Complex Analysis/rectifiable curve

Definition

Let $\gamma \colon [a,b]\to \mathbb {C}$ be a continuous curve. It is called rectifiable if its length

{\mathcal {L}}(\gamma ):=\sup \left\{\sum _{i=1}^{n}|\gamma (t_{i})-\gamma (t_{i-1})|\ {\bigg |}\ n\in \mathbb {N} ,a\leq t_{0}<\ldots <t_{n}\leq b\right\}

is finite, and ${\mathcal {L}}(\gamma )$ is called the length of $\gamma$ .

Approximation of path length by polygonal chain

The following image shows how a polygonal chain $P$ can be used to approximate the length of a curve $\gamma$ .

rectifiable curve - approximation of length by polygonal chain - created with Geogebra on Linux

Estimation of length

The length of the polygonal chain $P_{n}$ underestimates the actual length of a rectifiable curve $\gamma$ , i.e. ${\mathcal {L}}(P_{n})\leq {\mathcal {L}}(\gamma )$ . In general, ${\mathcal {L}}(P_{n})<{\mathcal {L}}(\gamma )$ . By applying the triangle inequality, we get $<$ if the path's trace is not a line.

Path length for differentiable paths

If $\gamma$ is continuously differentiable, then $\gamma$ is rectifiable. Let $a\leq t_{0}<\ldots <t_{n}\leq b$ , then there existsmean value theorem $\tau _{i}\in (t_{i-1},t_{i})$ such that

\sum _{i=1}^{n}|\gamma (t_{i})-\gamma (t_{i-1})|=\sum _{i=1}^{n}|{\gamma '(\tau _{i})}|\cdot (t_{i}-t_{i-1})

Riemann sum as length of polygonal chain

The right-hand side of the above equation for the polygonal chain is a Riemann sum for the integral $\int _{a}^{b}|\gamma {\,}'(t)|\,dt$ . If we take the maximum of the interval widths $\max _{i\in \{1,\ldots ,n\}}(t_{i}-t_{t-1})$ for $n$ to infinity, the length of the polygonal chains ${\mathcal {L}}(P_{n})$ converges to the length of the path ${\mathcal {L}}(\gamma )$

Length for continuously differentiable paths

Let ${\displaystyle \gamma$ be a continuously differentiable path, then

{\mathcal {L}}(\gamma )=\int _{a}^{b}|\gamma '(t)|\,dt

gives the length of the path $\gamma$ .

Note - Length for continuously differentiable paths

Since $\gamma$ is continuously differentiable, $\gamma {\,}'$ is a continuous function. Since $[a,b]$ is a compact interval, $\gamma {\,}'$ takes a minimum and maximum. Therefore, $\gamma {\,}'$ and $|\gamma {\,}'|$ are bounded, and we have:

{\mathcal {L}}(\gamma )=\int _{a}^{b}|\gamma '(t)|\,dt<\infty

Piecewise continuously differentiable curves

In general, piecewise $C^{1}$ -curves are always rectifiable, because we can apply the above considerations to the individual parts of the curve, which then additively give the length of the entire curve. In the further course of complex analysis, paths (e.g. over the triangle edge) are considered that only possess the property of continuous differentiability in a piecewise manner, for which we can then still calculate the length as the sum of the arc lengths.

Non-rectifiable curve

As an example of a non-rectifiable curve, consider $\gamma \colon [0,1]\to \mathbb {C}$ ,

t\mapsto \left\{{\begin{array}{ll}0&t=0\\t+it\cos t^{-1}&t>0\end{array}}\right.

Continuity - continuous differentiability

First, $\gamma$ is continuous and, on each interval $[\epsilon ,1]$ , even continuously differentiable. On these intervals, the length is given by

{\mathcal {L}}(\gamma |_{[\epsilon ,1]})=\int _{\epsilon }^{1}\left|1-{\frac {i}{t}}\sin t^{-1}\right|\,dt.

Calculation of improper integral

For $\epsilon >0$ , this converges to

\int _{0}^{1}\left(1+{\frac {1}{t^{2}}}\sin ^{2}t^{-1}\right)^{1/2}\,dt=\infty

so $\gamma$ is not rectifiable.

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Source: rektifizierbare Kurve - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/rektifizierbare%20Kurve
Date: 12/11/2024

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