Complex Analysis/Path of Integration

Smooth paths and path subdivision

The following definitions were abbreviated with acronyms and are used as justifications for transformations or conclusions in proofs.

• (WG1) Definition (Smooth path): A path $\gamma :[a,b]\to ℂ$ is smooth if it is continuously differentiable.
• (UT) Definition (Subdivision): Let $[a,b]$ be an interval, $n\in \mathbb{N}$ and $a=u_0<\dots <u_n=b$. $(u_0,\dots ,u_n)\in {ℝ}^{n+1}$ is called a subdivision of $[a,b]$.
• (WG2) Definition (Path subdivision): Let $\gamma :[a,b]\to ℂ$ be a path in $U\subseteq ℂ$, $n\in \mathbb{N}$, $(u_0,\dots ,u_n)$ a subdivision of $[a,b]$, ${\gamma }_k:[u_{k-1},u_k]\to ℂ$ for all $k\in \{1,\dots ,n\}$ a path in $U$. $({\gamma }_1,\dots ,{\gamma }_n)$ is called a path subdivision of $\gamma$ if ${\gamma }_n\left(b\right)=\gamma \left(b\right)$ and for all $k\in \{1,\dots ,n\}$ and $t\in [u_{k-1},u_k]$ we have ${\gamma }_k\left(t\right)=\gamma \left(t\right)\wedge {\gamma }_k\left(u_{k-1}\right)={\gamma }_{k-1}\left(u_k\right)$.
• (WG3) Definition (Piecewise smooth path): A path $\gamma :[a,b]\to ℂ$ is piecewise smooth if there exists a path subdivision $({\gamma }_1,\dots {\gamma }_n)$ of $\gamma$ consisting of smooth paths ${\gamma }_k$ for all $k\in \{1,\dots ,n\}$.

Integration path

• (WG4) Definition (Path integral): Let $f:U\to ℂ$ be a continuous function and $\gamma :[a,b]\to U$ a smooth path, then the path integral is defined as: $\underset{\gamma }{\int }f:=\underset{\gamma }{\int }f(z)dz:=\underset{a}{\overset{b}{\int }}f(\gamma (t))\cdot {\gamma }^{\prime }(t)dt$. If $\gamma$ is only piecewise smooth with respect to a path subdivision $({\gamma }_1,\dots ,{\gamma }_n)$, then we define $\underset{\gamma }{\int }f(z)dz:=\sum _{k=1}^n\underset{{\gamma }_k}{\int }f(z)dz$.
• Definition (Integration path): An integration path is a piecewise smooth (piecewise continuously differentiable) path.

Example

The following path is piecewise continuously differentiable (smooth) and for the vertices $z_1,z_2,z_3\in \text{Spur}(\gamma )$ the closed triangle path $\gamma :[0,3]\to ℂ$ is not differentiable. The triangle path is defined on the interval $[0,3]$ as follows:

$\gamma (t):=⟨z_1,z_2,z_3⟩(t):=\{\begin{array}{cc}(1-t)\cdot z_1+t\cdot z_2& \text{for }t\in [0,1]\\ (2-t)\cdot z_2+(t-1)\cdot z_3& \text{for }t\in (1,2]\\ (3-t)\cdot z_3+(t-2)\cdot z_1& \text{for }t\in (2,3]\end{array}$

Paths from convex combinations

The piecewise continuously differentiable path is formed from convex combination.The sub-paths

• ${\gamma }_1:=⟨z_1,z_2⟩$ with ${\gamma }_1:[0,1]\to ℂ,(1-t)\cdot z_1+t\cdot z_2$
• ${\gamma }_2:=⟨z_2,z_3⟩$ with ${\gamma }_2:[1,2]\to ℂ,(2-t)\cdot z_2+(t-1)\cdot z_3$
• ${\gamma }_3:=⟨z_3,z_1⟩$ with ${\gamma }_3:[2,3]\to ℂ,(3-t)\cdot z_3+(t-2)\cdot z_1$

are continuously differentiable.

See also

• Goursat's Lemma (Details)
• convex combination
• Convex combinations and interpolation on triangles in the plane

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• Source: Kurs:Funktionentheorie/Integrationsweg - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Integrationsweg
• Date: 11/20/2024