_3$, and put $v_4 = v_1$ for convenience.

From Boundary of Polygon as Contour, it follows that there exists a contour $C_n$ such that $\operatorname{Im} \left({C_n}\right) = \partial \triangle_n$.